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atjhttp  :  //books  .  qooqle  .  com/ 


E<Wjt,  I   ll«r^b,H5 


► 


HARVARD 
COLLEGE 
LIBRARY 

THE  GIFT  OF 

Miss  Ellen  Lang  Wentworth 
of  Exeter,  New  Hampshire 

5ooooooooooooooooooooooooooooooooooooooooooooo£ 


c 


3  2044  097  012  348 


*     THE 


Elements  of  Algebra. 


BY 


G.  A.  WENTWORTH,  A.M., 

AUTHOR  OF  A  SERIES  OF   TEXT-BOOKS  IN   MATHEMATICS. 


TEACHERS'  EDITION. 


BOSTON,  U.S.A.: 

PUBLISHED  BY  GINN  &  COMPANY. 

1895. 


HARVARD  COLLEGE  LIBRARY 

GIFT  OF 
MISS  ELLEN  L*WENTWORTH 

HAY  *  1W3 


Entered  according  to  Act  of  Congress,  in  the  year  1882,  by 

G.  A.  WENTWORTH, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington. 


Typography  by  J.  8.  Cvshino  &  Co.,  Boston,  U.S.A. 

PRESSWOBK  BY  OlNN  &  Co.,  BOSTON,  U.S.A. 


PREFACE. 


THIS  edition  is  intended  for  teachers,  and  for  them  only.  The 
publishers  will  make  every  effort  to  keep  the  book  from 
pupils;  and  teachers  are  urged  to  exercise  the  utmost  care  not  to 
lose  their  copies,  or  to  leave  them  where  pupils  can  have  access 
to  them. 

It  is  hoped  that  young  teachers  will  derive  great  advantage 
from  studying  the  systematic  arrangement  of  the  algebraic  work, 
for  such  attention  has  been  paid  to  this  as  the  limitation  of  the 
page  would  allow. 

It   is   also   expected   that  many   teachers,  who  are  pressed  for 

time,   will   find  great  relief  by  not  being    obliged   to  work   out 

every  problem  in  the  Algebra. 

G.  A.  WENTWORTH. 
Exetee,  N.  H., 

December,  1882. 


ALGEBEA. 


Exercise  I. 

Whena=l,  &  =  2,  c  =  3,  d=4,  e=5,  /=0: 

1.  9a  +  26  +  3c-2/  a    * 

=  9  +  4  +  9-0  &« 

=  22.  4s 

=  F5 


2.  4e-3a-36  +  5c 
=  20-3-6  +  15 
=  26. 


10. 


=  2. 

e«  +  6« 


3.  8afc  —  6crf  +  9cde  —  def  c*  —  6« 

=  48-24  +  540-0  ^  58  +  2 

=  564.  3*  -  2s 

125  +  2 

a    ^ac     86c     bed  D   g__g 

&         d         e  =127 
=  6  +  12-12 

=  6'  11.        6"  +  * 


5.  7e  +  6crf-^-^  2*  + 4s 

2ckj  =  - 

=  35  +  24-20  22o+4l78 

=  39.  8  +  64 

4  +  16-8 
G.  aSc'  +  ic^-de^+Z8  =6. 

=  18  +  96-20  +  0 
=  94.  ^        e«_* 


7.  e*  +  6e'&*  +  J*-4e»&-4e&3  c  +e4  +  ls 

=  625  +  600  +  16-1000-160  =      °  ~*     m 

=  81  5*  +  20  +  4s 

125-64 


8    8aa  +  3&*     4c2 +  65'     c*  +  d*  25  +  20  +  16 

8  +  12     36  +  24     9+16^ 


4  5  25  13.  100 +  80 -+-4 

=  5  +  12-1  =100  +  20 

- 16.  =  120. 


2 

ALGEBRA. 

14. 

75  -  25  x  2 

20. 

3a;  +  7y-*-7+aXj/ 

-  75  -  50 

=  9  +  35  +  7  +  2x5 

=  25. 

=  9  +  5  +  10 
=  24. 

15. 

25  +  5x4-10-5-5 
=  25  +  20-2 

21. 

6b-8y  +  2yxb-2b 
=  60-40-5-100-20 

=  43. 

=  60-f-20 

16. 

24-5x4-*- 10  +  3 

=  39$. 

=  24-20-5-10  +  3 

22. 

(66-8t/)-s-2yx6  +  26 

=  24-2  +  3 

=  (60-40)-*- 10x10  +  20 

=  25. 

=  20-5-100  +  20 
=  201. 

17. 

(24-5)x(4  +  10  +  3) 
=  19x($  +  3) 
=  19x4* 

=  64|. 

23. 

(66-8y)-r(2yX6)+26 

=  (60  -  40)  -5-  (10  x  10)  +  20 
=  20-5-100  +  20 
=  20*. 

24. 

Qb-(8y  +  2y)xb-2b 

18. 

zy  +  4ax2 
=  15  +  16 

=  60 -(40  +  10)  X  10-20 
=  60-40-20 

=  31. 

=  0. 

19. 

xy  — 156-5-5 

25. 

6&-*-(&-y)-3a;  +  fory-5-10a 
=  60 -5-(10-5)-9  + 150+20 

=  15-1^ 

=  15-30 

=  12-9  +  7* 

=  -15. 

=  10*. 

26.  Express  the  sum  of  a  and  b. 

a  +  6. 

27.  Express  the  double  of  x. 

2x. 

28.  By  how  much  is  a  greater  than  5? 

a -5. 

29.  If  x  be  a  whole  number,  what  is  the  next  number  above  it? 

z  +  1. 

30.  Write  five  numbers  in  order  of  magnitude,  so  that  x  shall 
be  the  middle  number. 

x—  2,     x  —  lt     x,     a?  +  l,     x  +  2. 

31.  What  is  the  sum  of  x  +  x  +  x  + written  a  times? 

ax. 

32.  If  the  product  be  xy  and  the  multiplier  z,  what  is  the 
multiplicand?  xy  +  x=y. 


teachers'  edition. 


33.  A  man  who  has  a  dollars  spends  b  dollars;  how  many 

dollars  has  he  left?  * 

a  —  o. 

34.  A  regiment  of  men  can  be  drawn  up  in  a  ranks  of  b  men 
each,  and  there  are  c  men  over:  of  how  many  men  does  the  reg- 
iment consist?     #  ab  +  c 

35.  Write,  the  sum  of  x  and  y  divided  by  c  is  equal  to  the 
product  of  a,  6,  and  m,  diminished  by  six  times  c,  and  increased 
by  the  quotient  of  a  divided  by  the  sum  of  x  and  y. 

c  %  +  y 

36.  Write,  six  times  the  square  of  n,  divided  by  m  minus  a, 
increased  by  five  b  into  the  expression  c  plus  d  minus  a. 

6n*    +5b(c  +  d-a\ 
m  —  a 

37.  Write,  four  times  the  fourth  power  of  a,  diminished  by 
five  times  the  square  of  a  into  the  square  of  &,  and  increased 
by  three  times  the  fourth  power  of  b. 

4o4-5a»&2  +  36*. 


Exercise  IT. 

3.  The  greater  of  two  numbers  is  six  times  the  smaller,  and 
their  sum  is  35.    Required  the  numbers. 

Let  x  =  smaller  number. 

Then  6  a?  =  larger  number, 

and  6  x  +  x  =  sum  of  numbers. 

But  35  =  sum  of  numbers. 

Therefore,  6a;  +  x  =  35,  x  =  5,  and  Qx  =  30. 

4.  Thomas  had  75  cents.  After  spending  a  part  of  his  money, 
he  found  he  had  twice  as  much  left  as  he  had  spent.  How  much 
had  he  spent  ? 

Let  x  =  number  of  cents  spent. 

Then      75  —  x  =  number  of  cents  left. 

But  2x  =  number  of  cents  left. 

Therefore,  75  -  x  =  2x,  -  3a;  =  -  75,  and  x  =  25. 


ALGEBRA. 


5.  A  tree  75  feet  high  was  broken,  so  that  the  part  broken  off 
was  four  times  the  length  of  the  part  left  standing.  Required 
the  length  of  each  part. 

Let  x  =  number  of  feet  left  standing. 

Then  4x  =  number  of  feet  broken  off, 

and  x  +  4x  =  number  of  feet  in  height. 

But  75  =  number  of  feet  in  height. 

Therefore,    5  x  =  75,  x  =  15,  and  4x  =  60. 

6.  Four  times  the  smaller  of  two  numbers  is  three  times  the 
greater,  and  their  sum  is  63.    Required  }he  numbers. 

Let  x  =  smaller  number. 

Then      63  —  x  =  larger  number, 
and  4  x  =  4  times  smaller ; 

also,      3  (63  —  x)  =  3  times  greater. 
.Ax  =3 (63  -a:),  4a?=189-3z,  7x=189,  x  =  27,  and  63 - x  =  36. 

7.  A  farmer  sold  a  sheep,  a  cow,  and  a  horse,  for  $216.  He 
sold  the  cow  for  seven  times  as  much  as  the  sheep,  and  the 
horse  for  four  times  as  much  as  the  cow.  How  much  did  he 
get  for  each  ? 

Let  x  =  number  of  dollars  received  for  sheep. 

Then  7x  =  number  of  dollars  received  for  cow, 

and  28  x  =  number  of  dollars  received  for  horse, 

and  x  +  Ix  +  28  a  =  number  of  dollars  received  for  all. 

But  216  =  number  of  dollars  received  for  all. 

.\a?  +  7a:  +  28a;=216,  36a  =  216,  x  =  6,  Ix  =  42,  and  28a  =168. 

8.  George  bought  some  apples,  pears,  and  oranges,  for  91 
cents.  He  paid  twice  as  much  for  the  pears  as  for  the  apples, 
and  twice  as  much  for  the  oranges  as  for  the  pears.  How  much 
money  did  he  spend  for  each  ? 

Let  x  =  number  of  cents  paid  for  apples. 

Then  2x  =  number  of  cents  paid  for  pears, 

and  4»  =  number  of  cents  paid  for  oranges, 

and   x  +  2x  +  4x*=  number  of  cents  paid  for  all. 

But  91  =  number  of  cents  paid  for  all. 

.\a:  +  2a>  +  4a:  =  91,  7a=91,  a?=13,  2x=26,  and  4z=52. 

9.  A  man  bought  a  horse,  wagon,  and  harness,  for  $350.  He 
paid  for  the  horse  four  times  as  much  as  for  the  harness,  and  for 
the  wagon  one-half  as  much  as  for  the  horse.  What  did  he  pay 
for  each  ? 


TEACHERS     EDITION. 


Let  x  =  number  of  dollars  paid  for  harness. 

Then  4  a;  =  number  of  dollars  paid  for  horse, 

and  2x  =  number  of  dollars  paid  for  wagon, 

and    x  +  4a;  +  2a;  =  number  of  dollars  paid  for  all. 

But  350  =  number  of  dollars  paid  for  all. 

.\a;  +  4a;  +  2a;  =  350l  7  a;  =  350,  x  =  50,  4a;  =  200,  and   2a;  =100. 

10.  Distribute  $3  among  Thomas,  Richard,  and  Henry,  so 
that  Thomas  and  Richard  shall  each  have  twice  as  much  as 

Henry. 

Let  x  =  number  of  dollars  Henry  has. 

Then  2x  =  number  of  dollars  Thomas  has, 

and  2  a;  =  number  of  dollars  Richard  has, 
and  a;  +  2a;  +  2a;  =  number  of  dollars  all  have. 

But  3  =  number  of  dollars  all  have. 

Therefore,  a;  +  2x  +  2a;  =  3,  5a;  =  3,  x  =  $,  and  2a;  =  1J. 

11.  Three  men,  A,  B,  and  C,  pay  $  1000  taxes.  B  pays  four 
times  as  much  as  A,  and  C  an  amount  equal  to  the  sum  of  what 
the  other  two  pay.    How  much  does  each  pay  ? 

Let  x  =  number  of  dollars  A  pays. 

Then  4  a;  =  number  of  dollars  B  pays, 

and  5  a:  =  number  of  dollars  C  pays, 

and  a;  +  4a;  +  5a;  =  number  of  dollars  all  pay. 

But  1000  =  number  of  dollars  all  pay. 

.\a;  +  4a;  +  5a;  =  1000,  10a;  =  1000,  a;=100,  4 x  =  400,  and  5 x  =  500. 

Exercise  III. 

1.  +16  +  (-11)  4.  -7  +  (+4) 
=  16-11  =-7  +  4 
=  5.  =-3. 

2.  -15  +  (-25)  5.   +33 +(+18) 
==_15_25  =33  +  18 

=  -40.  =51. 

3.  +68 +  (-79)  6.   +378 +  (+709) +  (-592) 
=  68-79  =  378  +  709  -  592 

=  -11.  =495. 

7.  A  man  has  $5242  and  owes  $2758.  How  much  is  he 
worth  ? 

f  5242  +  (-  $2758)  =  f  5242  -  $2758  =  $2484. 

8.  The  First  Punic  War  began  B.C.  264,  and  lasted  23  years. 
When  did  it  end  ? 

-  264  +  (+  23)  =  -  264  +  23  =  -  241 ;  i.e.  241  b.c. 


6  ALGEBRA. 


9.  Augustus  Caesar  was  born  B.C.  63,  and  lived  77  years. 
When  did  he  die  ? 

-  63  +  (+  77)  =  +  14  ;  i.e.  14  a.d. 

10.   A  man  goes  65  steps  forwards,  then  37  steps  backwards, 
then  again  48  steps  forwards.    How  many  steps  did  he  take  in 
all  ?    How  many  steps  is  he  from  where  he  started  ? 
65  +  37  +  48  =  150.        65  -  37  +  48  =  76. 

Exercise  IV. 

1.  5ab  +  (-bab)  7.   120my  +  (-95ray) 
=  5  ab  —  5  ab  =  120my  —  95  my 
=  0.  =  25my. 

2.  Smx  +  (-2mx)  8.  -33o62  +  (lla62) 
=  8mx-  2mx  =  -  33a&2  +  Ha6* 
=  6mz.                                               =  -22a&2. 

3.  — 13  mng  +  (-  7  mng)  9.  -  75<cy +  (+ 20zy) 
=  —  \$mng  —  *lmng  —  ~7oxy  +  20xy 
=  —  20mng.  =  —  55xy. 

4.  -5s2  +  (+8a?)  10.   +15a*s2  +  (-a2.T2) 
=  -  5ar»  +  8a2  =  15a2ar»  -  a2x2 

=  3r».  =  14a2s2. 

5.  25my2  +  (-18my2)  11.   -62ms  +  (+ 762m8) 
=  25my2  -  18my2  =  -  62ms  +  7  b2m* 
=  7my*  *662m8. 

6.  7a6  +  (-5ai)  "    12.  5a  +  (-36)+(+4a)  +  (-7&) 
=  7ab  —  5db  =5a  —  Sb  +  ia  —  7b 

=  2a6.  =  9a -105. 

13.  4a2c  +  (-  10zyz)  +  (+  6a2c)  +  (-9zyz)  +  (-  lla2c)  +  (+  20xyz) 
=  4a2c  —  lOxyz  +  6a2c  —  9xyz  —  11  a?c  +  20xyz 

=  —  a*c  +  xyz. 

14.  3rc2y  +  (-4a6)+(-2mn)  +  (+5rc2y)  +  (-ariy)  +  (-4a;2y) 
=  3a?y  —  4a6  —  2mn  +  52?y  —  x*y  —  4a:2y 

=  3a^y  —  4a6  —  2mn. 

Exercise  V. 

1.   5a  +  36+    c  2.         7a-46+    c 

3a  +  36  +  3c  6a  +  36-5c 

a  +  36  +  5c  —12a          +  4c 

9a  +  96  +  9c  a-    6 


TEACHERS     EDITION. 


3.  o+    b  —  c 

—  a+  b  +  c 
a—  b  +  c 
q  +    b  —  c 

2a  +  2b 

4.  a  +  26  +  3c 
2a  —    b-2c 

—  a  +    b  —    c 

—  a—    b  +    c 

a+    b  +    c 

5.  a-26  +  3c-4a* 
-2a  +  36-4c  +  5a* 
+  3a-46  +  5c-6a* 

—  4a  +  56  —  4c  +  7a* 


-2a  +  26 


+  2d 


Xs-   4ar2  +   5a?-   3 

2a* -14a?- 14a:  +    5 

-    ar*+    9x*  +      g?+    8 

2a*-    9^-  8a;  +  10 

11.       3a?2  —  xy  +  xz  — 

—  5a?2  —  xy  —  xz 
6a?2 


7.  «*-2a»  +  3a« 

x*  +    x2  +    x 
4a;4  +  5a:8 

+  2a?2  +  3a?-4 

-3s*-2a?-5 

5z4  +  4ar»  +  3ar2  +  2a?-9 

8.  a8  +  3ai2-3<z2&-    6s 
2a8-6a&2  +  6a*&  -76s 

a8-    a&2  +26* 

4a8-4a&2+2a*&  +    ft8 -76* 

9.  2a6  +  2a2a?-   Sax9 
12ab-6a*x  +  10ax* 

—  8a6  — 5a*a? +  ax* 

6ab  —  9a2a?  +    7oa?2  +  aa?8 

10.       c4-3c8  +  2c2-4<?  +  7 

2c*  +  3^  +  2^  +  5c  +  6 

-4c*  -4c2  -5 


-    c* 

3y2+    4yz-    z2 
+    5yz 


+    c  +  8 


-4a?2 


-2zy 


4yz 
-    5yz  +  3z2 
+    y2  +    3yz  +  3z* 


■6.J/-6Z 


-  2y2  +  llyz  +  5z2  -  6y  -  6z 


12.      m 


»5  — 3m4n  — 6m8n2 

5m*n  +    ra8n2  +    ra2n8 

+  7ra8n2  +  4m2n8  —  3mn* 

—  2m2n8  —  3mn4  +  4n5 
3m6  +  2mn4  +  2n6 

2m5  +  7m4n —    nft 

6m5—    m4n  + 2m8n2  +  3m2n8  — 4ran4  +  5n6 


Exercise  VI. 


1.  +25-(+16) 
=  25-16 

=  9. 

2.  _50-(-25) 
=  -  50  +  25 
=  -25. 


3.  _3i_(+58) 
=  -31-58 


4.   +107 -(-93) 
=  107  +  93 
=  200. 


8  ALGEBRA. 


5.  Borne  was  ruled  by  emperors  from  B.C.  30  to  its  fall,  a.d. 
476.     How  long  did  the  empire  last? 

476  -  (-  30)  =  476  +  30  =  506 ;  i.  c.  506  years. 

6.  The  continent  of  Europe  lies  between  36°  and  71°  north 
latitude,  and  between  12°  west  and  63°  east  longitude  (from 
Paris).  How  many  degrees  does  it  extend  in  latitude,  and  how 
many  in  longitude? 

71-(+36)  =  71-36  =  35. 
12-(-63)=12  +  63  =  75. 


Exercise  VII. 

1.  5x  —  (—4x)  8.  —  4ixy~ (—  5xy) 
=  5<c-f4a;  =  —  ixy  +  dxy 
=  9.x.                                             =  xy. 

2.  -3ab-(\-5ab)  9.   Sax-(-3ay) 
-3ab  —  5ab  =  8  ax  +  3  ay. 


=  -Sab. 

3a&2  -  (+  lOafc2)  =  2al2y  -  aby. 


10.   2ab*y-(+<iby) 


-S^lOaV  n    9a,  +  (5^_(+8rf) 

--(a0-  =  9s2-f5z2-8s2 

4.  15m2*2 -(-7m2*2)  =  6z*. 

=  22m2x2.  =5W  +  18x2/-10^y     ^ 

5.  -7at,-(-3ay)  =13^ 

=  -7ay  +  3ay  13.   17aa* -(-a*8) -(+ 24a*8) 

=  -4ay.  =17aaJ,  +  ar,-24aa!8 

6.  17axs-(-24arl)  =  -6aa* 

=  17ax8  +  24a;c8  14.   -3ab  +  (+  2mx)  -(-4mx) 

=  41  ax*.  =  —  3ab  +  2mx  +  4:mx 

=  —  3a6  +  6ma;. 

7.  5a2»-(-3a2ic) 

=  5a2x  +  3a2a;  15.  3a-(+  2&)-(-4c) 

=  8a2a\  =  3a -26+ 4c. 


Exercise  VIII. 

6a-2&~    c  2.    3a-26  +  3c 

2q-26-3c  2a-86-    c 

4o  +2c  a  +  66+4c 


TEACHERS     EDITION. 


3.  7x»-8aj-l  6.  x*-3xy-   y*+yz-2z* 
5x*-6x  +  3  x2  +  2xy-Sy2        -2z*  +  5xz 
2x2-2x-4t  -5zy  +  2y2  +  yz          -bxz 

4.  4** -33* -2a?-    7x+   9  7.       o8-3a2&  +  30&2-    6» 

x*-2x*-2x*  +    Ix-   9  -a8-+  3a26-3a62  +    6» 

3a?*-    a*            -14x+18  2a8-6o26  +  6aP-2P 

5.  2ics-2aa;  +  3aa  8.   x*-bxy+    xz-y2  +  7yz  +  2z- 

x2  —    ax+    a2  x2—   xy—    xz        -f2yzf3z* 

Xs—    ax  +  2a2  -4a?y +  2x2-y*  +  5y2-    2* 

9.   2ax2  +  3afcc-4&2a;  +  12&8 

clx2  —  4abx  —  5b2  x  +  bx2  —  x* 


ax2  +  labx  +    62a:  +  12  63  —  bx2  +  x3 

10.  ear1^^ +  4zy2-2y8-   5  a?  +    xy-   4y2  +  2 
8a^-7aay+    ay2-    y8  +    9s2-    xy  +    6y2-4 

-2ar»  +  3ay2-    y8- 14a?  +  2zy- 10y2  +  6 

11.  a4-    ft* 

-f4q86-6a*&*  +  4ay 

a*_    6*-4a86  +  6a26i-4a6s 

2q*-2&*-4a86+6q2&2  +  4ay 

-  a*  +    6*  -Satf1 

12.  a?y2-3x2y8  +  4sy*-    y5 

—  4sy*  —  4.^  —  a?  +  2ar*y 

ar»y2  -  Sx2^  +  8*y  +  3y*  +  a?  -  2a*y 

a?y2  —  SaPy*  +  4sy*  —    y5 

—  4zy*  —  4y*  —    a^  +  2ar*y 

x*y2-3x2y*  -Sy6-    ar>  +  2z*y 

a?y2  — SarV +  8sy*  +  3y5  +    xb—2x*y 

-Sxy*-8y*-2x*  +  lx*y 

13.  a252-    o26c-8a62c-a2c2+    a&^-G&V2 

+  2a2 be  -  5ab2c  +  2 ah?  -  562c2 

a^-Stf&c-Sa^c-aV-    a&c2-    &2<? 

14.  12a  +  36-5c-2rf=69  16.   2a?-y2-2a#  +    z2 
10a-    6  +  4c-3rf  =  45  a?-y2  +  2sy--    22 

2<z+46-9c  +    d  =  24  a?         -4ay+222 

15.  b-a. 

2a8-6o26  +  6a62-26s  17.     12ac+8ca*-9 

a*-*Ja2b -3&8  -lac-   9cd+  8 

as+    a2b  +  §at?+    6s  19ac  +  17cd-17 


10  ALGEBRA. 


18.  -   6a2  +  2a6-3c2  22.      3a;2  +  2&y-    y2 

4a24-6a6-4c2  -   s»-3ay  +  3,y2 

-10a2-4a6  +    c2  4s2  +  5a)y-4y2 

3s2  +  4sy  —  5y2 

19.  9sy-4a;-3y  +  7  **  +    ^  +    y2 

sy-2s-6y  +  l  +  ex2  -  <fr2 

o^  —  ty2  —  car2  +  dy1 

20.  —  a2  6c—    a62c-f    ab&—  abc 

a2bc  +    ab2c-    abc2+  abc   24.   as  +    bx  +    by  +  cy 

-2a2bc-2ab2c  +  2abc2-2abc  F~    bx~    hV  +  ^ 

2&c  +  26y 

21.  7a2-2a;  +  4  26.  5s2  +  4a-4y  +  3y2 
2^-f  3s--l  5s2-3s+3y  +  y2 
5a2  —  5a? +  5  Ix  —  7y  +  2y2 

26.  a262  +  12a6c-9ac* 

4a62  —  6acx  +  3  a2  x 

a2b2  +  12a6c  -  9ax*-  4a62  +  6acx  -  3a2a; 

27.      a2-2a6+c2-362  29.     2z*-2y2-    z2 

2a2-2a6         +  362  2x2  +  3y2-    22 

-5y2 
-x2-2y2  +  Zz2 
aJ_3y2-3z2 


-a2 

^-^-B^2 

a2  + 

6»  + 

c2* 

<p 

62  + 

c*  + 

d2 

a2  + 

62- 

c2- 

d2 

a2- 

62  + 

(*  + 

d2 

3a2  +  262  +  2c2  +  2d2 
62  +    c2*    cP 


S-    8;    J;    I  30.  -2a8  +    «Jc  +  2«* 


a2c  —    ac2 


—    a9  -\-    ac2 


-a2  +    b2+    <?  +    ri2 
a2  +  262  +  2c2  +  2d2 

3  a2  +  262  +  2c2  +  2d2  a8  -  2a2c  +  Sac2 

q2  +  2y  + 2c2  +  2d2  -    a8 +    ac2 

2a2  2a*-2a2c  +  2ac2 


Exercise  IX. 

1.  (a  +  &)  +  (&  +  c)-(a  +  c)  2.   (2a--6-c)-(a-26  +  c) 

=a+b +b+c—a—c  = 2o— 6 — c  — a+ 26— c 

«26.  -a +  6- 2c. 


teachers'  edition.  11 

3.  (2*-y)-(2y-2)-(2*-a?)     4.   (a-x-y)-(b-x+y)+(c+2y) 
=  2x  —  y  —  2y  +  z  —  2z  +  x  =a— x— y— 6  + a— y  +  c  +  2y 

=  3a?  — 3y  — 2.  =  a  — 6  +  c. 

5.  (2<e-y +  32)  +  (-<e-y-42)-(3a>-2y-z) 
=  2x  —  y  +  32  —  a;  —  y  —  42  —  3a;+  2y  +  2 

=  -2a\ 

6.  (3a  -  6  +  7c)  -  (2a  +  36)  -  (56  -  4c)  +  (3c  -  a) 
=  3a-6  +  7c - 2a - 36 -  56  4- 4c  +  3c -a 

=  -96  + 14c. 

7.  l-(l-a)  +  (l-a  +  aJ)-(l-a  +  a»-a») 
=  1  —  1  +  a  +  1  —  a+a*  —  1  +  a  —  a%+  a* 
=  a  +  a*. 

8.  a -{26 -(3c +26) -a}  10.   3a-{6  +  (2a-6)-(a-6)} 
=  a -{26 -3c -26 -a}  =*3a  -{6  +  2a-  6-a  +  6} 

=  a-26  +  3c+ 26  +  a  =  3a  —  6  —  2a  +  6  +  a-6 

=  2a  +  3c.  =  2a-6. 


11.   7a-[3a-{4a-(5a-2a)}] 
__     t,      v_    -26)}  =  7a-[3a-{4a-5a  +  2a} 

=  2a-{6  — a+26}  =  7a  — [3a-4a  +  5a- 2a 


=  2a  — 6  +  a  — 26  =  7a  — 3a  + 4a  — 5a  + 2a 

=  3a-36.  =5a. 

12.  2x  +  (y  -  32)  -  [(3x  -  2y)  +  z]  +  5x -  (4y  -  32) 
=  2x+y  —  32  —  [3z  —  2y  +  2]  +5x  —  4y  +  32 
r  — 32  — 3  a;  +  2y  —  2  +  5aj  — 4y  +  32 


13. 


=  2x  +  y- 
=  4a?  — y  —  2. 

.  {(3a-26)  +  (4c-a)}-{a-(26-3a)-c}  +  {a-(6-5c-a)} 
=  J3a  —  26  + 4c  —  a}  —  {a -26  +  3a  —  c}  +  {a  —  6  +  5c  +  a} 
=  3a  — 26  +  4c  — a  — a  +  26  — 3a  +  c  +  a  — 6  +  5c  +  a 
=  -6  + 10c. 

14.  a-[2a  +  (3a-4a)]-5a-{6a-[(7a  +  8a)-9a]} 
=  a- [2a  +  3a -4a]  -  5a - {6a- [7a  +  8a -9a]} 
=  a  — 2a  — 3a  +  4a  — 5a  — {6a  — 7a  — 8a  +  9a} 

=  a-2a  — 3a  +  4a  — 5a  — 6a  +  7a  +  8a  — 9a 
=  -5a. 

15.  2a-(36  +  2c)-[5_6-(6c-66)+_5c-{2a-(c  +  26)}] 


.36_2c-[56-6c  +  66+5c-{2a-c-26} 
-36-2c-[56-6c  +  66  +  5c-2a+.c  +  26] 
-36-2c-56  +  6c-66-5c  +  2a-c-26 


*2a-36-2c-[56-6c  +  66+5c-{2a-c-26}J 

=  2a-   "'      ~ 

=  2a- 
-4a-166-2ct 


12 


ALGEBRA. 


16.  a-[26+{3c-3a-(a  +  6)}  +  {2a-(6  +  c)}] 
=  a-[26  +  f3c-3a-a-6}  +  {2a-6-c{] 
=  a  —  [26  + 3c  —  3o  —  a  —  6  + 2a  —  6  —  c] 

=  a  —  26  —  3c+3a  +  a  +  6  —  2a  +  6  +  c 
=  3a-2c. 

17.  16 -a; -[7a; -{8a; -(9a; -3a; -6a;)}] 
=  16  -  a; -[7a; -{8a; -(9a;- 3a;  +  6a;)}] 
=  16-a;-  lx -{8a; - 9a;  +  3s -6a;}] 
=  16-a;-  lx  -  8a;  4-  9a;-  3a;  +  6a;] 

=  16  -  x  -  7x  +  8a;  -  9a;  +  3a;  -  6a; 
=  16 -12a;. 

18.  2a-[36  +  (26-c)-4c  +  {2a-(36-c~^~23)}l 


=  2a-[36  +  26-c-4c  +  {2a-(36-c  +  26)}] 
=  2a-  36  +  26-c-4c  +  {2a-36  +  c-26}] 
=  2a - [36  4- 26  -  c- 4c  +  2a- 36 +  c- 26] 
=  2a-36-26  4-c4-4c-2a4-36-«4-26 
=  4c. 


19.  a  -  [2 6  +  {3  c  -  3  a  -  (a  +  6)}  +  2 a  -  (6  +  3  c)] 
[26  +  {3c-3a-a-6  4-2a-6-3c] 
[26  +  3c-3a-a-6  +  2a-6-3c] 
26 -3c  4- 3a  4- a  4- 6 -2a  4- 6  + 3c 
*3a. 


a  —  |^o  - 
=  a-[2 
=  a-[2 
=  a-2fl 


20.  a-[56-{a-(3c-36)4-2c-(a-26-c)}] 
"""      't-3c  +  36+2c-d+26+c}] 
i4-3c-36-2c4-a-26-c] 
-3c  +  36  +  2c-a+26  +  c 


—  iou-ia  —  y 

•-  a  —  [5  6  —  {a  - 
=  a  —  [56  —  o-f 
*a  — 56  +  a  — 


Exercise  X. 


1.   2a-36-4c4-a*4-3e-2/        2.   a-  2a;  +  4y-3z-26  +  c 

=  (2a-36)-(4c-rf)+(3e-2/)  =(a_2a;)4-(4y-3z)-(26-c) 

=(2a-3'      •        -    -      —  -       -       -   "     -       -- 


-36-4c)4-(d4-3e-2/). 


a-^)  +  (4y- 
>-2a;4-4y)- 


(38  +  26-c) 


3.  a5  +  3a*-2a»-4a2  +  a-l 

=  (a5  4-  3a4)  -(2as  +  4a»)  4-  (a-  1) 
=  (a6  4-  3  a*  -  2  a»)  -  (4  a»  -  a  4- 1 ). 

4.  -3a-26  + 2c-5a*-e-2/ 

=  -(3a  +  26)  +  (2c-5rf)-(c  +  2/) 
=  -  (3a  +  26  -  2c)  -  (5c?  +  e  4-  2/). 

5.  aa;  —  by  —  cz  —  6x  +  cy  4-  az 

=  (ax  —  by)  —  (cz  +  bx)  +  (cy  +  az) 
*=  (ax  -  6y  —  cz)  —  (6a;  —  cy  —  az). 


TEACHERS*   EDITION.  13 

6.   2x*-3xiy +  ±x*y*-bx2y*  +  xyi-2y!> 

=  (2x*  -  3x*y)  +  (4ary  -SaPy*)  +  (*y*  -  2y*) 
=  (2a-5  -  3a*y  +  4ary)  -  (5^^-  :ry*  +  2jf). 

(1.)  2a-36-4c+a'  +  3e-2/=(2a-36-4c)  +  (d+[3c-2/]). 
(2.)  a-2a;  +  4y-3z-26  +  c  =  (a-2a?  +  4y)-(3z  +  [26-c]). 
(3.)  a5  +  3a*~2a8-4a2  +  a-l=(a5  +  3o4-2a8)-(4a2-[a-l]). 
(4.)  -3a-26  +  2c-5d-e-2/= — (3a  +  26-2c)-(5d  +  [e+2/]) 
(5.)  ax  —  by  —  cz  —  6o?  -f  cy  -f  az  =  (ax  —  by  —  cz)  —  (bx  —  [cy  +  az]) 
(6.)  2x*-3xty  +  4xfy*  —  5xty  +  xy*-2y6  =  (2x*-Zx*y  +  4xiyi) 

-(5a?ys-[xy*-2y«]) 

8.  2ax  —  6ay  +  46z  —  46a;  —  2cx  —  3cy 

=  (2a  -  46  -  2c)x  -  (6a  +  3c)y  +  4 6z. 

9.  ax—  bx  +  2ay  +  3y  +  4az  —  36z  —  2z 

=  (a  -  6)a?  +  (2a  +  3)y  +  (4a  -  3 b  -  2)z. 

10.  ax  —  2by  +  5cz  —  46a;  —  3cy  +  az  —  2cx  —  ay  +  46z 
=  (a  _  46  -  2c)a;  -  (a  +  26  +  3c)y  +  (a  +  46  +  5c)*. 

11.  12aa;  +  12ay +  46y-126z-15cx-f  6cy +  3cz 

=  (12a  -  15c)a;  +  (12a  +  46  +  6c)y  - (126  -  3c)z. 

12.  2ax  —  Zby  —  jcz  —  2bx  +  2cx  -f  8cz  —  2cx  —  cy  —  cz 
=  (2a  -  26  +  2c  -  2c)a  -  (36  +  c)y  -  (7  c  -  8c  +  c)z 
=  (2a- 26)x- (36 +  c)y. 


Exercise  XI. 

1.  -17x8  =  -136.  4.   -18x-5=+90. 

2.  -12.8x25 

=  -12.8xl00  +  4  =  -320.        5.  43x-6  =  -258. 

3  3  29 

5;49  6.  457x100  =  45700. 

2961 

1316  7.  (_358-417)X-79 
16*5  L-775X-79 

- 18.0621  =  61225. 

8.  (7.512  -  {-  2.894})  X  (-  6.037  +  {13.963}) 

-  (7.512  +  2.894)  x  (-  6.037  +  13.963) 

-  10.406  x  7.926 
=  82.477956. 


14  ALGEBRA. 


9.  13  x  8  x  -7  11.  -20.9  12.    -  78.3  x  -0.57     =  44.631 ; 

=  104x-7  -1.1  1.38  X -27.9     =-38.502; 

=  -  728.  "209  44.631  x  -38.502  =  -1718.382762. 

209 

10.    -  38  5^  13.  -2.906  x  -2.076=  6.032856 ; 

9  lA' yo  6.032586  X  -1.49  =  -8.98895544  ; 

~~^                 ^  -8.98895544  x    0.89  =-8.0001703416. 

" 6™  183.92 


2052 


Exercise  XII. 

9.   6ax-2a«-12tf.                 21.  lm*x 

3mx* 

10.  5mnx  9m  =  45m2n.  ^ — r^ 

11.  'SaxX— 4by  —  — 12 abxy.  —  2mq 

12.  -$cmxdn  =  -8cdmn.  -±2m*qa* 

13.  -7a&x2ac  =  -14a*&c.          ^  Q     _ 

22.  —  3pcf 

14.  5ra23X  3™^  =  15 m*a?.  6p*q 

15.  5a*x-2a»  =  -10a'»+»  "^iS 

16.  3a2a2x7a8aj4  =  21a5s6.  -  144p«g* 

17.  7ax-46  =  -28a6; 

-28a6x -8c  =  224a6c           23.  2a2m8j*x3amV=6a8m8a*; 

18.  8a&2x3ac  =  24a262c;  6 a»m8^x 4 a»m^=  24 a«m^. 

24 a2b2c  x  -  4c2  =  -  96  aW. 

M   ,                                  24.  6A28X-9ajJy222=-54aV25; 

27  ^  -54^26  X  -3atyz=l62afy*. 


19. 


-39mp 
243 
81 
- 1053  abmp 
18  ap 


25.  3aix  2amx— 4ma;X&a 
=  -24a262m2a2. 

26.  7am2x362w2 
=  21  ab2m2n*; 


8424 

1063 21  a&2m2n2  X  - 4a& 

- 18954  a2  bmp*  =  -  84  a2  6s  m2  n2 ; 

on          a   m  .3  -84a2&8m2w2Xa26n 

20.         6a&y  =  -84a4Mm2n8; 

26.r  -84a*Mm2n8X-262n 

I2ab*tf  =168a466w2n4; 

-5a2y  168a*66m2n4X-?nn2 

-60a8#y  =  -168a*66m8n« 


teachers'  edition.  15 


Exercise  XIII. 

5.  (4a2-36)x3o&  8.  (a*x-5a*x*  +  ax* +  2x*)xax*y 
=  l2a*b  —  9abi.  =  a4x*y-5a*x*y  +  a*x*y  +  2ax*y. 

6.  (8a2-9a6)x3a2 
=  24  a* -27  a8  6. 

7.  (3x*  -  4y2  +  5z*)  X  2x2y  9.  (-9a6+3a8&2-4aJ68-65)X  -3a6* 
=  6^^-80^^  +  \Qx*yz*.  =  27aW  -  9a466  +  12a867  +  3a&». 

10.  (3a^-2a^y-7ajy2  +  y8)x-5x,y 
=  -15a^y  +  lOarty  +  35  a1/  -  Sa^y*. 

11.  (-4w2+5x*y  +  8r,)x-3xay   12.  (-3  +  2ab  +  a262)x  -a* 
-  12aY  -  15x*y2  -  24a*y.  -  +  3a*  -  2a6b  -  a662. 

13.  (-z-2xz*  +  5x*yz2-6x*yi  +  3x*y*z)x-3x*yz 
=  Sx*yz*  +  6x*yz*-  lbx*tfz*  +  l$x*ifz  -  9x*y*z\ 


1.  ^-4 

a?  +  5 


+  5x2-20 

:e*  +    a^-20 

2. 

y-6 
y  +  13 

y2-   6y 
+  13y-78 

y2+    7y-78 

3. 

a*  +  aix*  +  ar* 

a6  +  a*a^  +  a2a^ 
—  a*  x*  —  a2  x*  —  x* 

a6                         -a* 

4. 

**  +  xy  +  y2 
x  -y 

■  aJ  +  afy  +  sy2 
—  any  —  xy2  —  y3 

Exercise  XIV. 

5. 

2x-y 
x  +  2y 

2a?  —    xy 

+  4xy  — 2y2 
2x2  +  3xy-2y2 

6. 

2x8  +  4x2  +  8x  +  16 
3a;  -6 

6x*  +  12x*+24x2  +  48x 
-123»-24a»-48x- 

-96 

6x* 

-96 

7. 

a^  +  x2  +  x  — 1 

a;  -1 

X*  +      X3  +  X2  —      X 

—     X8  — X2—     X  +  1 

tf*                    -2x  +  l 

8. 

x2  — 3ax 
x  +  3a 
ar'-Sax2 

+  3ax2-9a2x 

-9a2x 


16 

ALGEBRA 

9.       262  +  3a6-a2                       11. 

-56  +  7a 

a2  +  ab  +  62 
a  —b 

-1068-15a&2  +   baH 

+  14a62  +  21a26-7a3 

a*  +  a2b  -f-aJ2 
-a^-a^-fc3 

-106s- 

-     abi  +  26aib-7a* 

a3                      -ft3 

10.   2a  +  b 
a+26 

12. 

a2  -  a&  +  b* 
a  +b 

2a2  +    ab 

+  4a5  +  262 

a*  —  a*b  +  ab2 
+  a2b  —  ab2  +  J3 

2a2  +  5a6  +  262 

a8                      +#» 

13. 

2a&-562 
3a2-4a6 
6a86-15a262 

-    8a262-f20a&8 

6a8&-23a262  +  20a&8 

14. 

-a8  +  2a26-63 

4a2  +  8a& 

-4a5  +  8a*6-4a268 

-8a*  6               +  16a362-8a&* 

-  4a5              -  4a'2 ft3  +  16a3  62  -  8a&* 

15. 

a2  +  ab  +  b* 
a*  —  ab  +  b* 
a*  +  a36  +  a2&2 
-as6-a262-a68 

+  a2&2  +  a&8  +  # 

t 

a*  +  a262  +6* 

16.  as-3a26  +  3a62-&3 

a2-2a&    +  b2 

a5-3a*6  +    3a3  b2  -      a2b* 

-2a46  +    6a3 i2-   6a263  +  2a64 

+      a9b2-   3a263  +  3a&*-&* 


17. 


a5  -  5  a*  6  +  10a362  -  10a2  63  +  5a6*  - 

z-f  2y  —  3z 
x  —  2y  +  3z 

-&» 

x2  4-  2a^  —  3a?z 

—  2xy            —  4y2  +    6yz 

+  3a;z            +    6yz-9z2 

a2                        -4y2  +  12yz-9z2 

TEACHERS     EDITION.  17 


18.  2s2  +  3:ry-f  4y2 
Sx2  —  4sy  +  yz 

6tf4  +  9z3y  +  12a2y2 

-SaPy-WaPtf-lGxy* 

+  2a*yz  + Say's +  4y»g 

6a^  +    afy  -  16zy8  +  2x*yz  +  3sy2z  +  4y«2 

19.  x*  +  xy  +  y% 

a?  +  xz  +  z2 


ic*  +  ic*y  +  a?y2 

+  x*z  +  x2yz  +  xy*z 

+  a^z2  +  xyz2  +  y2zl 

x*  -f  s?y  4-  a^y2  +  ar*z  +  x*yz  -f  xy2z  +  a£z2  4-  xyz1  4  y2^ 

33.  a2  +  ab+.b* 


a2 

-a&  +  62 

-afc8 
-a&84- 

a* 

+  a86 
-a8& 

4-a262 
-a262- 

+  a2b2-\ 

ft* 

a* 
a* 

-a2£* 

4a262 

4     6* 

4 

6* 

a8 

+  a«62 
-a«#» 

4a4M 
-a*6*-o26« 

< 


a8  +a464  +$8 


34.  4a8-4a2&4-a# 

4a2  +  3a6  4  b* 

16a6    -16a*6  +    4a8  62 

+  12a*6  -12aW  +  3a*P 

4-   4a8&2-4a2684-a&4 

16a5    -   4a*6  -    4a8&2-    a?P  +  ab* 

2a2ft  +  y 

32a76-   8a662-   8a6 6s-  2d* 6*  4-  2 a8 6* 

+  16a568-4a«6*-4a86s-q2&64a&7 

32a76-   8<*ft»-  +   8a563-6a*6*-2a866-a266+a67 


18  ALGEBRA. 


) 


25.  ^-5^  +  13^- a? -» 

a?-2x  -2 

3^-53*  + 13s5-      z4-      a* 

-2a*  +  10^-260*4-    2a3  +  2x2 

-   2s6 +  103? -26s3 +  23? +  23? 

a-7  _  736  +  21  rr5  -  17a?  -  25a?  +  4a?  +  2a: 

26.  3?-2a?  +  33;-4 
43?  +  33?  +  2s  +  l 
4a?-8a?  +  123?-163? 

+  3a?-    6a?  +    9a?-12a? 

+    2a?4-   43?+    63^-83? 

+      a-3-    23?  +  33?-4 

"   19.  ^500  —  0  0* 
3a2  -4a& 


6a36-15a262 

-    8a262  +  20a63 

6a8&-23a262  +  20a&8 

14.  -a8  +  2a26-&8 

4a2  +  8a& 

_4a5  +  8a*6-4a2&8 

-Sa*b +  16a8&2-8a64 

ri^  _4a2&3  +  16a8&2-8a&4 

15.  a2  +  a&  +  62 
a^-afc  +  ft2 


at  +  tfb  +  an* 

-os6-a262-a&8 
+  a262  +  a&3  +  &4 

a*  +  a262  +  Z? 

16.  a8-3a2&  +  3a&2-&8 

a*-2a6    +  62 

a*-3a46  +    3a3  ft2-      a268 

-2a4&  +    6a862-   6a2&3  +  2a&4 

+      a362-    3a268  +  3a64-&» 
a5  -  5a46  +  10a3  62  -  10a2 b*  +  5a64  -  P 

17.  x  +  2y-3z 
x-2y  +  3z 

a?  +  23ry  —  3zz 

-2zy  -42/2+    6yz 

+  3  3:2 +    6yz  — 9z2 

&  -4y*  +  l2yz-9z* 


teachers'  edition.  19 


31.  x-3                                         32. 

.T-1 

x2  -  a?  +  1 

iC2  -+  JC  +  1 

x*-3x 
-    x  +3 

X*-4:X    +3 

X    +1 

a4- 

H 

x4- 
a8- 

-ar'  +  ic2 

har1  — a2  +  x 
+  x*  —  x  +1 
+  a*         +1 

ar»-4s2  +  3a; 

+    a2  —  4z  +  3 
ar1— 3a2  —    a;  +  3 
x  +3 
0^-3^-      a^  +  3a; 

+  33*-   9a?-3a?  +  9 

-z2  +  l 
1-a^  +  ar4 

-  a6  —  X*  —  X2 
+  ar4  +  x2  +  1 
+  **          +1 

x<            -10a2           +9 

33.  tf  +  ab+.V* 
a2  —  ab  +  b2 

a*  +  a86  +  a262 
-a8&  -a262-a&8 
+  a262  +  a&8  + 

6* 

a*             +  a262            + 

6* 

a8  +  a662+a46* 

-a*V-a*b*-a2b* 

+  a*M  +  a266  +  &8 

a8  +a4J4  +  $8 


34.  4a8-4a26+a6» 

4a2  +  3a6  +  b2 

16a6    -16a4  6  +   4as62 

+  12a46  -12a862  +  3a268 

+    4a8y-4a2y  +  a6* 

16a5    -   4a46  -    4as62-    tfP  +  ab* 

2q*&  +  y 

32a76~   8a662-   8a6&8-  2a4M  +  2a*P 

+  16g568-4a464-4a8y-a2&6  +  a&7 

32a76-   8a<62+   8a6^-6a46*-2a865-a266  +  ab1 


20  ALGEBRA. 


35.  x  +  a 

x+2 

a 

+ 

ax 
2  ax    +    2a% 

x*  + 

3ox    +    2a*  • 

X    — 

3a 

x*  + 

Sax'1   +    2a2x 

Sax2  -   9a2x  -6a8 

X8 

-    7a2x  -6a8 

x  — 

4a 

X4- 

7a2x2  —    6a*x 

+  28a8x  -4ax8+    24a* 

x4- 

7a2x2  +  22a3x  -4ax3+    24a* 

X    + 

oa 

X*- 

7a2x3  +  22a8x2-4ax4-f    24a4x 
20a2x8  -  SSa8^  +  5ax*  +  110a4x  +  120a6 

x5-27a2x3  -  13a3  x2  +    ax*+  134a4x  +  120a5 
or     x*+    ax4-27a2x8-13a8x2  +  134a4x  +  120a5 


36.     81a4-9a262  +  64  37.  y2-y2-2z2 

9a2  +  ft2 y*  +  yz-2z* 

729a6  -  81  a*b2  +  9a2b*  y*-ySz-  2y*z* 

+  81a462-9a264  +  66  +  rz~    y%zi-2yzi 

729  a6  +  66  -2yV  +  2.VZ8  +  4z* 

27  a8  +  J8  y  y 

27  a8  -ft8  y*-2yz-z* 

729a6  +    27a868  y2  +  2yz-z2 

-    27a863-&6  y*-2y*z-    ylz% 
729a6                    -66  2y32-4y222-2y23 

729a6  +  66 -    y*z*  +  2yz*  +   z4 

531441a12  -729  a6  66  y4  -ey2*2  +   z4 

+729a666-612  £ -by*z*  +4z* 

531441a12  -ft12  -    y2*8  -3Z4 


38.   3a2    -a&-3&2 
a26-2&2 
3a46-a862-3a263 


-6a2&2  +  2a&8  +  664 


3a4&-a8&2-3a263-6a262  +  2a&8  +  6&*     . 

-2aM-66» 

3a46-  a862  -  3a268-  6a262  +  2a&3  - 2a64  +  664  -  66* 


teachers'  edition.  21 


39.  a  +  h-c 

a  —  h  +  c 


a2+ab—ac 
—ab       —b2+   be 
+ac      +  bc—c2 
a2  -tf+zbe-c* 

-a+b+c 

-a*+ab2— 2abc+ac2 

+a2b-b*         +2b2c-  be2 

+a2c-  b2c+2bc2-c* 

-ai+ab2-2abc+ac2+a2b-b3+a2c+  b2c+  bc*-c* 
a+b+c 

a*+  a2l?—2a2bc+  a2c2+a*b-abs+a?c+  ab2c+  abc2-ac* 
+  a2b*+  a2bc  -a*b+aV        -2ab2c+  abc2        -b*+Pc+  6V-&C8 

+  a2be+  aV -a*c+  aVtc-2abc2+aci      -&c  +  b2c*+bc*-c* 

a*+2a2b2  +2a2c2  -b*        +2W        -c4 


40. 


a  +  b 
b  +c 
ab  +  b* 

+  ac  +  bc 

c  +  d 

a  +  d 
ac  +  ad 

+  cd+d2 
ac  +  ad  +  cd  +  d2 

a  +  c 
b-d 
ab  +  be 

—  ad—  cd 

ab  +  b2  +  ac  +  be 

ab  +  be  —  ad  —  cd 

(ab  +  b2  +  ac  +  be)  —  (ac  +  ad  +  cd  +  d2)  —  (ab  +  bc  —  ad—  cd) 
=  ab  +  b2  +  ac  +  be  —  ac  —  ad  —  cd  —  d2  —  ab  —  be  +  ad  +  cd 
=  b*  -  d\ 


41.  a  +  b  +  c  +  d 

a  +  b  +  c  +  d 


a2  +    ab  +    ac+    ad 

+    ab  +b2+    bc+    bd 

+    ae  +    be  +c2  +    cd 

+    ad +    bd         +    cd+d2 

a2  +  2ab  +  2ac  +  2ad  +  b2  +  2bc  +  2bd  +  c2  +  2cd  +  d2 


a—b—e + d 
a—b—c+d 


a2  —    ab—    ac+    ad 

—    ab  +b2  +    be—    bd 

—    ac  +     be  +  c2  —    cd 

+    ad -    bd         -    cd  +  d2 

a*  _  2ab  -2ac  +  2ad  +b2  +  2bc-  2bd  +  c2  -  2cd  +  d2 


22  ALGEBRA. 


a—b+c—d 
a  —  b  +  c  —  d 
a*  —    ab  +    ac—    ad 

—    ab                        +b2—    bc+    bd 

+    ac                     —be             +  c2 
—    ad                    +    bd 

-  cd 

-  cd+d* 

a2  -  2ab  +  2ac-  2ad  +  62  -  26c  +  2bd  +  c2 

a  +  6  — c  — a* 
a  +  6  —  c  —  d 
a2  +    aft  —    ac  —    ad 

+    a6                        +    6*-    be-    bd 
—    ac                       —be             + 
—    ad                     —   -bd 

-2cd  +  d* 

c*  +    cd 

+    cd  + 

d2 

ai  +  2ab-2ac-2ad+    b*-2bc-2bd  + 
a2  +  2a6  +  2ac  +  2ad  +    62  +  26c  +  26a*  + 
a2  -  2a6  -  2ac  +  2ad  +    62  +  26c  -  26a*  + 
a2-2a6  +  2ac-2ad  +    62-26c  +  26d  + 

c*  +  2cd  + 
c*  +  2cd  + 
c*-2cd  + 
<?-2cd  + 

a*2 
rf2 
d2 
d2 

4a2  +  462  +4C2  +  4d2 


42.  (a  +  6  +  c)2  —  a  (6  +  c  —  a)  —  6  (a  +  c  -  6)  —  c  (a  +  6  —  c) 
=  a2+62+c2+2a6+2ac+26c— a6— ac+a2— ab— 6c+62— ac— 6C+C2 
=  2a2+262+2c2. 


(a-6)a;-(6-c)a-{(6-a;)(6-a)-(6-c)(6  +  cft 
=  ax  —  bx  —  ab  +  ac  —  {(ft2  —  bx  +  ax  —  ab) — (62  —  c2)} 
=  ax  —  6a;  —  a6  +  ac  —  Jo2  —  6a;  +  ax  —  ab  —  62  -f  c2} 
=  ax  —  bx  —  ab  +  ac  —  o*  +  bx  —  ax  +  ab  +  6s  —  c2 
=  ac  — c2. 


44.   (m  +  n)  m  —  {(m  —  n)2  —  (n  —  in)  n\ 

»» m2  +  win  —  {m2  —  2mn  +  n2  —  n*  +  mn] 
=  m2  +  ron  —  ra2  +  2mn  —  n2  +  n2  —  mn 

=-  2mn. 


45.   (a  -  6  +  cy  —  {a  (c  —  a  —  6)  —  [6  (a  +  6  +  c)  —  c  (a  —  6  —  c)]} 
~ '  ab) 


(ab  +  62  +  6c)  -  (ac  -be-  c2)]} 
:lac  —  a2  —  a6) 


=  a2  —  2a6  +  2ac  —  26c  +  62  +  c2  —  {(ac-  a2  —  a6) 

=  a2  -  2a6  +  2ac  -  26c  +  62  +  c2 

—  [a6  +  6'2  +  6c  —  ac  +  6c  +  c2]} 
=»  a2  -  2a6  +  2ac  ~  26c  +  62  +  c2 

—  {ac  —  a2  —  a6  —  a6  —  62  —  26c  +  ac  —  c2} 
=»  a2  —  2a6  +  2  ac  —  26c  +  62  +  c2  —  ac  +  a2 

+  a6  +  a6  +  62  +  26c  —  ac  +  c2 
=■  2a2 +  26*  + 2c2. 


teachers'  edition.  23 


46.  {p2  +  q2)  r-(p  +  q)(p{r  -q}-q{r  -p}) 
=  p2r  +  q*r-(p  +  q)({pr  -pq}  -  {qr  -pq}) 
=p2r  +  q2r  —  (p  +q)(pr—pq  —  qr  +pq) 
=  tfr  +  q2r-(p+q)(pr-  qr) 
=p2r  -f  q2r  —  (p2r  —  q1r) 
=p2r  +  g^r  —  p2r  +  q*r 
=  2q*r. 


=  yx*yA-L6x*y*+W-i2xy—Zy*ittx9y+6xy°- 
=  9a?y2-13a?yH4>-{9aty-13a?yH4yf 
=  92*y2-l$x*y*+fy-9a*y*+lSx*if-4y 
=  0. 


48.  a2-{2o6-[-(a  +  {6-c}Xa-{6-c})  +  2a6]-46c}-(6  +  c)« 
=  a2-{2a6-[-(a  +  6-c)(a-6  +  c)  +  2a6]-46c}-(6  +  c)2 
=  a2-{2a6-[-a24^,-2Wc2  +  2a6]-46c}-(6+c)1 

=  a2-{2a6  +  a2-62  +  26c-<?-2a6-46c}-(6  +  c)2 
=  a2  -2a6-a2  +  6*-26c  + ^  +  206  +  46c-  bi-2bc-<* 
=  0. 

49.  {ac-(a-6)(6  +  c)}-6{6-(a-c)} 

=  {ac  -  (a6  —  6*  +  ac  -  be)}  -  6  (6  —  a  +  c) 
=  ac  —  ab  +  &  —  ac  +  be  —  b2  +  ab  —  be 
=  0. 

60.  5{(a-b)x-cy}-2{a(x-y)-bx}-{3ax-(5c-2a)y] 
=  5  {aa;  —  bx  —  cy}  —  2  {ax  —  ay  —  bx\  —  {3  ax  —  5  cy  +  2  ay] 
=  5ax  —  56a;  —  ocy  —  2  ax  +  2ay  +  26x  — 3aa?  +  5cy  —  2ay 
=  -3bx. 

51.  (a;-l)(a;-2}-3a!(a;  +  3)  +  2{(a;  +  2)(a;  +  l)-3} 
=  x2  -  Sx  +  2  -  Zx2  -  9a?  +  2  {x2  +  3a;  +  2  -  3} 

=  a?- 3a; +  2- 3a?- 9a;  +  2a?  +  6a; +  4-6 
=  -6a;. 

52.  {(2a  +  6)2  +  (a-26)2}{(3a-26)2-(2a-36)2} 

=  {(4a2  +  4a6  +  62)  +  (a2-4a6  +  462)}{(9a2-12a6  +  462) 

-(4a2-12a6  +  962)} 

=  {4a2  +  4a6  +  62  +  a2-4a6  +  462}{9a2-  12a6  +  46s 

-4a2  +  12a6-962} 

=  {5a2  +  56n{5a2-562} 

53.  4(a-36)(a  +  3  6)-2(a-66)2-2(a2  +  662) 

=  4(a2-962)-2(a2-12a6  +  3662)-2a2-1262 
-  4a2  -  366»  -  2a2  +  24a6  -  72 62  -  2a2  -  1262 
=  24a6-12062. 


24  ALGEBRA. 

54.  x2  (x2  +  y2)2  -  2x*y2  (x  +  y)  (x  -  y)  -  (a?  -  y3)2 

=  x2  (x4*  +  2a?2y2  +  y*)  -  2x2y2  (a*  -  y2)  -  (a»  -  y3)2 

=  a*  +  2**y»  +  a*y*  -  2x*y2  +  2x2y4  -  x6  +  2a?y3  -  y« 

=  3a?y4  +  2ic33/3-y6. 

55.  16(a*  +  62)(a2  -  b2)  -  (2a  -  3)  (2  a  +  3)(4a*  +  9) 

+  (2&-3)(2&  +  3)(4&2  +  l/j 
=  16(a*  -  ft*)  -  (4a2  -  9) (4 a*  +  9)  +  (4  J2  -  9)(46»  +  9) 
=  16a*-16M-16a4  +  81  +  1664-81 
=  0. 

Exercise  XV. 

7.  (s  +  y)2  19.  (Sabc-bcdf 

=  a*  +  2a?y+y2.  -9as-Pc>-6aftV<Z  +  Pc><*'. 

8.  {y-zf  20.  (4ar>-a!y2)2 

=  y2  -  2yz  +  z2.  =  160*  -  8x*y2  +  x*y*. 

9.  (2x+l)2  21.  (x  +  y)(x-y) 
=  4ar*  +  4a;  +  l.  =  x2-y2. 

10.  (2a  +  56)2  22.  (2a  +  6)(2a-6) 
=  4a2  +  20afc  +  2562.  =  4a2  -  b2. 

11.  (1-z2)2  23.   (3 -a?) (3+*) 
-l-23»  +  aJ*.  =  9-a*. 

12.  (3aa?-4^)2  24.   (3a6  +  26*)(3a&  -  2&2) 
=  9a2z2-  24aars  +  16a?*.  =  9a262  -464. 

13.  (l-7a)2  25.   (4a?2-3y2)(4a;2H-3y2) 
=  i  -  14a  +  49a2.  =  16ar*  -  9y4. 

14.  (5sy  +  2)2  26.  (a*x2-by*)(a*x2  +  by*) 
=  25x2y*  +  20  ay  +  4.  =  a6  or4  —  ^y8. 

15.  (ab  +  cd)2  27.  (6<ry-5y2)(6sy  +  5y2) 
=  a2  b2  +  2a6ca-  +  <*d\  =  36a*y2  -  25y*. 

16.  (3mn-4)2  28.   (4a*-  l)(4af>  +  1) 
=  9m2w2  -  24mn  +  16.  =  16x10  -  1. 

17.  (12+  5s)2  29.  (l+3a63)(l-3a&3) 
=  144  +  120s  +  25a*  =  1  -  9a26«. 

18.  (4ajy2-yz2)2  30.   (ax  +  by)(ax-by)(a2x2  +  b2y2) 
=  WaPy*  -  Sxfz2  +  y2z*.  =  a*xA  -  b*y*. 

Exercise  XVI. 

1.  (x  +  y  +  z)2  =  x2  +  y2  +z2  +  2xy  +  2xz  +  2yz. 

2.  (a;  -  y  +  z)2  =  x2  +  y2  +  z2  -  2a:y  +  2arz  -  2yz. 


teachers'  edition.  26 

3#  (m+n_p_<j)2  =  ra2+n2+pa+2*+2  mn— 2  rap-2  mq-2  np-2  nq+2pq. 

4.  (a?  +  2x -3)2  =  a?  +  4a?-2a? -12*  +  9. 

5.  (a?  -  6a;  +  7)2  =  a?  -  12a?  +  50a?  -  84a;  +  49. 

6.  (2a?2  -  7x  +  9)2  =  4a?  -  28a?  +  85a?  -  126a;  +  81. 

7.  (a?  +  y*  _  z*)*  =  a?  +  y4  +  z4  +  2 a?y2  -  2a?z2  -  2y2z2. 

8.  (a?-4a?y2+y4)2  =  a?+18a?y4  +  y8-8a?y2-8a?y«. 

9.  (a8  +  6s  +  c3)2  =  a6  +  66  +  c6  +  2a8  68  +  2a8  c8  +  26s  c8. 

10.  (a*-tf-Jf  =  x*  +  tf  +  z*-2x*y*-2x'*  +  2yi2*. 

11.  (a;  +  2y -  3z)2« a?  +  4y2  +  9Z2  +  4 scy  -  6a;z  -  12yz. 

12.  (a?  -  2y2  +  5z2)2  =  a?  +  4y*  +  25  z4  -  4a?y2  +  10a? z2  -  20y2z«. 

13.  (a?  +  2x-2)2  =  a?  +  4a?-8a;  +  4. 

14.  (a?  -  5a;  +  7)2  =  a?  +  39a?  -  10a?  -  70a;  +  49. 

15.  (2a?  -  3a;  -  4)2  =  4a*  -  12a?  -  7a?  +  24a;  +  16. 

16.  (a;  +  2y  +  3z)2  =  a?  +  4y2  +  9Z2  +  4an/  +  6a;z  +  12yz. 

Exercise  XVII. 

1.  (x  +  2)(a?+3)  =  a?  +  5a;  +  6.      12.  (x-4y)(x  +  y)  =  xi-3xy-4yi. 

2.  (a;  +  l)(x  +  5)  =  a?  +  6a;  +  5.     13.  (a-26)(a-56)  =  a2-7a6+1062. 

4.  (a;-8)(a;-l)  =  a?-9x>8.  ar-i-o*-y-t-    y 

5.  (*-8)(*  +  l)-**-7*-8.  La?- 2a?y  -  3z2y2. 

6.  (a;-2)(a;  +  5)  =  a?  +  3a;-10.  16    (aa._9)(oa;  +  6) 

7.  (x-3)(a;  +  7)  =  a?  +  4a;-21.  =  a2 a?  -  3 ax- 54. 

8.  (.-,(._4)-*-e.  +  a.    it.  ^y)faZ^m_A 


10.  (a:-2a)(a;+3a)=a?+aa;-6a2. 


19.   (a;  +  12)(a;-ll)  =  a?+a;-132. 


11.   (x-c)(x-d) 

=  a?_(c  +  cZ)a;  +  cd.  20.  (a;-10)(a;-5)  =  a?-15a;  +  50. 

Exercise  XVIII. 

1.  (x  +  a)s  =  a?  +  3a?a  +  3a;a2+a8.     3.  (a;  +  l)8  =  a?  +  3a?  +  3a;  +  1. 

2.  (a;-a)8=a?-3a;2a  +  3ara2-a8.     4.  (x-  l)8  =  a?-  3a?  +  3a;-  1. 

5.  (a;  +  a)4  =  a?  +  4a?a  +  6a;2 a2  +  4<ra8  +  a4. 

6.  (z-a)4  =  a?-4a?a  +  6a?a2-4a?a8  +  a4. 


26  ALGEBRA. 


7.  (x  +  l)*  =  z*  +  438*6s8  +  4a;  +  l. 

8.  (s-l)*  =  a?*-4aj8  +  6x2-4a;  +  l. 

9.  (x  +  y)5  =  i5  +  5z*y  +  10ar»y»  +  lO^y8  +  5sy*  +  y6. 

10.  (a;-y)5  =  a^-5a^y  +  10rBsy,-10x2y8  +  5a!y4-y5. 

11.  (x  +  l)6  =  x6  +  5s*  +  1038  +  10a*  +  5a;  +  l. 

12.  (x-l)5  =  a*-5a?*  +  10arJ-10(r8  +  5a;-l. 


4. 


11. 


Exercise  XIX. 

+  264  Aft        6. 
+  4  "66*         3 

-1.23 

9. 

-0.1123 

45)-424.35 

-61)6.8503 

345 

61 

-|648-581. 

-8 

793 

75 

690 

61 

1035 

140 

+  3840  =  128 

1035 

122  , 

-30 

183 

7. 

-11 

183 

-2568  =  214      + 

-24)- 264 
24^ 

+  12 

24 
24 

-21.7 

-49.)  1063.3       8. 

+  «A 

10. 

-0.022* 

98 

-85)- 3670 

+■  324)-  7.1560 

83 

340 

648 

49 

270 

676 

343 

255 

648 

343 

tt- 

■A 

AWt 

0.31831 

+    12. 

0.0101321+ 

-314159.)-100000.0 

314159)- 

-3183.10 

942477 

314159 

575230 

415100 

314159 

314159 

2610710 

1009410 

2513272 

942477 

974380 

669330 

942477 

628318 

319030 

410120 

314159 

314159 

4871  95961 


teachers'  edition.  27 


Exercise  XX. 

5.  «~-3m.  18.   30a^  =  _6£ 

2x  —  bx*y  x 

6    12g4=      ia3  19    ^tfwtx*  _4ma^ 


7. 


—  3a  5a?msx       5as 

10a*_6«   *  20.  M^^„6^ 

26c        c  7ayV 


8    _a^___l  21.  -S^*8^6-  3Wd" 


9. 


-x5         a?"  -a462crf»  a2 

12am     n  22    *  ^ am5  n*  ^  ^*  =  ^ am3  n 

—  2m  "     47»2ns|?*j5  j>g» 


in    35 a6c^     ,.  23.   (4a'&z»xl0a'ft»c)-i-5as'Psl 

""l^T  =  40a464«4-*-5a36222 

=  8a62*2. 

U*  ^t  =  ^~;  **•  (21a^2«-t-3sy2z)(-2a*y2Z) 

12.  -?^  =  -9a4.  —  M^A 

-3a3  25.   104a6V-*-(91aWB7-*-7a464a:) 

oA  ...  =  104a6sa*-*-13a62a* 

13.  -36ma?==  _£jg!.  =  86ar>. 

4aa^  4oa; 

26.  (24aW*  +  3a'P) 
14    a62c8_,2  +  (35a862a*  +  -5a86a;) 

^67  ~(8a86a0  +  (-7a»6a0 

=»  a*  6a;. 

15.  ^£j?L  =  m4a:2.  27.  85a**+i-i-  5a<— » 

mF^  =17a*"»+i-(4«-«) 

1<L-5i«hy  =17a3- 

36ay  y  28    84a""4 

nnc    ,        A  '     12a2 

,7.  225  m2y  _9m  =  7a»-4-2 

25  my2        y  =  7a«"«. 

Exercise  XXI. 

3.  (ISamy - 27bny  +  36cpy) -^ - 9y  =  - 2am  +  3 bn - ±cp. 

4.  (21aa;-186a;  +  15ca;)-*--3a;  =  -7a  +  66-5c. 

5.  (12a* - 8ar>  +  4a;)-*- 4a;  =  33*- 2x*  +  1. 


28 


ALGEBRA. 


6.  (3a?  -  63?  +  9z7 -12a?)-*- 3a?  =  a; -2a? +  33? -4a*. 

7.  (35rasy  +  28m2y2  -  14my8)  -+■  -  7  my  =  -  5m2  -  4my  +  2y2. 

8.  (4a*6  -  6as&*  +  12a* 6s)  -*-  2a2  6  =  2a2  -  3ab  +  662. 

9.  (12arV-  15a?y2  -  24a?y)  -*■  -  3a?y  =  -  4sy2  +  5a?y  +  8a*. 

10.  (12a?y*-24a?y2  +  363?y8-123?y2)^12a?y2  =  a?y2-2a?  +  3zy-  I 

11.  (3a*-2a56-a662)H-a*=3-2a6-a262. 

12.  (3a?yz2  +  Gatyz*  -  ldx^y2^  +  18zVz)  -5-  -  3a*yz 
=  —  z  —  2arz2  +  5a?yz2  —  6a?y2. 

13.  (-  16a8 52<?  +  8a*&V-  12a5 6V)  ■*-  - 4a2 6V 
=  4ac8-2a2c2  +  3a36c. 

Exercise  XXII. 


6. 

3?-7a;  +  12 
a?-3a; 

X- 

-3 

2a;-3 

11. 
12. 

13. 

7a?- 

7a?- 

a?- 
a?- 

-24a?+58a;-21|7a;-3 

3J-4 

-  3a;2                |a?-3a;+7 

7. 
8. 

-4a; +  12 
-4a;  +  12 

a?  +    x-72 

a?  +  9a; 
-8a; -72 
-8a;-72 

2a?-    a? +  2 

3J  +  9 
3J-8 

ia;-9 

-21 
-21 

-1 
-a? 

a?+58a;-21 

*?+  9s; 
49x-21 
49a;-21 

x-1 

2a?-33? 

3?  +  s;  +  3 
2a;+6 

a?  +  a?  +  a?  +  a?  +  3;  +  l 

9. 

2a?  +  3a;-9 
2a?-33; 

6a;-9 
6a;-9 

6a?+143?-  4a;+24 

6a?+18a? 

a8- 
a8- 

a? 

a? 

-1 

-3? 
3?-l 
3?-3? 

3?-l 

a?  — a? 

33?-2a;+4 

a?-l 

-  4a?-  4a;+24 

-  43?-12a; 

r-1 

a?  — a; 

x-1 

10. 

83J+5 

$x+'< 

9a?+33?+    3J-1 
9a?-33? 

24 
>A 

3. 

-2aP  +  P 
-a26 

x-l 
a  —  b 

3a?+2$+l 

a*  +  ab-  b2 

6a?+   x 

6  a?-2  x 

3x 

3^ 

-1 
-1 

a2 
a2 

b-2al 

b-    al 

—  al 

-al 

2  +  68 

2  +  68 
2  +  68 

TEACH HRS     EDITION. 


29 


14.   x*-8ly* 

a;-3.y                           15.   xA- 

-y5 
-x*y 

*-.v 

x*~ 3afy  a?+3a?y+9ay2+27iya            a?- 

xt+aPy +x2i/2  -fay1 +y4 

3x*y  —  Sly*                                         x*y 
3afy-9a;V                                        a^ 
9a?y2-81y* 
9^^-27^ 

27*^-8  ly4 
27a^-8b/ 

-y4 

—  afy2 
X*y*  -  y5 
arty2  —  xhp 

a;2^3  —  xy* 

xy'-f 
xy^-y1 

16.  a5  +  3265 

a  +  2b 

a5  +  2a*6 

a*-2a36  +  4a262-8a63  +  166* 

-2a*b 
-2a*b 

+  326* 
-4a362 

4a3&2  +  32#> 
4a862  +  8a2&3 

-8a2&3  +  32&* 
-8a263-16aM 

16a64+3265 
16a&*  +  326* 

17.   2a*  +  27a&3-8164|a-f  36 

2a*  +  6a*b              |2a3-6a26  +  18a&2-2768 

-6a36+27a63-8U* 
-6a36-18a262 

-Sib* 

18a262  +  27a&3- 
18a262  +  54a&3 

-  27a63-8U* 
-27a&3-816* 

18.  a^-5ar>+ 11a;2- 12&  + 6 

x2-3x  +  S 

a^-3a*  +    3»2 

a? -2a; +  2 

-2ar»+    8a;2-12a; 
-2ar*-f    6a*-    6a; 

2a;2-    6a? +  6 
2a?-   6a;  +  6 

19.  x*  +    x*-   9a;2-16x-4la;24-4a;  +  4 

a^  +  4ar>  +    4ar*                   ja^8  —  3x  —  1 

-3ar»-13a^-16a;-4 
-33*- 12a;2 -12a; 

—  a?—   4a;  — 4 
-a?-   4a?-4 

30 


ALGEBRA. 


^+5^jf6^J  x2  -  5  x  4-  6 
-5^-19^  +  36 

6a^  +  30a;  +  36 
6a?  4- 30x  + 36 


21.  x4  +  64  |g24-4g  +  8 

x44-4x84-   8g8|a2-4a;-f  8 
-4ar>-   8.jr2  +  64 
-4x8-16x2-32a; 

8x2  + 32x  +  64 
8x*  4-  32x4-  64 


22.  x*4-    x3-243r-35x4-57|x24-2x-3 
x44-2x8-    3a?  |a£-&-19 

-  x8-21x2-35x 

-  x8-    2x24-    3x 

-19x2-38x4-57 
-19x2-38x4-57 


23.   1-    x-3x2-x5|l4-2x4-x2 

1  +  2x4-    x2         |l-3x4-2x2-x8 
-3X-4X2-X5 
-Sx-ea^-Sx3 


2x24-3x8-x5 

2x24-4x34-2x4 

-  xs-2x*-xB 

-  x8-2x4-x5 


24. 


x«-2x84-l 

s^-^-fx4 


x2-2x4-l 


2x5-x4 
2x6-4x44-2x8 


x44-2x34-3x24-2x4-l 
2xs4-l 


26.  4x*-  ar»4-4x 
4x54-6x44-4x8 


3x4-4x34-l 
3xi-6x84-3x8 

2x3-3x2+l 
2x3-4xa4-2x 

x*-2x+l 
x2-2x+l 


25.   a4+2a2624-964 
a4-2a3&4-3a262 


a*-2a54-35* 
a24-2a&4-362 


2a»6-  a262+964 
2a36-4a2624-6a68 

3a*V-6dP+9& 
3a262-6a&8+9&4 


2x24-3x4-2 


2x8-3x24-2x 


-6x4— 5x34-4x 

4x84-6x24-4x 
4x84-6x24-4x 


27.  a5-243 
a6-3a4 


a-3 


a44-3as4-9a24-27a4-Sl 


3a*-243 
3a4-9a8  , 


9a3-243 
9a8-27a2 


27a2-243 
27a2-81a 


81a-243 
81a-243 


teachers'  edition. 


3i 


28 

18a;*  -  45  ar»  +  82a;2  -  67a;  +  40 .  3s2  -  4  a;  +  5 
18ar*  -  24ar*  +  30a;2                     Ua;2  -  7a?  +  8 

-21ar*  +  52x2-67a; 

-21arJ  +  28a;2-35a; 

24a;2 -32a; +  40 
24a?-32a;  +  40 

29. 

3ii-9xt-6xy-y2\xa  +  3x-{-y 
xt  +  SoP  +  xty          |a;2-3a;-v 
-  3ar»  -  9a;2  -  x2y  -  6xy  -  y2 
-3a? -9a;2            -3a;y 

-x*y-Sxy-y* 
—  x2y  —  3a?y  —  y2 

30. 

a*-6a*y +  9ary~4y4 

a^-Sar'y +  2a;2y2 

-3ar»y  +  7a;2y2-4y4 

-3ar,y  +  9ar!y2-6a^ 

-2a?y2  +  6a^ 

—  2x2y2  +  6a;y 

x2-3xy-\-2y2 

• 

x2  —  '6xy-2y2 

3 

8-4y4 
1-4.V* 

31. 

x4  +  a?2y2  +  y*   1  a;2  —  xy  +  y2 
a;4  -  a?y  +  x*y2 J  a?  +  xy  +  y2 

afy  +  y4 

a^y  —  a?y2  -faty3 

»2.V*  -  *y3  +  y4 
a?y2  —  ajy5  ■+  y* 

32. 

a?  +  ar*  +  oty-aPy2 -  2xy2  +  y*\x*  +  x-y 
a£  +  a?  —  a*y                                 |  a?2  +  an/  —  i/2 

a^y  +  a:2y  —  a^y2  —  2xy2  +  y3 

a**y  +  a;2y              —    xy2 

—  x*y2  —    xy2  +  y3 

—  sc'y8  —    xy2  +  y3 

33. 

2a?  +    xy—xz  —  Sy2  —  4yz  —  z2 
2a;2  +  3ary  +    a;z 

-  2arcy  —  2a;z  -  3y2  -  4yz  -  z2 
~2xy            -Sy2-    Vz 

-2xz            -Syz-z2 
-2xz             -Zyz-z2 

2a;  +  3y  +  z 

x-y-z 

32 


ALGEBRA. 


34.   12-38a;-f  82o;2-112j3+  106^-70^  1  3-5s+7a* 

12-20a?+28a?  1 4-6a;+8z2-10arJ 


-18a;+54aj2-112ar> 
-lSx+Wx2-  42ar» 


24a?- 
24a?- 


70ar*-fl06;c* 
40ar>+  56a^ 


-  30ar*+  50^-70.^ 

-  30^+  50**-7Qg» 


36. 


3* 

a5  —  x*y  +  x*y*  - 


+  3Z5 
a^y8  +  xy* 


x*  _  gJy  +  j?y2  —  ayy3  -f  y* 


s  +  y 

a*y  —  a^y2  -t-  a;2^3  —  xy*  +  y3 
a^y  —  a^V2  +  a^y8  —  %y*  +  y5 


36.  2z4-7a;8y  +  232y2-2zy8-y4 
2a^  —    x*y  +    a;2y2 

—  6ar*y -f    a?y2  —  2a*y8 
-6ar}y  +  3ac2y2-3.Ty3 


2a;2  —  ay-f  y2 


sP  —  Sxy—y2 


■l&y1^ 
-2ar»y2-f 


sy8- 
a;;/8- 


37.   16ar*  +  4x2y2  +  y* 
16g*-8ar*y  +  4a:2y2 


4iX2  —  2xy  -f  y2 


4a?2  +  2zy  +  y! 


8ar*y +  y4 
8ar,y-4a?2y2  +  2a;y8 


4a?y2  — 2#y8  +  y* 
4s2y2  —  2ay8  +  y4 


38.  32a56-56a462+  8  a8&8-4  a264-a65 1  -4  a26+6  a&2+68 


32a55-48a462-  8a868 

-  8a4&2+16a368-4a8&4 

-  8tt462+12a868+2a264 

4a363-6a264-a66 
4a868-6a264-a65 


-8a3+2a2&-a&2 


39. 


l  +  5ar»-6a^ 
l-a;  +  3a^ 


l-a?  +  3a:2 


l+a-2a>2 
x-3x2  +  5x*-6x* 
x—    x2  +  3S8 
-  2a? +  2ar»- 6a;4 
-2a^  +  2ar^-6a4 


teachers'  edition. 


33 


40.    l-51a3&3-52g464|-l+3a6  +    4a262 
l-3a6-4a262        \-  I -Sab-  lSa2b2 
Sab  +  4«262  -  51  a3 b3  -  52a464 
Sab-9a?b*-12a?b3 


13a2  b2- 
13a262- 


-39a363-52a464 
-39a363-52a464 


41.  x7y  —  xy7 

x1y-2x6y2  +  2x&y3-    x*y* 


x9y-2x2y2  +  2xyi-y* 


x4  +  2^2/  +  2a?y  +  ay1 


2x*y2  —  2xby*  +    a^y4  — an/7 

2aV-43*y3  +  4aV-23*,v* 

2a£y3-3z4y4  +  2z8y5-a*/7 
2s5y3— 4E4y4  +  4g3ys-2a2,y6 

a^y4-2ar,y5  +  2a;2i/a-an/7 
g4y4-2ariys-f  2a?2y6-a;y7 


42.  z*-6  afy +15  ay~20  afy8 +15  a?y-6  an/5 +y6 
3^-3^+  3a?V-     a^y3 

"  -3a^y+12a?4y2-19a^y3+15a;y 
-3a*y+  9a^,y2-  9ar»y3+  3afy4 

3a;y-10ar,y3+12arJy4-6an/6 
3**y»-  9a;y+  9x2y*-Sxy* 

-x*tf  +  3ay-3ay+y« 
-arV  +  SxfySxyt+y* 


a*-3a;2y+3an/2-y3 


3^-3^+3  a^2-^3 


43. 


ar  _  2a6&  -  2a4  J3  +  2a3 64  -  6a2 6s  -  3a66 

a3-2a2b~ab2 

a7-2a66-a5&2 

a562  -  2a463  +  2a3  A4  -  6a2b*  ■ 
a*b2-2a*V-    a364 

a4  +  a262  +  364 
-3a&« 

3a364-6a265- 
3a364-6a265- 

-3  aft6 
-3a6« 

44. 

Slafy-54ay  -183y+18ay-18ay-9y7 

81a*y  +27aV +27  *y 


-27^ 
-27a.y 


-9ay 
-9a?y 


3a*+ay+y4 


~54ay-27ay-18ay  -  9*y-18an/6-9y7 
-54sy  '   -18gy  -18ay 


27aj2y-18an/2-9y3 


-9y7 
-9£ 


34 


ALGEBRA. 


45.   a4  +  2a*b  +  Sa2b2  +  Sab*  +  lG64[a2  +  462 
a*  +  4a262 


a2  +  2a&  +  462 


2a36  +  4a262  +  8a63 

2a36 -+8a63 

4a262  +  166* 

4a262  +166* 


46.   -a*  +  2la*tf-24xy*  +  8y« 
—  aP  +  SaPy  —    xKy2 

-3x*y  +    tf*y«  +  21a*y» 
-So^y  +  dx^y2-    Sx^y3 


-x2  +  3xy  —  y2 


x*  +  3x*y  +  8x2y2-$y* 
24zy5  +  8y6 


-  $x*y2  +  24arY  -  24a*/5  +  83^ 

-  8x*y2  +  24s3  y3  -  8 x2y* 


8a;2 3/4-  24ay»  +  8/ 
8sV-24gn/b  +  8y6 


47.   16a*  +    8a262  +  96*  |4a2 

16a4-16a36  +  l2a2&2 


-4a6  +  362 


4a2  +  4a6  +  362 


16a36-   4a262  +  964 
16a36-16a262  +  12a&3 


12a262- 
12a262- 


12a634-964 
12a63  +  964 


48.   a3 


as  +  a?b  +  a2  c 


—  Sabc  +  ¥  +  c*\a  4-  b  +  c 


—  a2  6  —  a2  c  —  3  a6c  +  63  +  c3 

—  a2  6  —  a62  —    a&c 

-2abc  +  &  +  <* 


a&  —  ac  4-  62  —  6c  +  c2 


—  a?c  +  ab'1 
-a2c 


—  abc  —  ac2 

+  ab'2    —  abc  +  ac2  +  b*  +  c* 
+  ab2  +  63  +  b2c 


—  abc  +  ac2 

—  abc 


-  b2c  +  c3 

-  62c  -  6c2 


ac2 
ac2 


+  6c2  +  c3 
+  be2  +  (? 


49.  a3 


-6  abc+8  ¥+<* .  a2-2a6-ac+462-26c+c2 


a3-2  a26-a2c-f  4  a&2-2  abc+ac2 


a+2b+c 


+2a26+a2c-4a62-4a6c-ac2+863+c3 

+2a26        -4a62-2a6c        +863-462c+26c2 


+a2c 
+a2c 


—2  abc—ac2 
—2  abc—ac2 


+462c-26c2+c3 
+4&2c-26c2+c3 


teachers'  edition. 


35 


50.  o3  +  'Sa2b+3ab2  +  b*  +  <? 

a  +  b  +  c 

a3+    a2b              +  a2c 

a2  +  2ab  +  b'l-ac-bc  +  c2 

2a2b  +  3ab2-a2c  +  b*  +  c* 
2a*b  +  2ab2            +  2abc 

ab2-a2c-2abc  +  b*  +  c* 
ab2                       +b*  +  b*c 

—  a2c-2abc-b2c  +  c* 

—  a?c—    abc           —  ac2 

- 

abc  —  b2  c  +  ac2  -\- c9 
abc  —  b2c           —be2 

ac2  -{-be2  +  c* 
af  +  bct  +  c* 

Exercise  XXIII. 


a2  (b  +  c)  +  b2(a-c)  +  c2(a-b)+    abc\  a  +  Hc 
a2(b  +  c)  +  b2(a       )  +  c2(a       )  +  2abc\  a(b  +  c)-bc 


—  abc 

—  abc 


+  b2( 


2.  a^  —  (g +  6  +  c)a:2  -f  (ab  +  ac  +  6c)s- abc\ x2- (a  +  b)x  +  ab 
x*—(a  +  b      )x2  +  (ab \x___\x-c 


-ex2 
—ex2 


+  (ac-\-bc)x—abc 
+  (ac  +  bc)x—abc 


3.  x*-2axi+(a2  +  ab- 
x9—    ax2  -f  bx2 


■b2)x—a2b  -f  ab2\x  —  a  +  b 


-(a  +  b)x2  +  (a2  +  ab-b2)x-a2b  +  ab2 
-(g  +  6)rt  +  (a2         -b2)x 


Ix2— (a  +  b)x  +  ab 


+  abx— a2b  +  ab2 
+  afoc  — a26  +  a&2 


4.  a?4  —(a2  —  6  —  c) a2  —  (6  —  c) as  +  ftcls2  —  ax-\-c 
x4  +  (  +  c)a?  — aar»  |  a?  +  asc  +  6 


ax*- 
aa*- 


—  6      )  x2  —  (b  —  c)  oa;  +  be 
}jt? +  cox 


-{-bx2 
+  bx* 


—  box  +  6c 

—  bax  +  be 


36  ALGEBRA. 


5.  y3  —  (ra  +  n  +  p)  y2  +  (ran  +  mp  -f  r?p)  y  —  ronp  y  —p 


Vs  ~  (  +  P)  .V2 I  y2  -  (w  +  w)y  +  ran 

—  (m  +  n        )  ,y2  +  (ran  +  mp  +  rip)  y 

—  (m  +  n        )  y2  -f  (        +  mp  +  np)  y 

mny  —  winp 
mriy  —  mnp 


6.  x*  +  (5  +  a)  x8  -  (4  -  5a  +  b)  x2  -  (4  a  +  56)  x  +  4  b 
x*  +  (5        )  r»  -  (4 \x*_ 


a*  +  5x-4 
x2  +  ax  —  b 


or* -(    —  5a  +  b)x*  -(4a  +  5b)x 
ax*  +  (    +5a       )g*-(4q         )x 


-6x2  —  5&X  +  46 

-bx*  -56* +  46 

1  x2— (a+c)x-t-ac 
7.  a»-(&+a>+W 


a?*— (a+J+c+^Ja^+fai+ac+aa'+ic+ftrf+crf)^— (abc+abd+acd+bcd)x+abc<l 

x*— (a     +c      )as+(     +oc Jx*_ 

id  )xt—(abc + abd+acd+ bed  \x 

i.d)x*—(abc  +acd  )x 

bdx2—^      -\-abd  \-bcd)x+abcd 

bdx*-(       +abd         ~\  bcd)x+abcd 


— (  +6      -\-d)x*  \-(ab       +ad+bc-tbd+cd)x*—(abe+abd+acd+bcd 
— (  +6     +a*)s8+(a6       H-ad-i-frc       +cd)xt—(abc  +acd 


8. 

x*-(m— c)a^+(n— cm+d)xi+(r+m-dm)x2+(cr+dn)x+dr\xsj-mx2-{-nx+r 

x*-(m     )xi+(n )x*+(r )a? |x2+cx+d 

car*+r  — cra-f o*)ar,+(  -fen— dra^+^r+anb? 
cx*+(  —era      )x*+(  +cn         )x2+(cr        )x 

dx*  —dmx2         -\-dnx-\-dr 

dx* —dmx2  +dnx+dr 

9. 

a^-ma^  +  rtx8  — nx*  +  mx  —  1  x  — 1 

x*-x* 


x4  +(1  —  m)x*+(l  -ra  +  n)x*  +(1— m)x+\ 


(1  —  m)x*  +  nx*  —  nx2  +  ma;  —  1 
(1  —  m)  x*  —  (1  —  m)  x3 


(1  —  m  +  n)  x8  —  nx2  -f  mx  —  1 
(1  —  m  +  n)x3  —  (1  -ra  +  n)x2 


(1  —  ra)  a;2  +  raa;  —  1 
(1  —  m)x2  —  (1  —  ra)x 


x-1 
x-1 


10.  (x  +  y)»  +  3  (x  +  y)2«  +  3  (x  +  y)z2  -f  2s  1  (x  +  yf  +  2(x  +  y)z  +  z* 
(x  +  yY  +  2(x  +  yfz+    (x+y)z2         \x  +  y  +  z 


(x  +  y)22  +  2(x+y)22  +  2s 
(x  +  y)22  +  2(x  +  y)22  +  28 


TEACHERS     EDITION. 


Exercise  XXIV. 

1.  (y»  _  1)  +  (y  -  1)  7.   (l-8a»)+(l-2*) 
=  y*+y  +  1.  =1  +2x  +  ±x*. 

2.  (i3- 125) -*-(&- 5)  8.  (ar*-3265) -*-(*- 26) 

=  b*  +  56  +  25.  =  a*+2aM44aW+8*y+l(W. 

3.  (a* -216) -(a -6)  9.   (8a3.**8-  1)  +  (2az-  1) 
=  a2  +  6a  +  36.  =  4a2ar2  +  2aa;  +  1. 

4.  (*»- 343) -*-(*- 7)  10.   (l-27ary)  +  (l-3a!y) 
=  x2  +  7*  +  49.  =  1  +  3sy  +  9s*y2. 

5-   (**-y*)  +  (*-y)  U-  (64oW-27a£)-i-(4aft-3ar) 

=  x*  +  ar»y  +  a;2y2  +  xtf  +  y4.  =  16a2  b*  +  12ate  +  9  a*. 

6.  (o*-l)  +  (o-l)  12.  (243a5-l)  +  (3a-l) 

=  a4  +  a3  +  a2  +  a  +  1.  =  81a4  +  27a3  +  9a2  +  3a  +  1. 

13.   (32a5-24365)-*-(2a-3&) 

=  16a4  +  24a3&  +  36a262  +  54a&3+81&4. 


Exercise  XXV. 

1.  (s»  +  y»)  +  (*  +  y)  7.   (a*  +  3265)-5-(«  +  26) 

=  x*-xy  +  y2.  =  a4-2a36+4a262-8a&3+1664. 

2.  (a^  +  y^  +  fc+y}  8.   (512  a*y»  +  *»)  +  (8asy  +  e) 
=  x4  —  ar'y  -fa^y2  —  xy3  +  y4.  =  64  a^y2  — 8  xyz +  ?. 

3.  (l+8a3)-*-(l  +  2a)  9.  (729a3  +  21663)  +  (9a  +  66) 
=  1  -  2a  +  4a2.  =  81  a2  -  54a6  +  3662. 

4.  (27a3  +  ft3)-*- (3 a +  6)  10.  (64  a3  +  1000  J3)  +  (4  a  + 10  h) 
=  9a2  -  3a6  +  62.  =  16a2  -  40a&  +  1006*. 

5.  (8a3ar»  +  l)-i-(2az  +  l)     '         11.   (64 a363  +  27^)^(4 ab  +  3a) 
=  4a2a?  -  2az  +  1.  =  16a2  6»  -  12a&c  +  9a?2. 

6.  (a»  +  27ys)  +  (a  +  3y)  12.   (x»  +  343)  +  (*  + 7) 
=  a*  -  3zy  +  9y2.  =  x2  -  7x  +  49. 

13.   (27arY  +  828)-J-(3ary  +  22) 
=  9ar2y2~6a;y2  +  422- 


14.   (1024  a6  +  243  fr5)  -*-  (4  a  +  3  6) 
=  256a4  -  192a36  +  144  a2  62  - 


108a63+8164. 


38  '  ALGEBRA. 


Exercise  XXVI. 

1.  (x*-y*)  +  (X-y)  6.    (*»-81iV«)  +  (*   l-3y) 

=  x*  +  a^y  +  ay*  +  y3.  =  re3  -  3  a?y  +  9 xy2  —  2T?/3. 

2.  (aj*-y*)  +  (a?  +  y)  7.   (16**- l)  +  (2ar-l) 
=  x*-x2y  +  xyi-yi.  =  8a^  +  4ar*  +  2a:  +  1. 

3.  (a6 -a6)-*- (a -a?)  8.   (16a*  -  1)  -s-(2a:  +  1) 
=  ab+a*x+a?x2+a2a?+atfi+x&.  =  8  r1  —  4  a?  +  2a;  —  1 . 

4.  (a«-a*)-«-(a  +  a;)  9.   (81  a*x*-  1)-*-  (3az  -  1) 

=  cP-ah+aPtf-aW+ax'-x5.  =  27a3  ar,  +  9a2a*  +  3aa;+l. 

5.  (z*-81y*)  +  (x-3y)  10.   (81a**4- l)  +  (3a*  + 1) 

=  x*  +  3x2y  +  9xy2  +  27^.  =  27a3ar»  -  9a2a?  +  3ax  -  1. 

11.  (64a6-6«)-s-(2a-&) 

-  32a5  +  16a*6  +  8a362  +  4a2fc*  +  2a&*  +  6». 

12.  (64a6 -ft6)-*- (2a +  6) 

-32a5-16a*&  +  8a362-4a2&3  +  2a&*-&5. 

13.  (*«-729y«)^(a;-3y) 

=  a-5  +  3a^y  +  9ar»y2  +  27a?Y  +  81  xy*  +  243y*. 

14.  (x«-729y8)-*-(x  +  3y) 

=  x5  -  3x*y  +  9x3y2  -  27x2y3  +  81  xy*  -  243^. 

15.  (81  a*  -  16c*)  -h  (3a  -2c) 

-  27a3  +  18a2c  +  12ac*  +  8c* 

1G.    (81  a*  -  16c*)  h- (3a  +  2c)  18.  (256a*  -  10,000)  +  (4a  +  10) 

=  27a3-18a2c+12ac2-8c3.  =  64  a3- 160  a2 +400  a -1000 

17.   (256a*-  10,000)  -*-(4tf-  10)     19.   (625x*- 1)-?- (5x- 1) 
=  tiia*+  160a2+400a  + 1000.  =  125X3  +  25a?  +  5a;  +  1. 

Exercise  XXVII. 

1.  (aB  +  ys)  +  (««  +  yI)  6.   (*12  +  1)  -*-  (x*  +  1) 
«  x*  —  x2y2  +  y*.  =  xB  — x*  +  l. 

2.  (a6  +  1) -*- (a2  +  1)  7.   (64a*  +  y8)  -5- (4a;2  +  y2) 
=  a*-a2  +  l.  =16a^-4a?y2+y*. 

3.  (a10  +  y10)  -f-  (a2  +  y2)  8.  (64  +  a6)  -*-  (4  +  a2) 
=  a*-a*y2  +  ay-dY  +  if.  -  16  -  4a2  +  a*. 

4.  (6,0  +  l)  +  (6»  +  l)  9.   (729rf  +  ft*)  +  (9a*  +  M) 
=  ft«_Ji  + J4_  ji+  i,  =  81a*-9a262  +  fc*. 

5.  (a12  +  ft12)  -s-  (a*  +  &*)  10.   (729c«  +  l)-?-(9c2  +  l) 
=  a8  -  a*V  +  68.  --=  81  c*  -  9  c2  +  1. 


TEACHERS     EDITION. 


39 


Exercise  X;XVIII. 


5x-  1  =  19, 
5z  =  19  +  1, 
51  =  20, 


16a: -11  =  7x  +  70, 
16a; -7a  =70 +  11, 
9a;  =  81, 
x  =  9. 


3a; +  6  =  12, 
3rc  =  12-6, 
3x  =  6, 

a:  =  2. 


9.      24a: -49  =  19a: -14, 
24a; -19a;  =  49 -14, 
5a:  =  35, 

a:  =  7. 


24a:  =  7a;  +  34, 
24  a: -7a?  =  34, 
17a;  =  34, 
a;  =  2. 


10.       3a: +  23  =78 -2a:, 
3a:  +  2a:  =  78  -23, 
bx  =  55, 
a;  =11. 


4.  8rr-29  =  26-3a:, 

8a;  +  3a;  =  26 +  29, 
11a;  =  55, 
a;  =  5. 


11.        26-8a:  =  80-14x, 
14a; -8a:  =80 -26, 
6a;  =  54, 
x  =  9. 


12 -5a;  =  19 -12a:, 
-  5a;  +  12a;  =19 -12, 
7a;  »=  7, 
a;=l. 


12.        13  -  3a;  =  5a; -3, 
-5a;-3a;  =  -3-13, 
-8a;  =  -16, 
x  =  2. 


6.    3a;  +  6- 2a;  =  7a;, 
3a; -2a; -7a;  =  -6, 
-6a;  =  -6, 
*-l. 


13.       3a;  -  22  =  7a;  +  6, 
3a:-7a;  =  6  +  22, 
-4a;  =  28, 
x  =  -7. 


7.  5a; +  50  =4x  +  56, 

5a; -4a;  =  56 -50, 
a;  =  6. 


14.  8  +  4a;  =  12a; -10, 

4a:-12x  =  -16-8, 
-8a;  =  -24, 
x  =  3. 


15.  5a:  -  (3a;  -  7)  =  4a;  -  (6a;  -  35), 

5a;  -  3a:  +  7  =  4a;  -  6a:  +  35, 
-  4a;  +  5a;  -  3a;  +  6a;  =  35  -  7, 
4a;  =  28, 
a;  =  7. 


16.   6a;  -  2(9  -  4x)  +  3(5a?  -  7)  =  10z  -  (4  +  16a;  +  35), 
6a;  -  18  +  8a;  +  15a;  -  21  =  10a:  -  4  -  16a;  -  35, 
6a;  +  8a;  +  15a;  -  10a;  +  16a;  =  18  +  21  -  4  -  36, 
35  a;  =  0, 
a:  =  0. 


40  ALGEBRA. 


17.  9a; -3  (5a; -6) +  30-0, 

9  a?  -15a;  +  18  +  30  =  0, 

9a? -15a;  =  -18 -30, 
-6a?  =  -48, 

x  =  8. 

18.  a?  -  7  (4  as  -  11)  =  14  (a?  -  5)  -  19  (8  -  x)  -  61, 

x  -  28  x  +  77  =  14  x  -  70  -  152  +  19  x  -  61, 
a:  -  28a:  -  14a?  -  19a;  =  -  70  -  152  -  61  -  77, 
-60x  =  -360, 
x  =  6. 

19.  (a? +  7)  (a? -3)  =  (a; -5)  (a; -15), 

a?2  +  4z  -  21  =  x2  -  20a;  +  75, 
4a;  +  20  a;  =  75 +  21, 
24  a;  =  96, 
as-4. 

20.  (a;  -  8)  (a;  +  12)  =  (a?  +  1)  (a?  -  6), 

0^  +  43? -96  =  a?2 -5a; -6, 
4a?  +  5a?  =  -6  +  96, 
9a;  =  90, 
x  =  10. 


21.   (»- 


-2)(7-a?)+(a;-5)(x  +  3)-2(a;-l)  +  12  =  0, 
9a;  -  14  -  x2  +  x2  -  2a;  -  15  -  2a;  +  2  +  12  =  0, 
9a;  -  2a?  -  2a?  =  14  +  15  -  2  -  12, 
5a?  =15, 
a;  =  3. 

22.  (2a;-  7) (a;  +  5)  -  (9  -  2a?) (4  -a?)  +  229, 
2a?8  +  3a;  -  35  =  36  -  17a;  +  2a?2  +  229, 

20  a;  =  300, 
x  =15. 

23.  14-a?-5(a?-3)(a;  +  2)  +  (5-a;)(4-5a?)  =  45a;-76J 

14  -  a;  -  5a?2  +  5a?  +  30  +  5a?2  -  29a;  +  20  =  45a;  -  76, 
5  x  -  29  a?  -  45  a?  -  a?  =  -  7  6  -  20  -  30  -  1 4 , 
-70a?  =  -140, 
a?  =  2. 

24.  (a; +  5)2- (4 -a?)2  =  21  a?, 
(a?  +  10a?  +  25)  -  (16  -  8a?  +  a?2)  =  21a?, 

a?2  +  10a;  +  25  -  16  +  8a; -  a?2  =  21  a?, 

10a;  +  8a?  -  21  a?  =  -  25  +  16, 
_3a?  =  -9, 
a;  =  3. 

25.  5(a?-2)2  +  7(a?-3)2=(3a?-7)(4a?-19)+42, 
5(a?2-4a?+4)  +  7(a?2-6a;  +  9)=12a?2-85a;  +  133+42, 
50^-200? +  20  + 7a?2-42a?  +  63  =  12a?-85a?  + 133 +  42, 

23  a?  =  92, 
a?  =  4. 


teachers'  EDITION.    '  41 


Exercise  XXIX. 

6.  To  the  double  of  a  certain  number  I  add  14,  and  obtain  as 
a  result  154.     What  is  the  number  ? 

Let  x  =  the  number. 

Then  2a?  =  its  double, 

and  2  a;  +  14  =  its  double  increased  by  14. 

But  154  =  its  double  increased  by  14. 

Therefore,  2  a;  +  14  =  154,   2  a;  =  140,  x  =  70. 

7.  To  four  times  a  certain  number  I  add  16,  and  obtain  as  a 
result  188.    What  is  the  number  ? 

Let  x  =  the  number. 

Then  4  a;  =  4  times  the  number, 

and  4a;  +  16  =  4  times  the  number  increased  by  16. 

But  188  =  4  times  the  number  increased  by  16. 

Therefore,  4a;  +  16  =  188,   4a;  =  172,   a;  =  43. 

8.  By  adding  46  to  a  certain  number,  I  obtain  as  a  result  a 
number  three  times  as  large  as  the  original  number.  Find  the 
original  number. 

Let  x  =  the  original  number. 

Then  3  x  =  3  times  the  original  number. 

But  x  +  46  =  3  times  the  original  number. 

Therefore,  3  a;  =  a: +  46,   2  a;  =  46,   a;  =  23. 

9.  One  number  is  three  times  as  large  as  another.  If  1  take 
the  smaller  from  16  and  the  greater  from  30,  the  remainders  are 
equal.    What  are  the  numbers  ? 

Let  x  =  the  smaller  number. 

Then  3  x  =  the  larger  number, 

and  16  —  x  =  16  diminished  by  the  smaller  number  ; 

also,  30  —  3  a;  =  30  diminished  by  the  larger  number. 

Therefore,     16  -  x  =  30-  3a;,    2a;  =14,   x  =7,  3a;  =  21. 

10.  Divide  the  number  92  into  four  parts,  such  that  the  first 
exceeds  the  second  by  10,  the  third  by  18,  and  the  fourth  by  24. 

Let  x  =  the  first  part. 

Then  x  —  10  =  the  second  part, 

x  — 18  =  the  third  part, 
x  —  24  =  the  fourth  part, 
and  4  x  —  52  =  the  whole  number. 

But  92  =  the  whole  number. 

.\4a;-52=92,  4a;=144,  a;=36,  a;-10  =  26,  3-18  =  18,  a;-24  =  12. 


42  ALGEBRA. 


11.  The  sum  of  two  numbers  is  20;  and,  if  three  times  the 
smaller  number  is  added  to  five  times  the  greater,  the  sum  is  84. 
What  are  the  numbers  ? 

Let  x  =  the  greater  number. 

Then         20  —  x  =  the  smaller  number, 

5x  =  5  times  the  greater  number, 
3  (20  —  x)  =  3  times  the  smaller  number, 
5  a:  -f  3(20  —  a;)  =  5  times  the  greater  -f  3  times  th©  smaller. 
But  84  =  5  times  the  greater  +  3  times  the  smaller. 

/.5a;  +  3(20-a;)  =  84,   5a;  +  60-3a;  =  84,   2z  =  24,   a?=12,  20-x=S. 

12.  The  joint  ages  of  a  father  and  son  are  80  years.  If  the 
age  of  the  son  were  doubled,  he  would  be  10  years  older  than  his 
father.     What  is  the  age  of  each  ? 

Let  x  =  number  of  years  of  father's  age. 

Then         80  —  x  =  number  of  years  of  son's  age, 

2(80  —  x)  =  number  of  years  of  father's  age  +  10, 
x  +  10  =  number  of  years  of  father's  age  +  10. 
.\2(80-a:)  =  a;  +  10,  160-2x  =  a;+10,  -3a?  =  -150,  a;=50,  80-a;  =  30. 

13.  A  man  has  6  sons,  each  4  years  older  than  the  next  younger. 
The  eldest  is  three  times  as  old  as  the  youngest.  What  is  the 
age  of  each? 

Let  x  =  number  of  years  of  age  of  youngest. 

Then  x  +  4  =  number  of  years  of  age  of  second, 

x  +  8  =  number  of  years  of  age  of  third, 
x  +  12  =  number  of  years  of  age  of  fourth, 
x  +  16  =  number  of  years  of  age  of  fifth, 
x  +  20  =  number  of  years  of  age  of  sixth. 
3  x  =  3  times  age  of  youngest. 
/.  3«  =  a:  +  20,   2a;  =  20,   a;  =  10,   x  +    4=14,   x  +    8  =  18, 
x  +  12  =  22,   a; +  16  =  26,   x  +  20  =  30. 

14.  Add  #24  to  a  certain  sum  and  the  amount  will  be  as  much 
above  $80  as  the  sum  is  below  $80.     What  is  the  sum? 

Let  x  =  number  of  dollars  in  sum. 

Then  x  +  24  —  80  =  number  of  dollars  above  80, 
and  80  —  x  =  number  of  dollars  below  80. 

.-.a; +  24 -80  =  80 -a;,   2a;  =  136,   a:  =  68. 

15.  Thirty  yards  of  cloth  and  40  yards  of  silk  together  cost 
$330 ;  and  the  silk  twice  as  much  a  yard  as  the  cloth.  How  much 
did  each  cost  a  yard  ? 

Let  x  =  number  of  dollars  one  yard  of  cloth  cost. 

Then  2  a;  ^=  number  of  dollars  one  yard  of  silk  cost. 

30  a;  +  80  a;  =  number  of  dollars  all  cost. 
But  330  —  number  of  dollars  all  cost. 

/.30a;  +  80a;  =  330l    110a;  =  330,   x  =  3,   2a; -6. 


teachers'  edition.  43 

16.  Find  the  number  whose  double  increased  by  24  exceeds 
80  by  as  much  as  the  number  itself  is  less  than  100. 

Let  x  =  the  number. 

Then       2x  +  24  =  its  double  increased  by  24, 
2  x  +  24  —  80  =  excess  over  80, 

100  —  x  ==  difference  between  the  number  and  100. 
.\2x  +  24-80  =  100-*,   3a;  =  156,   x  =  52. 

17.  The  sum  of  $500  is  divided  among  A,  B,  C,  and  D.  A 
and  B  have  together  $280,  A  and  C  $260,  and  A  and  D  $220. 
How  much  does  each  receive? 

Let  x  =  number  of  dollars  A  has. 

Then        280  —  x  =  number  of  dollars  B  has, 
260  —  x  =  number  of  dollars  G  has, 
220  —  x  =  number  of  dollars  D  has, 
760  —  2x  =  number  of  dollars  all  have. 
But  500  =  number  of  dollars  all  have. 

.-.760  -2x  =  500,        -2x  =  -260,  x  =  130, 

280-*  =  150,   260-*  =  130,   220-*=  90. 

18.  In  a  company  of  266  persons  composed  of  men,  women, 
and  children,  there  are  twice  as  many  men  as  women,  and  twice 
as  many  women  as  children.    How  many  arc  there  of  each? 

Let  x  =  number  of  children. 

Then  2x  =  number  of  women, 

and  4*  =  number  of  men, 

7*  =  whole  number. 
But  266  =  whole  number. 

.-.7*  =266,  3  =  38,   2*  =76,   4*  =  152. 

19.  Find  two  numbers  differing  by  8,  such  that  four  times  the 
less  may  exceed  twice  the  greater  by  10. 

Let  x  =  greater  number. 

Then  *  —  8  =  smaller  number. 

4(3  _  8) -2*  =10. 

.\4*-32-2*=10,  2*  =  42,  *  =  21,  *-8  =  13. 

20.  A  is  58  years  older  than  B,  and  A's  age  is  as  much  above 
60  as  B's  age  is  below  50.    Find  the  age  of  each. 

Let  x  =  number  of  years  of  B's  age. 

Then  x  +  58  =  number  of  years  of  A's  age, 

(*  +  58)  —  60  =  number  of  years  of  A's  age  above  60, 
50  —  x  =  number  of  years  of  B's  age  below  50. 
.•.(*  +  58)  ^0  =  50-s,   2*  =  52,   x  =  26,   a; +  58  =  84. 


44  ALGEBRA. 


21.  A  man  leaves  his  property,  amounting  to  $7500,  to  be 
divided  among  his  wife,  his  two  sons,  and  three  daughters, 
as  follows :  a  son  is  to  have  twice  as  much  as  a  daughter,  and 
the  wife  $  500  more  than  all  the  children  together.  How  much 
was  the  share  of  each? 

Let  x  =  number  of  dollars  in  a  daughter's  share. 

Then        2x  =  number  of  dollars  in  a  son's  share, 
and  3  a?  =  number  of  dollars  given  to  all  the  daughters . 

also,  4  a;  =  number  of  dollars  given  to  all  the  sons. 

7  a;  =  number  of  dollars  given  to  all  sons  and 
daughters, 
7  a;  +  500=  number  of  dollars  given  to  wife, 
7  a;  +  7  a;  +  500=  number  of  dollars  in  whole  estate. 
But       7500  =  number  of  dollars  in  whole  estate. 
/.  7  a;  +  7a; +  500  =7500,  14a;  =7000,  x  =  500, 

2a;=1000,    7a;  +  500  =  4000. 

22.  A  vessel  containing  some  water  was  filled  by  pouring  in 
42  gallons,  and  there  was  then  in  the  vessel  seven  times  as  much 
as  at  first.     How  much  did  the  vessel  hold? 

Let  x  =  number  of  gallons  the  vessel  holds. 

Then  a?— 42  =  number  of  gallons  there  were  in  the  vessel, 
7  (x  —  42)  =  7  times  number  of  gallons  there  were  at  first. 
:.x  =7 (x -42),   a;=7a;-294,   -6a;  =  -294,   a;  =  49.  ' 

23.  A  has  $  72  and  B  has  $  52.  B  gives  A  a  certain  sum ;  then 
A  has  three  times  as  much  as  B.  How  much  did  A  receive 
from  B? 

Let  x  =  number  of  dollars  A  receives  from  B. 

Then  52  —  x  =  number  of  dollars  B  has  left, 
and         72  +  x  =  number  of  dollars  A  has. 
-.72  +  x  =  3(52  -a;),    72  +  x  =  156  -  3a;,   4x  =  84,   a;  =  21. 

24.  Divide  90  into  two  such  parts  that  four  times  one  part 
may  be  equal  to  five  times  the  other. 

Let  x  =  larger  number. 

Then  90— x  =  smaller  number. 
/.4a;  =  5(90-a?)J  4a;  =  450-5a;,  9a;  =  450,  a;  =  50,  90-a;  =  40. 

25.  Divide  60  into  two  such  parts  that  one  part  exceeds  the 
other  by  24. 

Let  x  =  lesser  part. 

Then  x  +  24  =  greater  part. 

,\a;  +  a;  +  24  =  60,   2a;  =  36,   .?  =  18,  a;  +  2i««42. 


teachers'  edition.  45 

26.  Divide  84  into  two  such  parts  that  one  part  may  be  less 
than  the  other  by  36. 

Let  x  =  lesser  part. 

Then  x  +  36  =  greater  part. 
.\a;  +  a;  +  36=84,   2a;  =  48,   x  =24,   84 -a;  =  60. 

27.  A  is  twice  as  old  as  B,  and  22  years  ago  he  was  three 
times  as  old  as  B.    What  is  A's  age? 

Let  x  =  number  of  years  of  B's  age. 

Then        2a;  =  number  of  years  of  A's  age ; 
also,      x  —  22  =  number  of  years  of  B's  age  22  years  ago, 
and     2  a;  —  22  =  number  of  years  of  A's  age  22  years  ago. 
.\3(a;-22)  =  2a;-22,   3a:- 66  =  2a;- 22,  a;  =  44,   2a;  =  88. 

28.  A  father  is  30  and  his  son  6  years  old.  In  how  many 
years  will  the  father  be  just  twice  as  old  as  the  sou? 

Let  x  =  number  of  years. 

Then  x  +  30  =  number  of  years  of  father's  age  x  years  hence, 
and         x  +  6  =  number  of  years  of  son's  age  x  years  hence. 
.\30  +  a:  =  2(a;  +  6),   30  +  x  =  2a;  +  12,  x  =  18. 

29.  A  is  twice  as  old  as  B,  and  20  years  since  he  was  three 
times  as  old.    What  is  B's  age? 

Let  x  =  B's  age. 

Then        2x  =  A's  age ; 
also,       x  —  20  =  B's  age  20  years  since, 
and     2  a;  —  20  =  A's  age  20  years  since. 

.\2a;-20  =  3(a;-20),   2a;- 20  =  3a;- 60,   a:  =  40. 

30.  A  is  three  times  as  old  as  B,  and  19  years  hence  he  will 
be  only  twice  as  old  as  B.    What  is  the  age  of  each? 

Let  x  =  number  of  years  of  B's  age. 

Then        3a;  =  number  of  years  of  A's  age  ; 
also,       x  +  19  =  number  of  years  of  B's  age  19  years  hence, 
and     3  a;  -f  19  =  number  of  years  of  A's  age  19  years  hence. 
.\3a:  +  19  =  2(a;  +  19),   3a;  +  19  =  2a;  +  33,   x  =19,   3a;  =  57. 

31.  A  man  has  three  nephews ;  his  age  is  50,  and  the  joint 
ages  of  the  nephews  is  42.  How  long  will  it  be  before  the  joint 
ages  of  the  nephews  will  be  equal  to  that  of  the  uncle? 

Let  x  =  the  number  of  years. 

Then  50  +  x  =  number  of  years  of  uncle's  age  x  years  hence. 
3  a;  +  42  =  number  of  years  of  nephews'  age  x  years  hence 
.-.3a; +42  =  50  +  *,   2a;  =  8,   a;  =  4. 


46  ALGEBRA. 


32.  A  sum  of  money  consists  of  dollars  xand  twenty-flve-cent 
pieces,  and  amounts  to  #20.  The  number  of  coins  is  50.  How 
many  are  there  of  each  sort? 

Let  x  =  number  of  dollars. 

Then  50— a;  =  number  of  quarters, 

and  x  +  — —  =  sum  in  dollars. 
4 
But  20  =  sum  in  dollars. 

...x  + 5^11?  =  20,  4a;-f50-a;  =  80,  3»  =  30,  as- 10,  50-3  =  40. 
4 

33.  A  person  bought  30  pounds  of  sugar  of  two  different 
kinds,  and  paid  for  the  whole  $2.94.  The  better  kind  cost  10 
cents  a  pound,  and  the  poorer  kind  7  cents  a  pound.  How  many 
pounds  were  there  of  each  kind? 

Let  x  =  number  of  pounds  of  the  better  kind. 

Then  30 —x  =  number  of  pounds  of  the  poorer  kind, 

and  10a:  +  7(30— x)  =  number  of  cents  he  paid  tor  all. 
But  294  =  number  of  cents  he  paid  for  all. 

.'.  10*  +  7(30  -x)  =  294,    10s  +  210  -  Ix  -  294, 
3a;  =  84,     a;  =28,    30 -a;  =  2. 

34.  A  workman  was  hired  for  40  days,  at  $  1  for  every  day 
he  worked,  but  with  the  condition  that  for  every  day  he  did  not 
work  he  was  to  pay  45  cents  for  his  board.  At  the  end  of  the 
time  he  received  #22.60.    How  many  days  did  he  work? 

Let  x  =  number  of  days  he  was  idle. 

Then  40— x  =  number  of  days  he  worked, 

and  45a;  =  number  of  cents  he  paid  for  board  ; 

also,     4000  —  100  a;  =  number  of  cents  he  received  for  work, 
(4000—  100a;)— 45a;  =  number  of  cents  cleared. 

But  2260  =  number  of  cents  cleared. 

.-.  4000  -100a;-  45a;  =  2260,   -  145a;  = -1740,   x  =  12,  40  -  x  =  28. 

35.  A  wine  merchant  has  two  kinds  of  wine ;  one  worth  50 
cents  a  quart,  and  the  other  75  cents  a  quart.  From  these  he 
wishes  to  make  a  mixture  of  100  gallons,  worth  $2.40  a  gallon, 
ilow  many  gallons  must  he  take  of  each  kind? 

Let  x  =  number  of  gallons  at  f  2. 

Then        100  —  x  =  number  of  gallons  at  $  3, 
and  2  a:  =  number  of  dollars  one  part  cost ; 

also,        3(100  —  x)  =  number  of  dollars  the  other  part  cost, 
and  2a:+3(100  —  a;)  =  number  of  dollars  all  cost. 

But  240  «=  number  of  dollars  all  cost. 

•  2a;  +  3(100 - x)  =  240,    2  a;  +  300 -3  a:  =  240,   a?  =  60,    100-a;  =  40. 


TEACHERS'    EDITION.  47 

36.  A  gentleman  gave  some  children  10  cents  each,  and  had 
a  dollar  left.  He  found  that  he  would  have  required  one  dollar 
more  to  enable  him  to  give  them  15  cents  each.  How  many 
children  were  there? 

Let  x  =  number  of  children. 

Then  10  x  =  number  of  cents  given , 

and         10  x  +  200  =  number  of  cents  required  to  give  each  15  cts. 
But  15  x  =  number  of  cents  required  to  give  each  15cts. 

.-.10a;  +  200  =15 x,   -bx  =  -200,   a;  =  40. 

37.  Two  casks  contain  equal  quantities  of  vinegar :  from  the 
first  cask  34  quarts  are  drawn ;  from  the  second,  20  gallons ;  the 
quantity  remaining  in  one  vessel  is  now  twice  that  in  the  other. 
How  much  did  each  cask  contain  at  first? 

Let  x  =  number  of  quarts  each  contained  at  first. 

Then  x  —  34  =  number  of  quarts  first  now  contains, 

and  x— 80  =  number  of  quarts  second  now  contains. 

2  (x  —  80)  =  twice  the  No.  quarts  second  now  contains. 


2(z-80)  = 
2  (a? -80)  = 


.\2(a:-80)  =  a;-34,   2a>-160  =  a;-34,   z=126. 

38.  A  gentleman  hired  a  man  for  12  months,  at  the  wages  of 
$90  and  a  suit  of  clothes.  At  the  end  of  7  mouths  the  man  quits 
his  service,  and  receives  $33.75  and  the  suit  of  clothes.  What 
was  the  price  of  the  suit  of  clothes? 

Let  x  =  number  of  dollars  the  suit  cost. 

Then  x  +  90  =  number  of  dollars  he  receives  by  the  year. 

and  — —  =  number  of  dollars  he  receives  by  the  month. 

12  J 

and  v     — l  =  number  of  dollars  he  receives  for  7  months. 

12 
But      x  +  33.75  =  number  of  dollars  he  receives  for  7  months. 

.?(«  + 30) -33.75    7a; +  630  =  405  + 12a;,   5a;  =  225,   a?  =  45. 
12 

39.  A  man  has  three  times  as  many  quarters  as  half-dollars, 
four  times  as  many  dimes  as  quarters,  and  twice  as  many  half 
dimes  as  dimes.  The  whole  sum  is  $7.30.  How  many  coins 
has  he  in  all  ? 

Let  x  =  number  of  half-dollar  pieces. 

Then  3  x  =  number  of  quarter-dollar  pieces, 

12  a;  =  number  of  dimes, 

24  a;  =  number  of  half-dimes. 

x  ,  3x  ,  12a?  ,  24a;      ,i        ,    ,  •     j  n 

-  +  —  +  -zr^r  + =•  the  whole  sum  in  dollars. 

2      4        10        20 

But  7.30  =  the  whole  sum  in  dollars. 

^  +  3£  +  12a;+24»  =  ^    1Qx  +  15a.  +  24a;  +  24a;  =  146, 

2      4        10        20 

73x  =  146,   a;  =  2,   3x  =  6,    12a;  =  24,   24a;  =  48 


48  ALGEBRA. 


40.  A  person  paid  a  bill  of  $  15.25  with  quarters  and  half- 
dollars,  and  gave  51  pieces  of  money  all  together.  How  many  of 
each  kiud  were  there? 

Let  x  =  number  of  half-dollars. 

Then  51 — *  =  number  of  quarter-dollars. 

50  a:  =  number  of  cents  in  half-dollars, 
25(51  —  x)  =  number  of  cents  in  quarter-dollars, 
50*  +  25(51  —  *)  =  number  of  cents  in  all. 
But  1525  =  number  of  cents  in  all. 

.-.  50s  +  25(51  -  x)  =  1525,   50*  +  1275  -  25a?  =  1525, 
25*  =  250,   a  =10,   51-*  =  41. 

41.  A  bill  of  £  100  was  paid  with  guineas  (21  shillings)  and 
hall-crowns  (2J  shillings),  and  48  more  half-crowns  than  guineas 
were  used.     How  many  of  each  were  paid? 

Let  *  =  number  of  guineas. 

Then  *  +  48  =  number  of  half-crowns, 

21*  +  2j  (*  +  48)  =  number  of  shillings  in  the  lot. 
But  2000  =  number  of  shillings  in  the  lot. 

.\21*  +  2£(*  +  48)=2000,    21*  +  5x  +  240  =  2000, 

42* +  5* +  240  =  4000,   47*  =  3760,  *  =  80,    *  +  48  =  128. 


Exercise  XXX. 

1.  5a* -15a  6.   6 a5 fc8  -  21  a4 b2  +  27a8 b4 
=  5a(a  -  3).  =  3a862(2a26  -  7a  +  9b2). 

2.  6a3  +  18a2 -12a  7.   54 x2  y6  +  108*V~  243*6  f 
=  6a  (a2  +  3a-  2).  =  27*2y8(2  +  4*2t/2  -  9*V). 

3.  49*2-21* +  14  8.  45*V°-90*5y7-360*V 
=  7(7**  -  3*  +  2).  =  45*V(*V  -  2*  -  8y). 

4.  4*^  -12*2y2  +  8*^  9.    70asy4-  140 a2 y*  +  210 ay6 
=  4*t/(*2  -  3*y  +  2y2).  =  70ay*  (a2  -2ay  +  3y2). 

10.   32a36«  +  96a668-128a869 
=  32a366(l  +  3a862-4a56s) 


y*  —  aw8  +  by2  +  cy 
=  y(y3-ay*  +  6y  +  c). 


Exercise  XXXI. 

1.  *2  —  a*  —  bx  +  ab  3.   be  +  bx  —  ex—  x2 

=  (*  —  a)  (*  —  b).  =  (b  —  *)  (c  +  *). 

2.  ab  +  ay  —  by  —  y2  4.   mx  +  mn  +  ax  +  an 
=  (a  —  y)(b  +  y)-  =  (m  +  a)  (a?  +  w). 


teachers'  edition.  49 


5.  cdx2  —  cxy  +  dxy  —  y2  8.   obey  —  b2dy  —  acdx  +  bd*  x 
=  (cx  +  y)  (dx  —  y).  =  (ae  —  ba )  {by  —  dx). 

6.  abx  —  aby  +pqx  —pqy  9.   ax  —  ay  —  bx  +  by 
=  (ab  +pq)  (x  -  y).  =  (a  -  6)  (x-y). 

7.  cdx1  +  adxy  —  bcxy  —  aby2  10.   cdz2  —  cyz  +  dyz  —  y2 
=  (ex  +  ay)(dx-by).  =  (c*  +  y) (cfe  - y). 


Exercise  XXXII. 

1.  s2  +  lla;  +  24  14.  a*  +  5a2  +  6 

=  (*  +  8)  (x  +  3).  -  (a2  +  3)  (a1  +  2). 

2.  3?  +  11*  +  30  15.  28  +  4zs  +  3 

=  (z  +  6)(s  +  5).  ~(*  +  3)(«»  +  l). 

3.  y2  +  lfy+60  16.  a262  +  18a6  +  32 

=  (y  +  12)  (y  +  5).  =  (aft  +  16)  (ab  +  2). 

4.  2*  +  132  *  12  17.  sV  +  7x*y*  +  12 

=  (z  +  12)  (2  +  1).  =  (**y2  +  4)  (x*y2  +  3> 

5.  a2 +  21  x  +  110  18.  210  +  102*  +  16 

=  (as  +  1 1)  (x  +  10).  =  (2s  +  8)  (2*  +  2). 

6.  y2  +  35y  +  300  19.   a2  +  9a6  +  2062 

=  (y  +  20)  (y  +  15).  =  (a  +  56)  (a  +  46). 

7.  6* +  236 +  102  20.  z8  +  9x8  +  20 

=  (6  +  17)  (6  +  6).  =  (*»  +  5)  (x9  +  4). 

8.  re2  +  3a  +  2  21.  a2*2  +  14a6x  +  3362 

=  (x  +  2)  (x  +  1).  -  (a*  +  11 6)  (ax  +  3  6). 

9.  2?  +  7x  +  6  22.  a2c2  +  7acz  +  10x2 

=  (a  +  6)  (a?  +  1).  =  (ac+ 5x)(ac  +  2x). 

10.  a2  +  9<x6  +  8 62  23.  x*y*z*  +  19zy2  +  48 
=  (a  +  86)  (a  +  6).  =  (xyz  +  16)  (xy2  +  3). 

11.  a?  +  13a*  +  36a2  24.   6V+  18a6c  +  65a2 

=  (x  +  9a)  (x  +  4a).  =  (6c  +  13a)  (6c  +  5a). 

12.  y*  +  19oy  +  48p2  25.   rV  +  23 rsz  +  90 22 

=  (y  +  16p)  (y  +  3p).  =  (rs  +  18  2)  (rs  +  52). 

13.  22+29a2  +  100g2  26.  m*n4  +  20m2n2pa  +  51p2o2 
=  (2  +  25a)  (2  +  4a).  =  (m2n2  +  17pa)(m2n2  +  3^o) 


50  ALGEBRA. 


Exercise  XXXIII. 

1.  a»-7«  +  10  13.   a2  b2  <?  -  13  abc  +  22 

-  (x - 5)  (a -  2).  =  (a&c  -  ll)'(a&c -  2). 

2.  a2 -29a? +190   *  14.  a?2 -15a? +  50 

=  (a? -19)  (a; -10).  =  (a?-*0)(a?-5). 

3.  a2  -23  a  +  132  15.   a?2-  20  a?  +100 

=  (a  -  12)  (a  -  11).  =  (a?  - 10)  (x  -  10). 

4.  &2-30&  +  200  16.   a^  -  21  oa;  +  54 

-  (6  -  20)  {b  -  10).  =  (ax  -  18)  (ax  -  3). 

5.  2a- 432 +  460  17.  a2a*-16a&a?  + 39  62 

=  (2  -  23)  (2-20).  =  (ox  -  13  b)  (ax  -  3  b). 

6.  a;2 -7a? +  6  18.   a2 c2  -  24 ac2  +  143 2* 

=  (a?  -  6)  (x  -  1).  =  (ac  -  13z)(ac -  llz). 

7.  a?*-4a2a?2  +  3a*  19.   a?2 -20a? +91 

=  (x*  -  3a2)  (a?2  -  a2).  =  (x  -  13)(a? -  7). 

8.  a?2-8a?  +  12  20.   a?*-23a:  +  120 

=  (a?_6)(a?-2).  =(x-15)(x-S). 

9.  22-57z  +  56  21.  22-  532  +  360 

=  (2  -  56)  (2  -  1).  =  (2-  45)  (z  -  8). 

10.  y6-7y8+12  22.  a?2-^  c)a? +  ac 
-(y»-W-3):  =(a?-a)(a;-c). 

11.  a?2y2-27a?y  +  26  23.  y222  -  28 abyz  +  187a2 62 
=  (xy-  26)  (ary  -  1).  =  (yz  -  17a6)  (yz  -  11  ab). 

12.  a*&6-lla2&8  +  30  24.   c2 d2  -  30 abed  +  221  a2 62 
=  (a2  6s  -  6)  (a2  6s  -  5).  =  (cd  -  17a6)  (erf  -  13  ab) 

Exercise  XXXIV. 

1.  r«  +  6.T-7      .  5.  22  +  ll2-12 

=  (a?  +  7)(a?-l).  =(2+12)(2-.l). 

2.  a?2  +  5a?-*84  6.  z2  +  132-140 

=  (x  +  12)  (a?-  7).  =  (2  +  20)  (2-7). 

3.  y2  +  7y-60  7.   a2  +  13a -300 

=  (y  +  12)  (y  -5).  =  (a  +  25)  (a  -  12). 

4.  y2  +  12y-45  8.   a2  +  25a- 150 

=  (y  +  15)  (y  -  3).  =  (a  +  30)  (a  -  5). 


TEACHERS*    EDITION.  51 

9.   &8  +  364_4  12.   c2  4- 17c -390 
=  (6*  +  4)  (V  -  1).  -  (c  +  30)  (c  -  13). 

10.  6V  +  3&C-154  13.   a2  +  a-  132 

=  (be  4-  14)  (6c  - 11).  =  (a  4- 12)  (a  -  11). 

11.  c">  +  15c5 -100  14.   x2y2z2  4- 9xyz- 22 

=  (c5  +  20)  (c5  -  5).  =  (xyz  +  11)  (syz  -  2). 

Exercise  XXXV. 

1.  s*-3a:-28  9.  y2-5ay-50a2 

=  (x  -  7)  (x  +  4).  =  (y  -  10a)  (y  4-  5a). 

2.  y2_7y_i8  10.  a262-3a&-4 

=  (y-9)(y  +  2).  -(o6-4)(a6  +  l). 

3.  a3_9a._36  11.  a2x2-3ax-54 

=  (x  -  12)  (x  +  3).  =  (ax  -  9)  (ax  4-  6). 

4.  z2-llz-60       "  12.  c2^2-  24ca*- 180 
=  (z  -  15)  (z  +  4).  =s  (erf  -  30)  (cd  4-  6). 

5.  2a_l3z-14  13.  a«c2-a»c-2 

=  (z  - 14)  (z  +  1).  =  (a8c  -  2)  (a*c  4-  1). 

6.  a2 -15a -100  14.   y8z*-5y*z2- 84 

=  (a  -  20)  (a  4-  5).  =  (y4z2  -  12)  (y*z2  +  7). 

7.  c10-9c5-10  15.   a262-16a&-36 

=  (c5  -  10)  (c5  +  1).  =  (ab  -  18)  (a&  +  2). 

8.  x2  —  8x-20  16.   x2-(a-6)x  —  ab 
=  (x  -  10)  (x  +  2).  =  (x  -  a)  (x  +  6). 

Exercise  XXXVI. 

1.  x2  4- 12x4- 36  6.  z*  +  14z2  +  49 
=  (x  +  6)2.  =  (z2  +  7)2. 

2.  x2*  28x4- 196  7.  x2  +  36xy  4-  324y2 
=  (x  +  14)2.  =  (x4-18y)2. 

3.  x2  4- 34x4- 289  8.   y*  4-  16VZ2  4-  64z* 
=  (x  +  17)2.  =(y2V8z2)2. 

4.  z24-2z4-l  9.   yi  +  24y8  +  144 
-(z-fl)2.  =(3/3  +  12)2. 

5.  y2  4-  200  y  4-  10,000  10.   x2z2  4-  162  xz  4-  6561 
=  (y  4- 100)2.  =  (xz  4-  81)2. 


52  ALGEBRA. 


11.  4a2  +  12aft2  +  9ft4  13.    9z2  +  \2xy  +  4y2 
=  (2a+3ft2)2.  =  (3z  +  2y)2. 

12.  9x2y*  +  30ary2z  +  25z2  14.   4 a4 a?2  +  20a2ar>y  +  25 z*y2 
=  (3zy2  +  52)2.  =  (2a2a;  +  5a?2y)2. 

Exercise  XXXVII. 

1.  a2 -8a  +  16  13.   16*«-8a^y2  +  a^y4 
=  (a-4)2.  =(4^ -ay2)2. 

2.  a2 -30a  +  225  14.   a**4-  2asftx2y4  +  ft2y* 
^(a-15)2.  =(a8a;2-6y*)< 

3.  i2  -38*  +361  15.   36r»y2-60zys  +  25y4 
=  (s-19)2.  =(6sy-5y2)2. 

4.  a2 -40a; +  400  16.   l-6aft8  +  9a2ft* 
=  (a;-20)2.                   '  =(1-3  oft3)2. 

5.  y2-100y  +  2500  17.  9m2n2-24mn  +  16 
=  (y-50)2.                   .  =(3mn-4)2. 

6.  y4-20y2+100  18.   4ft2sa-  12 bx*y  +  9 x*y2 
=  (y2-10)2.  =  (2ftx-3xyj*. 

7.  tf-50yz  +  625z*  19.   49a2-  112 aft  +  64 ft2 
=  (y-25z)2.  =  (7a -8ft)2. 

8.  ar4-32*2y2  +  256yi  20.   64x*y«-  IMsttfz  +  lOOz4*2 
=  (z2  -  16y2)2.  =  (8x*y*  -  10  A)2. 

9.  z6-  342s  +  289  21.   49  a2  ft2  c2- 28  aftcz  +  4  a? 
=  (2»-17)2.  =  (7aftc-2a:)2. 


10.   4a4y2  -  20arly8z  +  25y4z2 
=  (2a?2y-5yV. 


22.   121x4-286r,y+169y2 
=  (llr«-13y)2. 


11.  lGaJy*-8jrys22+y2z4  23.   289  afyV-^zyVo^yVd1 
=  (4*y2  -  yz2)2.  =  (17zyz-3yza7. 

12.  0a262c2-6a62c2a'  +  62c2a'2  24.  361xsy2z2-76a6cayz+4a262c2 
=  (3aftc  -  bed)*.  =  (19 xyz-2abc)*. 


Exercise  XXXVIII. 

1.  a2 -ft2  ,  3.   4a2 -25 

-  (a  +  ft)  (a  -  ft).  =  (2  a  +  5)  (2  a  -  5). 

2.  a2 -16  4.   a* -ft4 

-  (a  +  4)  (a  -  4).  =  (a2  +  ft2)  (a  +  6)  (a  -  6). 


TEACHERS*    EDITION.  53 


5.   a*-l  10.    1-49*2 

=  (a2  +  l)(a  +  l)(a  -  1).  =  (1  +  7*) (I  -  7*). 

6-   a8-*8  11.   a*  -25b2 

=  (aH6*)(a2+62)(a+6)(a-6).  =  (ai  +  db){a2  _  56). 

7'  t  (^-l)(a2+l)(a+l)(a-l).  *•   L^- 6~)-cH(a-6)  + 

8.  36s2 -49y2  =  (a-  6-c)(a-6  +  c). 

=  (Gx  +  7y)(6x-7y). 


9.   100a^y2-121a262 

=  (10a;y+lla6)(103y-lla6). 


13.   x*-(a-b)* 

=  {x-(a-b)}{x  +  (a-b)} 
=  (a;  —  a  +  6)  (a?  +  a  —  b). 


14.  (a  +  6)2-(c+rf)2 

=  [(«  +  &)  +  (c  +  <*)]  [(«  +  &)-(«  +  <*)] 
=  (a  +  6  +  c  +  d)  (a  +  6  -  c  -  d). 


15.   («  +  yf  -  (x  -  y)2 

=  {(*  +  y)  +  (*-y)M(z  +  y)-(z-y)} 

=  (x  +  y  +  x  -y)(x  +  y  - x  +  y) 
=  4xy. 


16.  2a6  -  a2  -  6*  +  1  19.   a2  +  126c  -  462  -  9c2 

=  1  -  (a2  -  2ab  +  b2)  =  a2  -  (46*-  126c  +  9c2) 

=  l-(a-6)2  =a2-(26-3c)2 

=  {1  +  (a  -  6)}{1  -  (a  -  6)}  =  {a  +  (2  6  -  3  c)}{a-(2 6-3  c)} 

~(1  +  a  -  6)(1  -  a  +  6).  =  (a  +  26-  3c)(a-26  +  3c). 

17.  s?-2y*-y*-^  20.   a2-  2ay  +  y"-3i-2«-  22 

=  a*  _  (y2  +  2y2  +  22)  =  (aa_2  av+y2)-(ic2+2a;2+22) 

=  s2-(y  +  2}2  ={a-y)*-(x  +  z? 

=  (a;  +  (y  +  z)}  {x  -  (y  +  2)}  =  {(a-t/)+(a?+2)}{(a-y)-(ir+2)} 

=  (x  +  y  +  2)  (x  -  y  -  2).  =  (a-y+x+z)(a-y-x-z). 

18.  s*-2zy  +  y2-22  21.   2xy  -  a^-y2  +  22 
=  (z2  -  2 ay  +  y2)  -  z2  =  z2 -(x2  -  2xy  +  y2) 
'(x-yY-z2  =z2^lx_yy 

=  te  +  (x-y)\{z-{x-y)} 
=  (z+x-y)(z-x+y). 


=  (x-y  +  z)(z-y-z). 


22.  x*  +  y2-z2-d2-2xy- 
=  (x*-2xy  +  y2)-(d2- 


■2dz 
l  +  2dz  +  z2) 
=  (x-y)2-(d  +  z)2^ 
=  fa  ~  V)  ~(d  +  z)}{(x  -  y)  +  (d  +  2)} 
=  (x  —  y  -  d  —  z)  (x  —  y  +  d  +  2). 

23.   »2-y2+22-a2-2a;2  +  2ay 

=  (xi~2xz  +  z2)  -  (a2  -  2ay  +  y2) 
=  (x-zf-(a-y)2 
=  ](x-z)-(a-y)}{(x-z)+{a-y)} 
—  (a;  —  2  —  a  +  y)  {x  —  z  +  a  —  y). 


54  ALGEBRA. 


24.  2a6  +  a24-62-c2 

=  (a2  +  2a&  +  62)-c2 

=  (a  +  6)2  -  c2 

=  (a  +  6  -h  c)  (a  +  6  —  c). 

25.  2xy-x2-y2  +  a2  +  &2-2a& 

=  (a2  -  2a&  +  i2)  -  (x2  -  2xy  +  y2) 

=  l(a-4)  +  (x-y)}{(a-6)-(x-y)} 
«=  (a  —  b  +  x  —  y)  (a  —  6  —  x  +  y). 

26.   (ax  +  6y)2-l  27.   l-x2-y*  +  2xy 

=  (ax  +  by  +  l)(ax  +  by-l).  =  1  -  (x2  -  2xy  +  y2) 


=  l-(x-y)2 

=  (l+x-y)(l-x+y). 


28.  (5a-2)2-(a-4)2 

=  {(5a  -  2)  +  (a  -  4)}{(5a  -  2)-(a  -  4)} 

=  (5a  -  2  +  a  -  4)  (5a  -  2  -  a  +  4) 

=  (6a  -  6)  (4a  +  2)  =  12  (a- 1)  (2a  +  1). 

29.  a?-2ab  +  V-x2  32.   d2-x*  +  4xy  -4y* 
»(a-6)»-^  =  d2-(x2-4xy  +  4y2) 
=  (a-6+x)(a-6-x).  =d*-{x-2yf 

=  (d  +  x  —  2y)(d—x  +  2y). 

30.  (x  +  l^-ty+l)2  33.  c?-b*-2bc-<? 
«(s  +  l+y  +  l)(a>  +  l-y-l)  =  a2-(62  +  26c  +  c2) 
=  (x  +  y  +  2)(x-y).  =a2-(6  +  c)2 


=  (a  +  6  +  c)(a  —  b  —  c). 

31.  (x  +  l)2-^-!)2  34.  4x*-9x2  +  6x-l 

=  (x  +  H-y-l)(x  +  l-y+l)  =4s*-(9x2-  6x4-1) 

=  (x  +  y)(x-y  +  2).  =  4x*-(3x-l)2 

=  (2x2+3x-l)(2x2-3x+l). 

Exercise  XXXIX. 

1.   a3 -ft3  4.  f-\2b 

=  (a  -  6)(a2  +  a&  +  62).  =  (y  -  5)(y2  +  5y  +  25). 

2.'s»-8  5.  y»-216 

=  (x  -2)(x2  +  2*  +  4).  =  (y  -  6)(y2  +  6y  +  36). 

3.  x8 -343  6.   8x3-27y3 

-  (x  -  7)(x2  +  7x  +  49).  =  (2x-3y)(4x2  +  6xy +  9y2) 

7.  64y3  -  lOOOz3  =  (4y  -  10z)(16y2  +  40yz  +  100s2). 

8.  729X3  -  512y3  =  (9x  -  8y)(81x2  +  72xy  +  64y2). 

9.  27a3  -  1728  =  (3a  -  12)(9a2  +  36a  +  144). 

10.    1000a3-1331&3  =  (10a-ll&)(100a2-l-110a6  +  12U2). 


teachers'  edition.  55 


Exercise  XL. 

1.  x*  +  y*  4.   y:j  +  64z3 

=  (x  +  y)(a*-xy  +  y2).  =  (y  +  4*)(y2  -  4y*  -f  16*2). 

2.  ar»  +  8  5.   646s  +  125c8 

=  (a;  +  2)  (a2  -  2  x  +  4).  =(46+5  c)(16  &2-20  6c+25  c2). 

3.  a? +  216  6.   216  a8  +  512  c8 

=  (x  +  6)  (a?  -  6  a;  +  36).  =  (6  a+8  c)(36  a2-48  OC+64C2). 

7.  729a*  +  1728^  =  (9a;  +  12y)  (81  x2  -  lOSxy  +  144y2). 

8.  ar5  +  y5  =  (a;  +  y)(a?*  —  a^y  +  a^y2  —  xy3  +  y4). 

9.  a?7  +  y7  =  (a;  +  y) (a8  —  ar>y  +  ar4 y2  —  a^y8  +  a^y4  —  a^/6  +  y6). 

10.   3265  +  243c5  =  (26  +  3c)(16&4-  2463c  +  3662c2-546c8  +  81c4). 

Exercise  XLI. 

1.  a6  +  b«  -  (a2  +  62)  (a4  -  a2  62  +  ft4). 

2.  a10  +  /;10  =  (a2  +  i2)  (a8  -  a6&2  +  a464 -  a266  +  6s). 

3.  a;12  +  y12  6.  a12+l 

=  (ar4  +  y4) (a*  -  x*y*  +  y8).  =  (a4  +  l)(a8  -  a4  +  1). 

4.  ft6  +  64c6  7.  64a6  -h  a^ 

=  (&*  +  4c2)(&4-462c2  +  16c4).        "   =  (4a2+a?)(16a4-4a2a*+a*). 

5.  a*+l  8.   729  +  c6 

=  (as" +  !)(** -«■  +  !).  =(9  +  c»)(81  -9<?  +  *»). 


Exercise  XLII. 

1.  a4  +  a2 J2  +  &4  =  (a2  +  ab  +  &2)(a2  -  a&  +  &2). 

2.  9a4  +  3z2y2  +  4t^  =  (3s2  +  3a;y  +  2y2)(3z2  -  3a*/  +  2y2). 

3.  16a?4  -  17ar2y2  +  y4  =  (4ar*  +  3a;y  -  y2)(4a;2  -  3ay  -  y2). 

4.  81a4  +  23a2 52  +  1664  =  (9a2  +  lab  +  462)(9a2  -  7a6  +  462). 

5.  81a4-  28a2 62  +  16&4  =  (9a2  +  10a&  +  462)(9a2  -  10a&  +  462) 

6.  9a?*  +  38a?y2  +  49y4  =  (3a;2  +  2xy  +  7y2)(3a;2  -2xy  +  7y2). 

7.  25a4  -9a2 62  +  16M  =  (5a2  +  7a6  +  468)(5a2  -  7a&  +  462). 


56  ALGEBRA. 


8.  49m4  +  110m2«2  +  81  n*  =  (7m2  +  4mn  +  9n2)(7m2  - 4mn  +  9n2). 

9.  9a4  +  21  a2c2  +  25c4  =  (3a2  +  3ac  +  5c2)  (3a2  -  3ac  +  5c2). 

10.  49a4-15a262  +  121&4  =  (7a2  +  13a&  +  1162)(7a2-13a6  +  n&2). 

11.  64a?4  +  128sV  +  81/  =  (8a2  +  4ay  +  9y2)(8o2  -  4oy  +  9y2). 

12.  4a4  -  37*V  +  9^  =  (2a2  +  5oy  -  3y2)  (2a2  -  5oy  -  3y2). 

13.  25a*  -  41  x*y*  +  16y4  =  (5a2  +  zy-  4y2)  (5a2  -  ay  -  4y2> 

14.  81a4-34a2y2+y4«=(9a2  +  4ay-y2)(9a2-4ay-y2). 


Exercise  XLIII. 

1.  12a2  -  5a  -2  13.   6a2a2  +  aa-l 

=  (4a+l)(3a-2).  =  (2aa  +  l)(3aa- 1). 

2.  12a2-7a  +  l  14.   6&2-76a-3a2 

=  (3a-l)(4a-l).  =  (36 +a)(26-3a). 

3.  12a2  -  a-  1  15.   4a2  +  8a +  3 

=  (4a  +  l)(3a  -  1).  =  (2a  +  l)(2a  +  3). 

4.  3a2 -2a -5  16.   at-ax-Gx2 

=  (a  +  l)(3a  -  5).  =  (a  +  2a)(a  -  3a). 

5.  3a2  +  4a-4  17.   8a2  +  14a&- 1562 

=  (a  +  2)(3a  -  2).  =  (2a  +  5 6)(4a -  36). 

6.  6a2  +  5a -4  18.   6a2  -  19ac  +  10c2 

=  (3a  +  4)(2a  -  1).  =  (3a  -  2c)(2a-  5c). 

7.  4a2  +  13a +  3  19.  8a2  +  34ay  +  21  y2 

=  (4a  +  l)(a  +  3).  =  (4a  +  3y)(2a  +  7y). 

8.  4a2-flla-3  20.   8a2-  22ay-  21y2 

=  (a  +  3)(4a  -1).  =  (4a  +  3y)(2a  -  7y). 

9.  4a2 -4a -3  21.   6a2  +  19ay-7y2 

=  (2a  +  l)(2a-  3).  =  (2a  +  7y)(3a-y). 

10.  a2-3aa  +  2a2  22.   11a2-  23a6  +  2ft2 
=  (a-  a)(a-  2a).  =  (11a-  b)(a-2b). 

11.  12a4  +  a2a2-a4  23.  2c2  -  13ca*  +  6d2 
=  (3a2  +  a2)(4a2  -  a2)  =  (2c-  d)(c  -6d). 

12.  2a2  +  5ay  +  2y2  24.   6y2  +  7yz-3z2 

=  (2a  +  y)(a  +  2y).  =  (2y  +  3z)(3y-z). 


teachers'  edition.  57 


Exercise  XLIV. 

1.  a*  +  Zatb+Zdt>lT&  5.  «*-4a"  + 6a?-4*+ 1 
=  (a  +  6)3.  ={x-lf. 

2.  o8+3a»  +  3a+l  6.  a4-4a8c  +  6a2c2-4ac8  +  c4 
=  (a  +  l>».  -(a-c)». 

3.  a8-3a2  +  3a  -  1  7.  st?  +  2xy  +  y2  +  2xz  +  2yz+zi 
=  {a-lf.  =  (x+y  +  zf. 

4.  x4  +  4ar,y +6ic2y*  +  4a^s+3/4        8.  x*  —  2xy  +  y2  —  2xz  +  2yz+z* 
=  (*  +  y)4.  -{x-y-zf. 

9.   a2  +  8"  +  c2  +  2a&-2ac  -26c 

=  (a  +  b  -  c)\ 


Exercise  XLV. 

1.  2xi-5xy  +  2y*-17x+13y  +  21. 

2  a?  -  5  zy  +  2  yJ  =  (a;  -  2  y )  ( 2  a;  -  y ) , 
2a* -17a;  +21    -  (a;-  7) (2 x-  3), 
2y2  +  13y  +  21    =  (-y-3)(-  2y -7). 

a;  — 2y,   a;  — 7,      —  2y  —  7; 

2x-y,   2a;--3,  -y-3. 
(x-2y-7)(2a>-y-3). 

2.  6a?-37ary  +  6y2-5a;-5y-l. 

6a*-5a;-l  =  (6a;  +  l)(a;-l), 
6y2-5y-l  =  (6y  +  l)(y-l), 
6x*  -  Wxy  +  6y7  =  (6a:  -  y)  (z  -  6y). 

6a;  — y,   6a; +  1,       1  —  y; 

a;-6y,   -6y-l,  *-l. 
(6x-y +  l)(a;-6y-l). 

3.  6a?-5a#-6y2-a;-5y-l. 

6  a?  -  5xy  -  6y2  =  (2a;  -  3y)  (3a;  +  2y\ 
6a£-*-l-(3as  +  l)(2s-l), 
-6y2-5y-U(-3y-l)(2y  +  l).  ' 

2a;-3y,   2a;-l,    -3y-l; 
3a;  +  2y,   3a;  +  1,    2y  +  1. 
(2a;-3y-l)(3a;  +  2y  +  l). 


58  ALGEBRA. 

4.  5a?-8a^4-3y24-7a;-5y  +  2. 

5  a?  -  Sxy  +  3y2  =  (5a;  -  3y)  (a;  -  y), 
5a?  4- 7a;  4- 2  =  (5x4- 2)  (a; +1), 
3y2-5y4-2  =  (3y-2)(y-l). 

5a; -3y,    -3y4-2,   5x4-2; 

*-y»      -y  +  j>    *  +  i- 

(5x-3y4-2)(x-y  +  l). 

5.  2a?-ary-3y2-8aj  +  7y +  6. 

2x2  -  8x  -  3y2  =  (2x  -  3y)  (as  4-  y), 
2a?  -  8x  +  6  =  (2x  -  2)  (as  -  3), 
-3y*  +  7y  +  6  =  (-3y-2)(y-3), 

2x-3y,    2a; -2,    _3y-2; 
x4-y,        a?-3,     y-3. 
(2x-3y-2)(a;  +  y-3). 

6.  a?-25y2-10x-20y4-21. 

a?  -  10  a;  4-  21  =  (x  -  7)  (a?  -  3), 

-  25y2  -  20y  4-  21  =  (5y  -  3)  (-  by  -  7), 

a?-25y2  =  (x4-5y)(x-5y). 

x  —  7,   x  —  5y,    —  5y  — 7; 

x  —  3,   x4-  5y,   5y  —  3.  ' 
(x-5y-7)(x  +  5y-3). 

7.  2a?  —  5xy  +  2ys  —  xz  —  yz  —  z2. 

2  a?  -  5xy  +  2y2  =  (2x  -  y)  (a;  -  2y), 
2a:2  —  xz  —  z2  =  (2a:  4-  z)  (x  —  z), 
2y2  -yz  -z2  =  (2y  4-  z)  (y  -  z). 

2a;  — y,   2a;  +  z,   —  y+z; 

x  — 2y,   x  — z,      —  2y  — z. 
(2x-y  +  z){x-2y-z). 

8.  6x24-xy-yi-3xz4-6yz-9z2. 

6a?4-xy-y2  =  (3x-y)(2x4-y), 
6a?-3xz-9z2  =  (3x4-3z)(2x-3z), 
-y2-6yz  +  9z2  =  (-y  +  3z)(y-3z). 

3a; -y,   3x4-3z,   -y  +  3z; 

2x4-y,   2x  —  3z,   y— y  — 3z. 
(3x4-3z-y)(2x-3z4-y). 

9.  6a;2  —  7xy  +  y2  +  35 xz  — 5yz  — 6z2. 

6  a:2  —  7xy  +  y2  =  (6  a;  —  y)  (x  —  y), 
6a?  4-  35  xz  -  6z2  =  (6a;  -  z)  (a;  4-  62), 
y2  -  5yz  -  Gz2  =  (y  4-  z)  (y  -  6z). 

6a;  — y,   6a;  — z,   —y  —  z\ 
x  — y,      x  4-  6z,   —  y -h  6z. 
(6x-y-z)(x-y  4-  62). 


teachers'  edition.  59 


5x*-8xy  +  3y2  =  (5a;  -  3y)  (x  -  y), 
5x2-3a;z-2z2  =  (bx  +  2z)(x-z)t 
3y2  +  yz-2z2  =  (3y-2z)(y  +  z). 

5a: -3y,   5x  + 2z,    -  3y  +  2z  ; 

x  —  y,        x  —  z,        —y  —  z. 
(5x  —  3y  +  2z)  (x  —  y  —  z). 

11.   2x*-xy-3y2-5yz-2z2. 

2a?  -  2xy  -  3y2  =  (2a;  -  3y)  (a;  +  y), 
-  3y  -  byz  -  2z2  =  (-  3y  -  2z)  (y  +  z), 
2a;*  -  2z2  =  (2a;  -  2z)  (x  +  z). 

2x-3y,   -3y-2z,   2x-2z; 

x  +  y,        y  +  z,  x  +  z. 

(2a;  -  3y  -  2z)  (x  +  y  +  z). 

.12.   Ba^-lSxy +  6y2  +  12xz-13yz+6z2. 

6X2  -  13xy  +  6y2  =  (3x  -  2y)(2y  -  3y), 
6x*  +  12xz  +  6z2  =  (3x  +  3z)  (2a;  +  2z), 
6y*  -  13yz  +  6za  =  (3y  -  2z)  (2y  -  3z). 

3x-2y,   3x  +  3z,   -2y  +  3z; 

2x-3y,   2x+2z,    -3y  +  2z. 
(3  x  -  2y  +  3z)  (2a;  -  3y  +  2z). 

13.  x»—  2xy  +  y2  +  5x  —  by  14.   2x2  +  5xy-3y2-4xz  +  2yz 

=  (x2  -  2xy  +  f)  +  (5a;  -  5y)  =  (2a?+5a?y-3y2)-(4a;z-2yz) 

=  (x-y)2  +  5(x-y)  =  (2x-y)(x+3y)-2z(2x-y) 

=  (a;-y)(a;-y  +  5).  -  (xH-3y-2z)(2x-y). 


Exercise  XL VI. 

1.  5a?— 15a;— 20  4.   a2  +  2ax  H-x2  +  4a  +  4a; 

=  5(x2-3x-4)  =  fa2  +  2ox  +  x2)+(4a  +  4x) 

=  5(x  +  l)(x  —  4).  =  (a  +  x}2  +  4(a  +  x) 

=  (a  +  a;Ha  +  a;  +  4V 

2.  2x5-16x*  +  24xs  V       *a«  +  »  +  *;- 

=  2x*(x*  -8a;  +  12) 
=  2x3(x-2)(x-6). 

3.  ZtfV-Sab-W  =(a2-2a6  +  62)-c2 
=  3(a262  -  3ab  -  4)  =  (a  -  6)2-c2 

-  3  (aft  -  4)  (a&  +  1).  =  (a  -  6  +  c)  (a  -  6  -  c). 

6.  x2-2xy  +  y2-c2-{-2ca*-rf2 

=»  (a;2  -  2xy  +  y2)  -  (c2  -  2ca*  +  d2) 

-  {(*  -  y)  +  (« -  <*)}  {(*  -  v)  -i«  -  <*)> 

=  (a;  —  y  +  c  —  d)  (x  —  y  —  c  -f  a). 


60  ALGEBRA. 


7.  4-x2-2.is-.r4  13.   a? - x* - ab - br 

=  4-(x2  -f  2ar*  +  at)  =  (a2  -  x2)  -  (a6  +  6a;) 

=  4-(a;  4-  ar1)2  =  (a  4-  a:)  (a  -  a;)  -  6  (a  4-  a;) 

=  (2  +  x  +  a;2)  (2-x-x*).  =  (a  +  x)(a-x-  6). 

8.  a2-62-a-6  14.  a2 -2ax  +  x2 +  a-x 

-  (a2-  62)  -(a  +  6)  =  (a2  -  2a*  4-  x2)  +  (a-x) 

=  (a  +  6)  (a  -  6)  -  (a  4-  6)  =  f  a  -  a-)  (a  -  x)  +  1  (a  -  .> 

=  (a  4-  6)  (a  -  6  -  1).  =  (a  -  a;)  (a  -  x  +  1). 

9.  a*  +  a2  +  l  16.   3a*- 3y2- 2a;  4- 2y 


r) 


=  (a44-2a24-l)-a 

=  (a2  +  l)2  -  a2 

=  (a2  +  a  4- 1)  (a2  -  a  4-  1). 


=  (3a;- 3y2)- (2a; -2y) 
-a^-y2)-^-*) 
=  3(a;-t/)(x  +  y)-2(x-  y) 
=  (a;-y)(3a;4-3y-2). 


10.  a;2  —  y2  —  xz+yz 
=  (x2-y2)-(xz-yz)  16.   a^  +  ar»  4-  x2  4-  a; 

=  (a;  4-  y)  (x  - y)  -  z(x  - y)  =  x* (x  4- 1)  4-  x (x  +  1) 

=  (x  -y)(x  4-  y  -z).  =  (a?  +  a;) (a?  +  1) 

11       ^  w^t.  -*(x»  +  l)(*+l). 

11.  a6  —  ac  —  62  +  6c 

=  (ab  -  ac)  -  (62  -  be)  17.  a4*4  -  a8 a*  -  a2a?  4-  1 

=  a  (6  —  c)  —  b  (6  —  c)  =  a?.t?(ax—l)—(ax+l)(ax—l) 

-  (a  -  b)  (b  -  c).  =  (ax  -  1)  (a^ar1  -  aa;-  1). 

12.  3a;2  -  Sxz  -xy  +  yz  18.   3ar»-  2a:2y-  27a#2  +  18y* 
**(3x2-3xz)-(xy-yz)  =  a*(3a;-2y) -9y2(3a?-2y) 
=  Sx(x-z)~y(x-z)                       =  (x2  -  9y2)  (3a;  -  2y) 

=  (Zx-y)(x-z).  =  (a;-3y)(a?4-3y)(3a;-2y). 

19.  4a*-a;2  +  2a;-l 

=  4a^-(a;2-2x  +  l) 

=  4a*-(a;-l)2 

=  (2a?4-a;-l)(2a?-a;  +  l). 

=(2x-l)(z  +  l)(2x2-x+l). 

20.  a;6-*/8 

=  (a<+y»)(a^-y») 

=  (x  +  y)  (x2  -  xy  +  y2)  (x  -  y)  (ar»  +  a#  4-  y2). 

21.  a^  +  y8 

=  (x2+y2)(a^-a^y2+y*). 

22.  729  -a* 

=  (27  + a*)  (27 -a*) 

=  (3  4-  a;)(9  -  3  a;  +  a?)  (3  -  a;) (9  +3a;  +  a?). 

23.  x12y  +  yl*  25.  x2  4-  4a;  -21 

=  y(x12  +  y12)  =(a;  +  7)(ar-3). 

=  y  (a*  +  y*)(a?  -  a^y4  4-  y8). 

24.  c(a4-c*)  26.  3  a2  -21a6  4-  306* 

-  c  (a2  +  J)  (a2  -  c2)  -  3  (a2  -  7a6  4- 10  62) 
=  c(a24-c2)(a  \-  c)  (a  -  c).                 =  3  (a~26)  (a- 56V 


TEACHERS     EDITION.  61 


27.  2x*-4x*v-6x2y2  33.  x2-2xy-2xz  +  y2  +  2yz  +  z2. 
=  2a*  (2x*-  2xy  -  3y2)  ^-2^  +v2  =  (x-t/)(x-t/) 
=  2x*(x-3y)(x  +  y).  ^^M'-SK 

28.  4a2-4a6  +  62  *+2y. +*-&  +  .&  +  .). 

-(2a-w  *-*•  *"'»  -y-«; 

v         '  *-y,  *-*,  -y-«. 

29.  le^-SOw  +  lOOy1  (»-y-*)(*-y-«). 
=  4(4x2-20xy  +  &y2 

=  4(2x-5y)».  34.  ^-06-6^-40  +  126 

=  (a-a6-662)-4(a-36) 

30.  36a*x*y*-25b*x*yi  =  (o-36)(a  +  26)-4(a-36) 
=  aty»(36a1-25ftV)                      =»(o-36)(a+26-4). 

=  xV(6a  +  6fec)(6a  —  5bx). 

«-     ^  •  j.     ™     •        «^.  *     35.  x*  +  2xy+ y2-x-y- 6. 

31.  9x2y4-30xy1*  +  25z2  «  ,_  0  *     J     ,     ,   ' 
-dx^-hh                                   x2  +  2x  +  y2  =  (x+y)2, 
-(dxy*     5z).                                   x2-x-6t(x  +  2)fx-3), 

32.  i6x*-x  y2-y-6  =  (y  +  2)(y-3). 

=  x(16x*-l)  *  +  y>   *+2,   y  +  2; 

=  x(4x2  +  l)(4x2-l)  *  +  y.   «-3,  y-3. 

-*(4a»  +  l)(2»  +  l)(2*-l).  (x  +  y  +  2)(x+y-3). 

36.  (o  +  6)4-c* 

=  {(o  +  6)s  +  cn«a  +  6)»-c«} 

=■  ((<*  +  &r  +  07  («  +  &  +  c)(a  +  &  -  c) 

=  (a2  +  2a6  +  6*  +  c*)(o  +  b  +  c)(a  +  6  —  c). 

37.  x2-xy-6y2-4x  +  12y  38.   l-x  +  x«-xs 

=  (x»-xy-6y«)-4(x-3y)  -  (1  -art  +  x»(l  -x) 

=  (x  +  2y)(s-3y)-4(x-3y)  =  (1  -x)(l  +  X2). 

=  (x  +  2y-4)(x-3y).  39.  3x,_llxy  +  6y, 

=  (3x-2y)(x-3y). 

40.  x2  +  20x  +  91 

=  (x  +  7)(x  +  13). 

41.  (a;-y)(x2-z2)-(x-2)(x2-y«) 

-  (*  -y)(*  +  *)(*-  *)-(*-*)  (x  -y)(x  +  y) 
-(x-y)(x-z)(z-y). 

42.  x*-5x-24 

=  (x-8)(x+3). 

43.  (x«-y*-za)s-4y2z2 

=  {(**  ~  y2  -  **)  +  SyzMx2  -  y2  -  z2)  -  2yz} 
=  fo»  _y«  _2»  +  2yz)(x2  -  y2  _  z2  -  2yz) 

-  {«■  -  (y2  -  2yz  +  z2)}**2  -  (y2  +  2y*  +  *2)} 
-p-(y-*-(y  +  z)2} 

=  (x  +  y  -  z)  (x  -  y  +  z)  (a;  +  y  +  z)  (x  -  y  -  z). 


62 


ALGEBRA. 


44. 


48. 


49. 


jx3y2  +  5x2yz-  60 xz2 
=  5x(a?y*  4-  xyz—  I2z2) 
=  5a;  {xy  +  4z)(rry  —  3z). 


45.   Sxt-xt  +  Sx-l 

=  a»(3ar-l)  +  (3*-l) 
=  (a*  +  l)(3a;-l). 


46.  x2  —  2mx  +  m2  —  n2 

=  (a?  —  2mi  +  m2)  —  n2 

(a;  —  m)2  —  w2 

=  (a;  —  m  +  n)  (a;  —  m  —  ?i). 

47.  4a2&2-(a2  +  62-c2)2 

!2ab  +  (a2  +  62  -  c2)}{2a&  -  (a2  +  62  -  c2)} 
2a&  +  a2  +  62  -  c2}{2a&  -  a2  -  J2  +  c2} 
(a2  +  2a6  +  62)  -  c*}{e*  -  (a2  -  2ab  +  62)} 
(a +  6)" -<?}{(? -(a -6)*} 
a  +  6  +  c)(a  +  6  —  c)(c  +  a  —  b)(c  —  a  +  b). 


a7  -ha? 

=  a6(a2  +  l). 

l-14asa:  +  49a8a:2 
=  (l-7a*a>)2. 


50.  y2-4y-117 

=  (y-13)(y  +  9). 

51.  x«  +  6a;-135 

=  (a;+15)(a;-9). 


52.  4a2-12a6  +  962-4c2 

=  (4a2-12a6  +  962)-4c2 

=  (2a-36)2-4c2 

=  (2a -  36 +  2c)(2a- 36 -2c). 

53.  (a  +  3&)2-9(6-c)2 

=  {(a  +  36)  +  3(6-c)}{(a  +  36)-3(6-c)} 
=  (a  +  3&  +  3&-3c)(a  +  3&-36+3c) 
=  (a  +  6&-3c)(a+3c). 


54.  9x2-4y2  +  4yz-z2 

=  9x2-(4y2-4yz  +  z2) 

=  9x2-(2y-z)2 

=  (Sx  +  2y-z)(Sx-2y  +  z). 

55.  662x2-76a8-3x* 

=  x2(6&2-7&x-3x2) 
=  x2(3&  +  x)(2&-3x). 


56.  a*-&-Sab(a-b) 

=  a8-3a26  +  3a62-6» 
=  (a-6)8. 

57.  x*  +  y*  +  Zxy(x  +  y) 

=  x8  +  3x2y  +  3xy2  +  y2 
=  (x  +  y)8. 


58.    tf-6»-a(a»  — F)+6(a  — &)* 

=  a8-&8-a8  +  a^  +  a26-2a&2  +  &8 
=  a26  — afc2 
=  a&  (a  —  6). 


59.  9x*y2-Sxy*  —  6y* 
=  3y2(3x2  —  xy  —  2y2) 
=  3y2(x-2/)(3x  +  22/). 


60.  6x2  +  13xy  +  6y2 

=  (3x  +  2y)(2x  +  Sy). 


TEACHERS     EDITION. 


63 


61.  Ga262-a68-126* 

=  62(6a2-a6-1262) 
=  62(3a  +  46)(2a-36). 

62.  a2  +  2ad  +  <22-462  +  126c-9c2 
=  (a2  +  2ad  +  d*)  -  (462-  126c  T  9c*) 
=  a  +  d)2-(26-3c)2  V*. 
-  {(a  +  d)  +  (26  -  3c)}{(a  +  d)  -  (26  - 
=  (a  +  rf  +  26-3c)(a+d-26  +  3c) 
=  (a  +  26  -  3c  +  d)  (a  -  26  +  3c  +  d). 


-k 


3  c)} 


63.  x*-2xiy  +  4xy2-$yi  64.   4a8*2  -Sabx  +  362 

=  a?  (a;  -  2y)  +  4y2  (a; -  2y)  =  (2ax  -  b)(2ax  -  36). 

=  (a*  +  4y2)(z-2y). 

65.  18ar2-24zy  +  8y2  +  9a;-6y 

=  (18s2  -  24  ay  +  8^)  +  (9x  -  6y) 
=  2(9^  -  12xy  +  4y2)  +  3  (3a?  -  2y) 
=  2(3s-2y)24-3(3a;-2y) 
=  (6a;-4y  +  3)(3a:-2y). 

66.  2ar2  +  2sy-12y2  +  6sz  +  18yz 
=  2(3*  +  xy  -  6y2  +  3a»  +  9yz) 
=  2(a?*+  xy  -  6y2)  +  3z  (x  +  3y) 
=  2(x  +  3y)  (*  -  2y)  +  3z  (x  +  3y) 
=  2(a;+3y)(a;-2yH-3z). 

67.  (x  +  y)»-l-ary(a:  +  y  +  l) 

=  (x  +  y  +  l)(x  +  y  -  1)-  xy  (x  +  y  +  1) 
=  (x  +  y  +  1)  (x  +  y  -  ary  -  1). 
=  (x  +  y+l)(l-x)(y-l). 

68.  .r2-y2-z2  +  2yz  +  a?  +  y-z 

=  x2  —  (y2  —  2yz  +  z2)  +  x  +y  —  z 
=  x*-(y-z)2  +  (x  +  y-z) 
•■(x  +  y-z)(x-y  +  z)  +  (x  +  y- 
z)(x-i 


*(x  +  y- 


-y  +  z  +  1). 


2x*  +  ±xy  +  2y2  +  2aa;  +  2ay   72.   12  az2  -  14 axy  -6 ay: 


70. 


*  2(x2  +  2xy  +  y2)  +  2a(a;+y) 
=  2(x  +  y)2  +  2a(a;  +  y) 
=  2(x  +  y  +  a)(x  +  y). 

16a26+32a6c  +  126c2 
=  46(4a2+8ac  +  3c2) 
=  46(2a  +  3c)(2a  +  c). 


=  2a  (6a* -7 
=  2a(3a;  +  y) 


xy  —  3y2) 
(2*-3y). 


71.  m2»  —  w?q  —  ri*p  -\-n2q 
=  (m2 p  —  m2q)  — (n2p  —  n2  q) 
=  f(p-q)-n*(p-q) 
=  (m2-n2)(p-q) 
«=  (m  +  w)  (ra  —  n  J  ( p  —  q). 


73.  2x*  +  4x*-10x 

=  2x(x2  +  2x-35) 
-2*(*  +  7)(*-5). 

74.  16a3z-2aT* 

=  2z(8aJ-ss) 

=  2x(2a-x)(4a2  +  2az  +  arJ) 

75.  326ar»-46y» 
=  46(8ar8-y8) 

=  46(2a,--y)(4s2  +  2*y+y2) 


64  ALGEBRA. 


76.  a- 27** 

-3aO(l  +  3aj  +  9s2). 


-« (1-27  Xs) 

-*(i-r  *  A 


77.  a^-y12 

-  C^+y*)  (*4-aV+y*)(*+y)  (^-sy +y2)  (*  -y)  (*»+3y+y2). 

78.  49m2-  121  na  81.  a^-^  +  oj-l 

-  (7m  +  lln)  (7m  - 11  n).  -  (a8  -  a2)  +  (x  -  1) 

-rf(*-l)  +  (*-l) 

79.  16-81  y*  -(rf  +  l)(»-l). 
=  (4  +  9y2H4-9y2) 

=  (4  +  9y2)(2  +  3y)(2-3y).    gg.  ^  +  2*  +  1  _y* 

=  (z  +  2a:  +  l)-y2 

80.  12«*-22-6  -(s  +  ip-y* 

-(a?  +  l+y)(*+l-y). 


=  (3z2  +  2)(422-3) 

83.  49(a -6)2- 64 (m-n)* 
=  {7(a-6)  +  8(m-7i)} 
-  (7a  -  76  +  8m  -  8n)  (W -  76  -  8m  +  8n)V 

84.  1 


it*  l/y      Ut  ^7/»  ">f 

7 (a  -  6)  +  8  (m  -  n)}{7(a  -  6)  -  8(m  -  n)} 
w'      "      _8n)(7a-r      "       "  "   x 


V       2a6       J 

-fl  I  g'  +  ^-^Vi    a'  +  y-^N 

V  2a&       A  2a&       / 

=  /2a6  +  a2  +  62  -  <*\  (2ab  -  a*  -  6'  +  c*\ 
\  2ab  )\  2ab  J 

_  /(a2  +  2a6  +  62)  -  c*\  (<*  -  (a2  -  2a6  +  62)\ 

V  2a6  )\  2ab  ) 
_  ((a  +  6)«  -  <A  /V  -(a  -  6)*\ 

V  2a6        JV        2a6        ) 

-  /(«  +  6+c)(o  +  6-c)\  / (c  +  a-6)(c-a  +  6)\ 

V  2a6  JV  2a6  / 

86.  a^-53a;  +  360  87.   2a6-2&c-ae  +  c6  +  2&2-6« 
=  (x  -  8)  (x  -  45).  -  (2a6-26c+262)-(ac-c«  i-fte) 

=  26(a-c  +  6)-e(a-c  +  &) 
86.  x*-2x*y  +  z2-4.z  +  8y-4  =  (26-e)(a  +  6-c). 

-  (38-2z2y+z2)-(4z-8y+4) 

-  x^(*-2y+l>-4(*-2y+l)  88.   125a*  +  350r»y2  +  245  w4 

-  (a*-4)(jp-2y+l)  =  5x(25x*  +  70a*y2  +  49y*) 

-  (aM-2)(a-2)(»-2y+l).  =  5ar(5a^  +  7y2)2. 


TEACHERS*   EDITION.  65 

89.  a8  +  a5  6  +  a*62  +  a863  +  a264  +  aP 

=  a(ab  +  a46  +  a362  +  a2  6s  +  a&*  +  6s) 
=  a{a3(a2  +  ab  -k6*)  +  6s  (a2  +  a6  +  6*)} 
=  a(as  +  68)(a2  +  a6  +  62) 

-  a  (a  +  6)(a2  -  ab  +  68)(a2  +  a6  +  6»). 

90.  2a*x-2a3cx  +  2acix-2c4'x 
=  2a8a?(a  — c)  +  2c8  a;  (a  —  c) 
=  {2a*x  +  2(?x){a-  c) 

=  2a;(as  +  cs)(a-c) 

=  2x  (a  +  c)  (a2  —  ac  +  <?)(a  —  c). 

91.  6a?-5a#-6y2  +  3a:z+15yz-9z2. 

6a2  -  bxy  -  6y2  -  (3a?  +  2y)(2a;  -  3y), 
6a?  +  3a;z  -9Z2  -(3o?-3*)(2*  +  3«), 
-  6y2  +  15yz  -  9z2  =  (2y  -  3z)(-  3y  +  3z). 

3a;  +  2y,   3x-3z,   2y-3z; 

2x-3y,   2a;  +  3z,   _3y  +  3z. 
(3a:  +  2y  -  3z)(2s  -  3y  +  3z). 

92.  4z2-9zy  +  2y2-3a;z-yz  -  z2. 

4ar*  -  9a?y  +  2y2  -  (4a;  -  y)  (a?  -  2y), 
4a?  -  3a?z  -  z2  =  (4a;  +  z)  (a?  -  z), 
2y2-ys-z2  =  (-2y-z)(-y  +  z). 

4a;  — y,   4a;  +  z,   —  y  +  z;  s 

x  —  2yy  x  —  z,     —  2*y  —  z. 
(4a;-y  +  z)(x-2y-z). 

93.  3a* -7a6  + 26s +  500-560  +  2^. 

3a2-7a6  +  262  =  (3a-6)(a-26), 
3a2  +  5ac  +  2c2  -  (3a  +  2c)  (a  +  c), 
2^2  _  5bc  +  2ct  =  (_  26  +  c) (-  b  +  2e). 

3a-b.   3a  +  2c,   -6  +  2c; 

a  — 26,  a+  c,        -26  +c. 
(3a-6+2c)(a-26  +  c). 

94.  a^-2a?  +  a?-8a;+8  95.   5a?-8a^  +  3y2-5a;  +  3y 
=  **  -  2a?  +  a?  -  (8a;  -  8)  =  (pat-Sxy  +  3y2)-(5a;-3y) 

=  x*(x2-2x  +  l)-S{x-l)  =  (5a;-3y)(a;-y)-(5a;-3y) 

=  a?(a;-l)2-8(a;-l)  =  (5a;-3y)(a;-y- 1). 

=  (ar»^a?-8)(a;-l). 

96.  a2-2a^H-rf2-462  +  126c-9c« 

=  (a1  -  2ad+  d*)  -  (462  -  126c  +  9c2) 
=  (a-a7-(26-3c)2 

-  {(a  -  d)  +  (26  -  3c)}{(a  -  d)  -  (26  -  3c)} 
=  (a-d  +  26-3c)(a-d-26  +  3c) 

=  (a  +  26-3c-a*)(a-26  +  3c-d). 

97.  (a?-a;-6)(a?-a;-20) 
-(a;-3)(a;+2)(a;-5)(a:+4). 


66  ALGEBRA. 


Exercise  XLVII. 

1. 

18a62c2rf=32x2a62c*d, 

5.  a2-62  =  (a  +  6)(a-6), 

36a26cd2  =  32x22a26cd2. 

a8-68  =  (a-6)(a2+a6  +  62). 

.-.  H.C.F.  =  18  abed. 

.:  H.C.F.  =  a  -  6. 

2. 

Ytvej*  =  17  ©a2, 

6.  a2  —  a?  =  (a  +  a?)  (a  —  x), 
(a  —  3;)*=  (a  —  art  (a  —  x). 

.*.  H.C.F.  =  a  —  x. 

7.  c^  +  x8  *=(a  +  x)(a2— ax  +  x2) 
(a  +  x^^a  +  x)8. 

/.  H.C.F.  =  a  +  x. 

Zlfq  -  17  X  2/fy 

51pV  -  17  x  3oV. 

/.  H.C  J1.  -  17^. 

3. 

gxV^-^xxVz4, 

20x*yV  =  22x5xVz2. 

8.  9X2 -1    =  (3x  +  l)(3x-l) 
(3x  +  l)2=(3x  +  l)2. 

.-.  H.C.F.  =  4x2y2z2. 

.-.  H.C.F.  =  3x  +  l. 

4. 

30a; V  =  2  X  3  X  5xV, 

90*V-2  x32x5xV, 

9.  7x2-4x     =  x(7x-4), 

120arly*  =  2Jx3  x  5*V. 

7a2x-4a2=a2(7x-4). 

/.  H.C.F.  =  30arV. 

.\H.C.F.  =  7x-4. 

10.   12a8x2y-4a8xy2  =  4a8xy(3x-y), 

30a2ary  -  10a?x*if  =  10a2xy  (3x  -  y). 
.-.H.C.F.  =  2 


2a2xy(3x  — y). 


11.   8a362c-12a26c8  =  4a26c(2a6-3c2), 
6ab*c  +  4a68c2  =  2a63c (36  +  2c). 

.-.  H.C.F.  =  2a6c. 


-2z-3  =  (x-3)(x  +  l),    13.   2a8-2a62  =  2a(a  +  6)(a-6), 

-x-12   =  (x-3)(x  +  4).  46(a  +  6)2  =  46(a  +  6)(a  +  6). 

.-.  H.C.F.  =  x  -  3.  .'.  H.C.F.  =  2  (a  +  6). 

14.  12x8y(x-y)(x-3y)  =  22x3x8y(x-y)(x-3y), 
18x2(x-y)(3x-y)    =  2x32x2(x-y)(3x--y). 

/.  H.C.F. -^(x-y). 

15.  3x8  +  6x2-24x  =  3x(a,-2  +  2x-8) 

=  3x(x  +  4)(x-2), 

6a*-96x  =  6x(x*-ie) 

=  6x(x-4)(x  +  4). 

.-.H.C.F.  =  3x(x  +  4). 

16.  ac  (a  —  6)  (a  —  c)  =  ac  (a  —  b)  (a  —  c), 
6c  (6  -  a)  (b  -  c)  =  6c  (a  -  6}  (c  -  6). 

.-.  H.C.F.  =  c  (a -6). 


17. 


10x3y-60x2y2-f5xy8  =  5xy(2x2-12xy  +  y2), 
Sa^y^-Say-lOOy*  =  5y*(x2-xy-20y2) 
=  5y2(x-5)(x  +  4). 
.-.  H.C.F.  =  5y. 


TEACHERS     EDITION. 


67 


18.  x(x  +  lf  =  x(x  +  l)\ 
a?(a?-l)  =  a?(a;-fl)(a;-l), 
2a;(a?-a:-2)=  2a;(a;-2)(ar+l). 

.-.  H.C.F.  =  a;(a;+l). 

19.  3a?-6a:  +  3=3(a:-l)2, 
6a?  +  6a:-12  =  6(a;+2)(a;-l), 
12a?-12  =  12(a:-l). 

.-.  H.C.F.  =  3(a;-l). 


20. 


21.  x*-y2  =  (x  +  y)(x-y). 
(x  +  yf  =  (a;  +  yft 
x*+'dxy  +  2y*=:(x+y){x+2y). 

.-.  H.C.F.  =  a;  +  y. 

22.  ^-j£-(*  +  y)(*-y). 

x*  -  r  =  (x-y)(xi+xy+y2) , 
a?-7ay+6y2  =  (a?-y)(ar-6y). 

.-.  H.C.F.  =  x-y. 


6(a-6)*  =  6(a-W 
8(a8-&2)2  =  8(a  +  &)2(a-&)! 
10(a*-&*)=10(a2+&2)(a+&)(a-&). 
.-.H.C.F.  =  2  (a -6). 


23. 


a?-l  =  (s-l)(a;  +  l), 
a?-l  =  (a;-l)(a?  +  a:  +  l), 
a?  +  a;-2Ji(a;-l)(a:+2). 

.-.H.C.F.  =  a;-l. 


Exercise  XLVIII. 


1. 

5a?  +  4a;— 1 

5a?-    x 

5x-i 
5a; -1 

20  a? +  21  a; -5 
20a?  +  16a;-4 

4 

5x-l 

x  +  1 

1 

.-.  H.C. 

F.  =  5a;-1. 

2. 

2a?-4a?-13a:-7 

2a?  +  4a?  +    2a; 

-8a? -15a; -7 
-8a?-16a;-8 

6a? -11a? -37a: -20 
6a? -12a? -39a; -21 

3 

a?  +    2ar+    1 
a?  +      x 

2a;-8 
a;+  1 

a;  +  l 

a;+    1 
x  +    1 

.-.H.( 

3.F.«a>+l. 

3. 

a)6a*+25a3-21a2+4a 
6a3+25a2-21a+4 
6  a8—  5a2+     a 

2a)24a4+112as-94a2+18a 

Reserve  a. 

12as+  56a2-47a  +  9 
12as+  50a2-42a  +  8 

2 

30a2-22a  +4 
30a2-25a  +5 

6a2-  5a  +  1 
6a2-  2a 

a  +  5 
2a-l 

3a -1 

-  3a+  1 

-  3a  +  1 

.-.  H.C.F. . 

«a(3a-l). 

4. 

9a?  +  9a?  -4a;  -4 
9a?           -4a; 

9a?          -4 
9a?          -4 

45  a?  +  54  a?  -20  a:  -24 
45  a?  +  45  a? -20  a; -20 

5 

9a:2             -    4 

x  +  1 
1 

.-.  H.C.] 

F.  =  9a?-4. 

68 


ALGEBRA. 


5. 

3a?)27a?-3a?  +  6a?-3a? 
9s4-   x*  +  2x  -1 
9s4  +  6s8  +  3s 

_6a?-    a?-a;-l 

-6a?-4a?       -2 

3a?-a;  +  l 


6. 

10)  20  a? -60  a? +  50  a; -20 
2a?-  6a?  +  5a?-  2 
2a?-    6a?  +   4a; 

a;-    2 

*-   2 


6a;)162a?+48a?-18a?+6g 
27a?  +  8a?-3a;  +1 
27a?-3a?  +  6a?-3a; 


3a?  +  2a?       +1 
3a?-   a?  +  a; 


3a?-a;  +  l 
3a?-a?  +  l 


Reserve  3  x. 
Sx 

3a;-2 

x  +  l 


,H.C.F.  =  3a;(3a?-a;  +  l). 


4s)32a?-92a?+68a?-24a; 
8a?-23a?+17a?-  6 
8a?-24a?+20a;-  8 
a?-  3a;  +  2 
a?-  2a; 
-     x  +  2 
.-.  H.C.F. 


Reserve  2. 
4 

2x 
x-1 

=  2(a;-2). 


4a?-    8a; -5 

4a? -10a? 

2a; -5 
2a;-5 


12a?-   4a;- 65 
12a? -24a; -15 


10)20a?-50 
2a;-   5 


2a; +  1 
H.C.F.  =  2a;-5. 


8. 

q)3g8-5a*a;-2aa? 
3a2-5oa;  -  2a? 
3a*-6aa; 

ax  —    2a? 

ax  -   2a? 


g)9as  -  8a»  a; -20  oa? 

9a*-   8ax-20a? 

9a*-15aa;-   6a? 

7a;)7aa?-14a? 

a   -   2x 


Reserve  a. 
3 

3a  +  a? 


,  H.C.F.  =  a(a-2a?> 


10a?  +    a?- 
10a? -5a;2 


9a; +  24 
+  15 


3)6a?-9a;  +    9 
2a?-3a;  +    3 


20a?  -  17a?  +  48a; -3 
20  a?  +  2a?- 18a?  +  48  a; 


-2a?  + 

-2a?  + 


a?  -3 

3a?-3a; 


-2a?  +  3a;-3 
-2a?  +  3a;-3 


2x 

-5 

—  x  ■ 


H.C.F.  -2a?  -3a;  +  3. 


TEACHERS     EDITION. 


69 


10. 

2)8a?-4a?  -32s  -182 
4a? -2a? -16a;-  91 
4a? -2a? -42a; 

13)  26  s-    91 
2a;-     7 


3)36a?- 

-84a?-llla;-126 

12a?- 
12a?- 

-28a?-   37a;-  42 
-   6a?-   48a;-273 

-11)- 

-22a?  +    liar +  231 

2a?-        x-   21 
2a?-     7a; 

6a;-   21 
6a;-   21 

2x 
x  +  3 


H.C.F.-2a;-7. 


11. 

5a?(12a?  +  4a?  +  17a;-3) 
12a?  +  4a?  +  17a; -3 
12a?  +  4a?-      x 

3)  18a; -3 
6a;-l 


.H.C.F.  =  5a;(6a;-l). 


10a;(24a?-52a?  +  14a;-l) 
24a?-52a?  +  14a;-l 
24a?  +    8a?  +  34a;-6 


-5)-60a?-20a;  +  5 


12a?  +   4a;-l 
12a;2-   2x 


6a;-l 
6a;-l 


Reserve  5x. 
2 


x 

2x  +  1 


12. 

2y)  18a?y-18a;V-2ayy3-8,v* 


9a?-9a?y-xy2~  4^ 

9a?  -xy2-20yi 

y)-9a?y         +  16y3 

-9  a?  +16  y2 

-9a?+12a?y 

-12ay+16y2 
-12ajy+16y2 


xy)9x*y  —  a?^8  —  20ayy* 
9a?-      sy2-20y8 
9a?-16ary2 
5y*)15xy*-20yi 
3a;    -   4y 


Reserve  y. 


-3a;-4y 


•.H.C.F.  =  y(3a;-4y) 


13. 

6a?_      a;-15 

03?-10a; 

9a; -15 
9a;  -15 


9a?- 
2 

-3a;- 

20 

18a?- 
18a;2- 

-6a;- 
-3a;- 

40 
45 

" 

-3a;  + 

5 

■2a; -3 
H.C.F  =  3a;-5. 


70 


ALGEBRA. 


14. 


12a?- 
2 

-    9a?  + 

5a; +  2 

24  a?- 18a?  +  10a?  +  4 
24a?  +  10a?  +      x 

-28a?  + 
-6 

9a; +  4 

168  a?- 
168  a?  + 

54  a: -24 

70a?  +    7 

-31)- 

124  a; -31 

15. 

3)6a? 

+  15a?- 

4a;  +    1 
■6a?  +  9 

2a? 
11 

+    5a?- 

-2a?  +  3 

22a?  +  55  a? 
22  a?  +  56  a? 

-  22a?  +  33 
-30a; 

-a? 
-a?- 

+    8a? +  33 
-   3a? 

11a;  +  33 
11a; +  33 


16. 

4a?—    a?y  —    xy1  —  5y* 
4a?  +  4a?y  +  4a?y2 

—  5a?y  —  5a?y2  —  5y* 

—  5a?y  —  5a?y2  —  5y* 


17. 


2a8 
2 

-2a2- 

3a- 

2 

4a3 
4a3 

-4a2- 
+  5a2- 

6a- 
26  a 

-4 

-9a2  + 
4 

20a- 

-4    - 

-36  a2  h 
-36  a2- 

-80a-    16 
-45  a +  234 

125) 

125  a 

-250 

24a?  +  10a;  +  1 
24a?  +    6a; 


4a?  +  1 
4a; +  1 


a?+7 
6a?  +  l 


.-.  H.C.F.  =  4a?  +  l 


3)9a?  +  6a?-51a?  +  36 
3a?  +  2a? -17a; +  12 

2 

6a?  +    4a?- 34a;  +    24 
6a?  +  15a?-   6a;  +      9 

-  11a?- 28a;  +    15 

-  11a?  +  88  a;  +  363 
-116)-116a?-348 

x+     3 


Reserve  3. 


-2x 
11 


-a?  +  ll 
•.  H.C.F.  =  3(x  +  3). 


7.t?  +   4a?y  + 
4 

4a?y2- 

-  3y» 

28a?  +  16a?2y  + 
28a?-   7a?y- 

16  xy2  - 

7a?y2- 

-35y« 

23y)23a?y  + 

23a?y»- 

f23y3 

a?   + 

xy 

+     f 

3a3- 

-    a2 

-2a- 

■16 

2 

6a3- 

-2a2 

-4a- 

-32 

6a3- 

-6a2 

-9a- 

-   6 

4a2 

+  5a- 

-26 

4  a2 

-8a 

13a- 

-26 

13a- 

-26 

4a?~  5y 
.-.  H.  C.F.  =  a? +  a?y  +  y*. 


a 
-9 

4a +  13 


.-.  H.C.F.  =  a-2. 


TEACHER8     EDITION. 


71 


18. 

2)12ys  +  2_y2-94y-60 

2)48y3- 

-24y2- 

-348.y+   30 

Reserve  2. 

6y3+      y2_47y_30 

8 

24y3-12y2- 
24y3  +   4y2- 

-174y+    15 
-188y-120 

4 

48y*  +   8y2-376y-240 
48y»-42y2-405y 

-16y2  +    14y  +  135 
-16y2-   40y 

-3y-25 

50y2  +   29y-240 

54y  + 135 
54y  + 135 

-8y  +  27 

400y2  +  232y-1920 
*            400y*-350y-3375 

291)582y  +  1455 

2y  +       5 

.-.  H.C.F.  =  2(2y  + 5). 

19. 

9a;(2a?-6a?-a?  +  15a;-10) 

2a?-6a?-a?  +  15a;-10 

_9 

18a?-54a?-  9a?+135a;-90 
18a?-  2a?-45a?+     5a; 


2)-52a?+36a?+130a;-90 
-26a?+18a?+  65a;-45 
9 


-234a?+162  a?+585  a;-405 

-234a?+  26  a?+585  x-  65 

68)  136  a;2-  340 


2  a?- 
20. 

15a?  +  2a?- 75a?  +    5a;  +  2 
15a?  -  75a?  +  15a; 


6  a?  (4  a?+6  a?-4  a?-15  x-15) 
4a?  +  6a?-4a?-15a;-15 
4a?  -  12a?- 2a?  +  30  a;- 20 


18a?- 
18a? 


-2a?-45a;  + 
-45  a; 


-2a;2 
-2a? 


Reserve  3a;. 
2 

a?-13 
9aj-l 


H.C.F.  =  3a;(2a?-5) 


2a? 

2a? 


-lOaj  +  2 
-10a; +  2 


.-.  H.C.F.  =  a?-5a;  +  l. 

21. 

21a?-32a?-54a;-7 
5 

105  a?  -160  a?  -270a;  -35 
105a?  +    99a?  +    12a; 


35a?+  a?-175a?+30a;+  1 
3 


105a?+  3a?-525a?+90a;  +  3 
105a?+14a?-525a?+35a;  +14 


-ll)-lla? 


+55a;-ll 


-  5a;  +  1 


7 

15a; +  2 


-259  a?- 
5 


282a; -35 


-1295  a? -1410a; -175 
-1295  a? -1221a; -148 


-27)- 189a;-   27 


7a; + 
H.C.F.  =  7a;+l. 


21a?- 
21a?- 

-    4a?- 
-32a?- 

15a?  -  2x 
54a?  -7* 

28  a?  +  39a?  +  5a; 
3 

84  a? 
84  a? 

+  117a? 
-128  a? 

+    15  x 
-216a;- 

28 

7)  245  a;2 

+  231  x  +  28 

35  a? 
35  a? 

+    33a;  + 
+      5a; 

4 

28  a;  + 
28  a;  + 

4 
4 

3a;-37 
5a:  +  4 


72 


ALGEBRA. 


y)9a?y-22a?y3-3a;y4+10y& 
9a;4  -22a?y2-3xy3+10y4 

J 

18a?-44a?y2-     6ay»+2(y 
18a?-  9a?y2+105a?y8-69a?y 


y)69a?y-35a?y2-l  lla^^Oy4 


69a? 
2 


-35a?y  -llla#2+20y3 


138a?-  70a?y-222ay2+  40^ 

138a?-529a?y-  693^+805^ 

153y)  459a?y-153a3/2-765y» 

3a?  -      ay  —    5y2 

23. 

4a?  +  2a?-18a?  +  3a;-5 
4a?-8a?  +    2a?-2a; 

10a?  -  20  a?  +  5a; -5 
10a?  -  20  a?  +  5a; -5 


xy)  9xby-Qxiy2+xiya—25x^ 

9a?-0a?y+a?y2  -25y4 

9j4       -22afy2-3jy3+10y4 

-y)-  ear'y +23a?y2+3sy3-35y4 

6a?  -23a?y  -3ay»+35y" 

63^  -  2a?y  -10a?y* 

-7y)-21a?y  +7ay2+35y8 

3a?    —  xy  —  by* 

3a?    —  xy  —  5y2 


Res.y. 

1 


3x 
23 

2ar 


24. 

3a?-7oa?  +  3a2a;-2as 


2 

6a?- 
6a?- 

14aa?  + 
9oa?- 

6a2a;- 
6a2a; 

-4a3 

-a)- 

-5aa?  + 

12a2a;- 

-4a3 

5a?- 
2 

12az  +4  a2 

10a?- 
10a;  - 

24  ax  + 
\bax~ 

8  a2 
10  a2 

-9^ 

-9aa;  + 

18  a2 

x-    2a 


H.C.  F.  =  y  (3x8  -  xy  -  by*). 


6a?-4a;4-lla?-  3a?- 
2 


3a;-l 


12a?-8a?-22a?-  6a;2-  6a;-2 
12  a?+6  a?-54a?+  9a?-15a; 


-14a?+32a?-15a?+  9a;-2 
2 


-28a?+64a?-  30a?+18a;-  4 
-28a?-14a?+126a?-21a;+35 


39)78a?-156a?+39a;-39 


2a?-    4a?+    x-  1 


3a? 


-7 


2a;  +  5 


H.C.F.  =  2a?-4a?+a;-l 


a?—   ax3—   o¥-   a8a?— 2  a4 
3 


3  a?-3  aa?-3  oV-3  a9x-6  a* 
3a?-7aa?+3a2a?-2a8a: 


4oa?-6a2a?-  a8a?-6a4 
3 


12aa?-18a2a?-  3a8a;-18a4 
12aa?-28a2a?+12a3a;-  8a4 


5a2)10a2a?-15a3a;-10a4 
2a?-  3aa;-  2a2 
2a?-  4a.x 

ax—  2a2 
ax—  2a* 


4a 


3a; 

5 

2aj  +  a 


,  H.C.F.  =  a;-2a 


TEACHERS     EDITION. 


73 


Exercise  XLIX. 


1.  2x*+x-l  =  (x+l)(2x-l), 
x*+5x+4  =  (x+l)(x+4), 
a^+l  =  (a;+l)(ir2-a:+l). 

.%  H.C.F.=  x+1. 


2.  y8-y2-y+i  =  ;/i(y-i)-(y-l) 
=  (y2-i)(y~i), 

3y2-2y-l=(y-l)(3y+l), 

y'-y'+y-i  «  y*(y-i)+(y-i) 
=  (y2+i)(y-i). 

.-.  H.C.F.-  y-1. 


3.  3s -4a? +  9* -10 
x*-2x*  +  5x 
-2x2  +  4x-10 
-2x2  +  4x-10 


x*-2x  +  5 


.  H.C.F.=  x*-2x  +  5. 


Xs  +  2a;2-   3a; +  20 
x*-4x2  +    9a; -10 


6)6art-12x  +  30 
a?-    2x  +    5 

a^  +  Sx2-   9x  +  35 
x*-2x*  +    5x 


7x2-14x  +  35 
7x2-14x  +  35 


x-2 
x  +  7 


4.  Xs-   7x*-f    16x-12 

_7 

7xs-49xa  +  112x-84 
7x8-32x2  +    36x 

-17x»+    76x-84 

7 

-119x2  +  532x-588 

-119x*  +  544x-612 

-12)-    12x-f    24 

x-     2 


.%H.C.F.=  x-2. 

5.  ys_5y*  +  lly-15 
y3-    y2+    3y  +    5 
_4W4t/2  +    " 


-*)-4y2 


+    8y-20 
2y+    5 


3x»- 
3x»- 

-14x«  +  16x 
~21x2  +  48x-36 

5x»- 
5x»- 

7x2-32x  +  36 
7x2-14x 

-18x  +  36 
-18x  +  36 

10x*  +  7x-14 
-lOx2 

7x-14 
7x-14 

-2y  +  5 


,  H.C.F.-y2-2y  +  5. 


y8-    y*  +  3y  +  5 
y»-2jy2-f  5y 


y2  -  2y  +  5 
y'-2y  +  5 

2y»_7y2  +  16y 
2y8-4y2  +  10y 


-15 


-3y2  + 
-  3y*  + 


6y-15 
6y-15 


3 

x  — 17 
7x-18 

6x*  +  7 


1 
y  +  l 


2y-3 


74 


ALGEBRA. 


6.    2x2+3_-5  =  (2x  +  5)(x-l). 
3x2-x-2  =  (3x  +  2)(x-l), 
2x2+x-3  =  (2x  +  3)(x-l). 

.-.  H.C.F.  =  x~l. 


7.  x»-l 

X8  +  X2  +  X 


-a^-x-l 
-x*-x-l 


7?  +  X  +  1 


x*-x*-x-2 
x9               -1 

_aJ._a._l 

2x*-    .ra-    x- 
2x*+2x2  +  2x 

-3 

-3^-32!- 

-Sa^-Sx- 

-3 
-3 

3  +  1 


2x-3 


H.C.F.-^  +  x-fl 


8.   x»-3x  -2 

_2x2-4x-2 
_2x2-4x-2 


x2  +  2x  +  l 

x2-t-     x 

x  +  1 
x  +  1 


2xs  +  3x2 
2x* 


-1 
6x-4 


3)3x*  +  6x  +  3 
x2  +  2x  +  l 


x«  +  l 

x5  +  2x2  + 


-2a,-2-    x  +  1 

-2a?-4a?-2 

3)3x  +  3 

x+1 


x-2 


x-2 


x  +  1 
H.C.F.  =  x  +  l 


9.   12(*-y*)-12fa*  +  jfl(a*-y") 

=  12(xa  +  ya)(x  +  y)(x-y); 
10(a»  -  y«)  _  10(x»  +  y3)  (x8  -  y3) 

:  10(x  +  y)(xa  -  xy  +  y2)(x  -  y)(x*  +  xy  +  y2) 


8(x*y  +  xy4)  = 


SxyfaJ  +  y8) 

(*  +  y)(a 

\  H.C.F. 


8  xy  (x 


xy  +  y2). 

2(x  +  y). 


10.        a^  +  xy^x^  +  y3) 

=  x(x  +  y)(x*  —  xy  +  y2) 


x»y  +  y*  = 
a,4  +  x*y*  +  y*  = 


y(x  +  y)(x*-xy  + 
(x4  +  2x2y2  +  y*)--x2y2 


(aJ  +  yX^-xv  +  y2); 


ar'+y^-xy 

x2  +  xy  +  y2)  (x2  -  xy  +  y2). 

.-.  H.C.F.  -  x*-xy+y\ 


teachers'  edition.  75 


11.   2(x2y-xy2)  =  2xy(x-y)f 


A)ty  ~  *$  =  ^i*  +  y  )(*  -  y), 

i  ly - «yj)  =  4zy (s - y)(x*  +  ^  +  ^ 
5(afy  ~  sy5)  =  5xy  (x  +  y)(x  -  y) (x2  +  y«). 

.-.  H.C.F.  =  xy(x-y). 

Exercise  L. 

i.  4asx=22xa8xa:,  10.   x2-  1  =  (x  +  l)(x-  1) 
6a2x2  =  3X2xa2x*2,  *-*-*(*  + 1) 

2ax2  =  2XaXs2.  *■- l-(»-l)(a> +  *  + 1). 

.-.  L.C.M.  =  12  a8  x2.  .-.  L.C.M.  =  a?(a?  +  l)(x»-l) 

2.  18ax2  =  32x2xaxa:2,  n    2a +1-2/14-1 

W-8»X?X.xy,  4a.-l-(2a++i)(2a-l), 

12^  =  3  x  9  X  x  X  y.  8a»  +  j  .  (V2a+lX&'-2a+l). 

•••  LC.M.  =  72<w»y«.  ...  L.C.M.  =  (8a»+l)(2a-l). 

..L.C.M.- **(«  +  *).  .-.L.C.M.  =  («  +  &)"  (a -6). 

.•.L.aM.-.(,+i)(—i,.      Sii-StSii+Sa-.). 

5.  a2-&'  =  (a  +  &)(a-&),  .'.  L.C.M.  =  4(1  +  x)(l -x> 
a2  +  a&  =  a(a  +  6).                    ^  ^_1 

.•.L.C.M.-a(a+6)(a^).  x2  +  x +*1  =  a-2  +  s  +  1, 

6.  2x-l  =  2x-l,  a3- 1  =  (s- l^x2  +  x  +  1). 
4s2  -  1  =  (2s  +'l)(2x  -  1).  .\  L. C. M.  =  x8  -  1. 

.•.L.C.M.  =  (2x4-l)(2x-l).    15,  *>_y.  =  (iC  +  y)(iC__y), 

7.  a-f-6  =  a4-6,  (*  +  y)'  -  (*  +  y)a. 
a8+J3  =  (a  +  6)(a2-  a&  +  62).           (*  -  tf*  "  (*  ~  V)  • 

.-.  L.C.M.  =  (a+6)(a2-a&+&»).  •*•  L- C-  M.  -  (*  +  y)2(*  -  y)2. 

8.  x2  -  1  =  (X  +  l)(x  - 1),  16.  * \-  y* «  (*  +  y)(«  -  y), 
rf  +  i-Wi,        *  3(*-y)8  =  3(x-v)2, 

x*  -  1  =  (x2  +  l)(x  + 1)  (*  - 1).       12(x*4yj=  ^(x+yX^-xy+y2). 
.'.  L.C.M.  =  (x2+l)(x4-l)(x-l).        •'•  L.C.M.  =  12(x»+y')(x-y)2 

9.  x2-x  =  x(x-l),  17.   er^  +  xyJ^Sxfx  +  y), 


^-l-(*-l)(a-  +  af  +  l)f  8(xy-y2)  =  8v(a-y), 

xs+l  =  (x4-l)(x2-x+l).  10(x2-y2)=l0(x  +  y)(o;-y). 

.%L.aMf  =  x(x8  +  l)(r,-^J).     .-.  L.C.M.  =  120  xy(x  +  y)(x-y) 


76  ALGEBRA. 

18.  x»  +  5x  +  6  =  (*  +  3)(x  +  2),     20.  x'  +  llx  +  30  =  (x  +  6Kx+5), 

x*  +  6x  +  8  =  (x  +  2)(x  +  4).  x»-f  12x+35  =  (x+5)(x+  7). 

.-.  L.C.M.  =  (x  +  2)(x  +  3)(x  +  4).        .-.  L.C.M.  =  (x+5)(x+6)(x+7). 


19, 


>.   a*-a-20=-(a-5)(a+4),     21.  x»-9x-22  =  (x+2)(x-ll), 
a*  +  a  -  12  =  (a  +  4)(a  -  3).  x*  - 13*  +  22  =  (a- 2)  (x-11). 

.%  L.C.M.  =  (a-3)(a+4)(a-5).    .%  L.C.M.-(x  +  2)(x-2)  (x-11). 

22.  4a&(a2-3a&+2&*)=4a&(a-26)(a-6), 
5a*  (a2  +  a&  -  65*)  -  5a*  (a  +  3i)(a  -  25). 

.-.  L. C. M.  -  20a*6  (a -  b) (a  -  2&)  (a  +  36). 

23.  20(xa-l)  =  20(x  +  lXx-l), 

24x«-x-2)  =  24(x-2)(x  +  l),    26    (a„&)(a_c)==  (<*_&)(«- c), 

(6-a)(M  =  -(a-5)(6-c), 
.(c-a)(c-&)  =  (a-c)(6-c). 

L.C.M.  =  (a-b)(a-c)(b-cy 


XS~ X   i rfS>  i^X+ 1  '    26-  (a~5)(a-c)=(a-6)(a-c, 
16(x*+x-2) -  16(x+2)fx-l).  ?*-«)  (i-c)  -  -\a-b)Q>-c), 

.-.  L.C.M.  =  240  (x+1)  (x-l)(s+2)  (x-2).  (c  -  a)  (c  -  6)  =  (a  -  c)  (6  -  c). 

24.  12xy(xs-y*)=12xy(x+y)(x-y), 
2x*(x+y)*=2x»(x+y)(x+y), 
3y'(x-y)'=3y*(x-y)(x-y).      ^    rB»_4^+3x  =  x(x»-4x+3) 

.-.  L. C.  M.  -  12xy(x-y)*(x+y)*.  =  x(x-3)(x-l), 

x*+x»-12x*=  x*(x*+x-12) 

25.  (a  -  b)  (b  -  c)  -  (a  -  6)  (6  -c),  -  x*(x+4)  (x-3), 
(6_c)(c-a)=-(a-c)(5-c)f           x5+3x*-4xs=x3(x«+3x-4) 
v(c_a)(a_&) (a-i)(a-c).  =  x»(x-l)(x+4). 

.-.  L.C.M.  =  (a-&)(6-c)(c-a).   .-.  L.C.M.  =  xs(x-l)  (x-3)  (x+4). 


xty-xy*  =  xy(x~y\ 
3x(x-y)*  =  3x(x-y)*t 
4y(x--y)s  =  4y(x-y)8. 


.-.  L.C.M.  =  12xy(x-y)8. 


29. 


,.  (a  +  &)«  _  (c  +  d)2  =  (a  +  6  +  c  +  d)(a  +  b  -  c  -  a*), 
(a  +  c)»  _  (6  +  df  =  (a  +  b  +  c  +  d)(a-b  +  c-d\ 
(a  +  df-  (6  +  c)*  =  (a  +  i  +  c  +  d )  (a  -  &  -  c  +  d). 

.-.  L.C.M.  =  (a+b+c+d)(a+b-c-d)(a-b+c-d)(a-b-c+d). 


(2  x  -  4)  (3  x  -  6)  =  2  (x  -  2)  X  3  (x  -  2), 
(x-3)(4x-8)  =  (x-3)x4(x-2), 
(2x-  6)(5x-  10)  -  2(x-  3)  X  5(x-  2). 

.-.  L. C.  M.  =  60(x  -  2)*  (x  -  3). 


TEACHERS     EDITION. 


77 


Exercise  LI. 

1.  6x*-x-2  =  (3x-2)(2x  +  l), 

21  x*  -  17*  +  2  =  (3*  -  2)  (7x  -  1), 
14x*  +  5x  -  1  =  (2x  +  l)(7x  - 1). 

.-.  L.C.M.  =  (3x-2)(2s  +  l)(7x-:i) 

2.  ^-l-(*  +  l)(s-l), 

x2  +  2  x  -  3  -  (x  +  3)  (x  -  1), 
6x2-x-2  =  (3x-2)(2x  +  l). 

.-.  L.C.M.  -  (2x  + l)(3x -2)(x-  l)(x  +  l)(x  +  3). 

3.  xs-27  =  (x-3)(x*  +  3x  +  9), 
x*  -  15x  +  36  -  (x  -  3)(x  -  12), 

a?  _  3x2  _  2x  +  6  =  x2  (x-  3)-2(x -  3) 
=  (x2-2)(*-3). 
.-.  L.C.M.  =  (x -  3) (x -  12) (x» - 2)(x*  +  3x  +  9). 

4.  5x2  +  19x-4  =  (5x-l)(x  +  4), 
lOx2  +  13x  -  3  =  (5x  -  l)(2x  +  3). 

.-.  L.C.M.  =  (5x-i)(x  +  4)(2x  +  3). 

5.  12x*  +  xy-6y2  =  (4x  +  3y)(3x-2y), 
18x2+  18xy-20y*  =  2(3x-2y)(3x+5y). 

.-.  L.  C.  M.  =  2  (3 x -7  2y)  (3x  +  by)  (4x  +  3y). 


6.  x)x*-2x*  +  x 
x3-2x2  +  l 
x8-2x  -1 

~2)-2x2-f2x   +2 

X2—     X    —  1 


2)2x4-2x8-2x  -2 

X4—      Xs—      X  —  1 

x4-2x8+    x 

x8-2x  -1 

Xs—      X*—  X 

X2-     X  -1 

X*  —     X  —  1 


1 

x  +  1 


Hence, 
and 


x4  -  2x»  +  x  =  (xa  —  x  -  1) X  x(x  —  1), 
2x4-2x3-2x-2  =  (x2-x-l)x2(xa+l). 

.-.  L.C.M.  =  2x(x*-x-l)(xHl)(x-l). 


7.  12x2  +    2x-    4  =  (6x  +  4)(2x-l)  =  2(3x  +  2)(2x-l), 
12x2  - 42x  -  24  =  (6x  +  3)(2x  -  8)  =  6(2x  +  1)  (x  -  4), 
12x*  -  28x  -  24  =  (6x  +  4)(2x  -  6)  =  4(3x  +  2)  (x  -  3). 

.-.  L.C. M.  -  12(3x  +  2)(2x  -  l)(2x  +  l)(x  -  4)(x  -  3). 


78 


ALGEBRA. 


8.   a*  -  6a;2  +  11a? -6 
a?  —  5a?  +    6x 


-  a?+    5a;- 

—  ar2  +    5a;- 

-6 
-6 

X* 

a?2 

-5a? +  6 
-3a; 

-2a; +  6 
-2a; +  6 

x3 

X3 

-9a*  +  26a?- 
-6a?  +  lla?- 

-24 
-   6 

-3) 

a*- 

X3- 

-3a* -f  15a;- 

-18 

a*- 

-8a*  + 
-5a?  + 

-    5a?  +    6 

19a; -12 
6a; 

3a*  + 
3a*  + 

13a?- 
15a?- 

12 
18 

-21- 

2a? + 

6 

X  — 

3 

a;-l 


a?-3 


a;-2 


Hence,  x*-  6a*  +  11a;-  6  =  (a;-  1) (x  -2) (a? -3), 
a?  -  9a?  -f  26a;  -  24  =  (a;  -  2)(x  -  3)(a?  -  4), 
a*-8a?  +  19a?-12  =  (a;-l)(x-3)(x-4). 

.-.  L.C.M.=(a;-l)(a;-2)(a;-3)(a;-4). 


9.  x*-4a*-(x  +  2a)(x-2a\ 

x9  +  2ax*  +  4a2  a;  +  8  a8  =  x*(x  +  2a)  +  4a*  (a;  +  2a) 

=  (a?  +  4a2)(a;  +  2a)l 
a?  -  2oa,J  +  4a2 a;  -  8a5  =  x*(x -  2a)  +4a2 (a;  -  2a) 

-  (a?  +  4a2)  (a;  -2a). 

.-.  L.C.M.«(a;  +  2a)(a>-2a)(a*  +  4aI). 


10.  ar,  +  2a?y-a^2-2y8 
x*-2x2y-xy*  +  2y* 


=  a?(a;  +  2y)-y2(a;-f  2y) 

=  (^-y2)(a;  +  2y), 

=  a*(a;-2y)-y2(a?-2y) 

-(*-t/)(*-2y). 

.-.  L.C.M.  =  (a*-y*)(a?  +  2y)(*-2y). 


11.   l+p+p*=*l+p+p*,  12.   1- 

1  -p  +  p2  =  l-7>  +  7>2, 

l+^+p' = (1  +p+p>)  ( l-p+p*). 

.•.L.C.M.  =  l+jpI+/ 


(l-ar-tt-aVl-a), 
(l-a)8=(l-a)(l-a)(l-a). 

.-.  L.C.M.  =(1- a)8. 


13.  (a  +  c)J 


(a  +  c)2  -  b1  =  (a  4-  6  +  c)(a  -  b  +  c), 
(a  +  i}2  -  c2  =  (a  +  6  +  c)(a  +  6  -  c), 
(6  +  cf  -  a2«=  (a  +  b  -f  c)(-a  +  6  +  c). 


.-.  L.C.M.  =  (a  +  &  +  c)(a  +  &-c)(a-6  +  c)(-a+6  +  c) 


TEACHERS     EDITION. 


79 


14.  cH<*-ch/-Sci/* 
4c* -cy  -3y* 

3cs-3c2y  +  cy2-ys 

9 
36c2-   9ey-27y* 

12c3-12c*y+    4cy2-4y8 
12c8-    3c2:y-    9cy2 

3c 

36c2-52cy  +  16y* 

-V)~  9«2.V  +  13cy2  -  4y* 

4 

43_y)43cy-43.y2 
c   —      y 

9c2    -13cy  +  4y* 
9c*    -    9cy 

9c-4y 

-  4cy  +  4y* 

—  4cy  +  4y* 

Hence  4c8  — c*y— 3cy*  =  (c  — y)(4c*  +  3cy), 

3  c8-  3c*y  +  cy*-tf  =  (c-  y^c2  +  y2). 

.-.  L.C.M.  =  c(c-y)(4c  +  3y)(3c*  +  y2). 

15.  7n8-8m  +  3  =  (m+3)(m2-3m+l), 
m6  +  3m6  +  m  +  3  =  m5  (m  +  3)  +  (m  +  3) 

=  (m*  +  l)(m  +  3). 
.-.  L.  C.  M.  -  (m  +  3)(ro*  -  3m  +  l)(m*  +  1). 

16.  20n4  +  n*  - 1  =  (5n2  -  l)(4n*  +  1), 

25n*  +  5n8  -  n  -  1  =  (5n*  -  l)(5n2  +  n  +  1) 

.-.  L.C.M.  =  (5n2-  l)(4n*  +  l)(5n*  +  n  + 1). 


17.  46s- 

7 


126* +  96-1 


286s-   846*+    636-7 
286s- 1606* +  1326 


766*- 

7 


696-7 


6*-268  +  6*-86  +  8 
4 


4M_  86a+  45»_ 
46*-1263+  96*- 


326+32 
6 


468-,56*-316+32 
468-1262+  96-  1 


532  62-   4836-     49 
5326* -30406 +  2508, 


76*-406+33 
76*-  76 


2557)25576-2557j 
6-       1| 
Hence,  46s -126* +  96 

6* 


-336+33 
-336+33 


6  +  1 

46  +  76 
76-33 


l-(6-l)(46*-86  +  l), 
268  +  62  -  8  6  +  8  =  (6  -  l)(b* - 6* -  8). 


18. 

2r)2r5-8rA+12r8-8r2+2r 
r*-47*+  6r*-4r  +1 


L.C.M.  =  (6-l)(462-86  +  l)(68-6*-8). 


-  2r* 


+1 


-4r)-4r3+  8r*-4r 

r2-  2r  +1 


3 

r)3r*- 

-6r»  + 

3r 

r4- 
r4- 

-27^  +  1 

-27*+    ra 

27*- 

27*- 

37^  +  1 

4r*  +  2r 

r2-2r 
r*-2r 

+1 
+1 

Reserve  r. 

1 

r*  +  2r  +  l 


Hence,  2r>-87**  +  127* -8r*  +  2r=  2r(r-l)*, 
3r*-6r8  +  3r  =  3r(r*-l)2. 

.-.  L.C.M. 


=  «r(r-l)*(r+l)2. 


80 


ALGEBRA. 


Exercise  LII. 


1. 


a?»-l 


4a;(a?  +  1) 

4a;(a?+l) 
_a?-l 
4a? 


a?2-9a?  +  20 

a?2-7a?+12 

(a; -3)  (a? -4) 
%-3* 


3. 


a?-2x-3 
x1-  10a-  +  21 
_(*-3)(*  +  l) 

(*_7)(*-3) 
=  aM-J. 

a; -7" 


a?4  +  a;2  +  1 
x2  +  .T  +1 
_(x2  +  x  +  l)(x2-x  +  l) 

a^  +  x  +  l 
-aj»-a:  +  l. 


6. 


a»  +  l 


as  +  2a2  +  2a  +  l 
_(g  +  l)(aa-a  +  l) 

(a  +  l)(a2  +  a  +  l) 

a*  —  a  +  1 


a*  +  a  +  1 


5    x*  +  2x*y*  +  if 
x*-tf 
(x8  +  ^)(x»  +  .v») 
(^  +  ys)(^-y») 
^*»  +  .y» 
x'-y* 


a»-.a_20 
a2  +  a  -  12 
_(a-5)(a  +  4) 

(a_3)(a+4) 

a  — 5 
~a-3* 


8.  x*-4x*  +   9a;- 10 

x*  +  2x*-   3a? +  20 

-6)-fijg  +  12y-30 

x2  -   2a?  +    5 


a?8 
a? 

+  2x»- 
-2a?1 

-3*  +  20 
+  5a? 

4a?*- 
4a?2- 

-8a? +  20 
-8a?  +  20 

H.C.F.  =  arl-2a;  +  5. 

y»-4a?»+9a;-10  _  (x2  -  2x  +  5)(x~2)  _  x- 2 
xs  +  2x2-3x  +  20     (x2-2x  +  5)(x  +  4)     x  +  4* 


TEACHERS     EDITION. 


81 


9. 

a*-5o?  +  lla;-15 

x*-2x2  +    5x 
-3a;2  +    6a; -15 
-3a*  +    6a; -15 

x3-    x*+    3x+    5 
^-5a;2  +  lla;-15 

1 

4)4  a1-    8* +  20 
a;2-    2a;  +    5 

a;-3 

.-.  H.C.F.=  x*-2x  +  5. 

z3_5a?  +  lla;- 
a^  — ar,  +  3o?  +  « 

15_(a?-3)(xa-2a;  + 
3        (x  +  l)(x*-2x  + 

*)  =  • 
5)     . 

c-3. 
r  +  l' 

10. 

X4- 

X* 

-tfy-xyt-y* 

+  afy* 

-  x*y  —  x2y2  —  xy*  —  y4 

-x*y            -xy* 

-*Y         -y4 

—  x^y1           —  y4 

x*  +  x*y  +  xy*  —  y4 
x*  —  x*y  —  a^y8  —  y4 

1 

2xy)2x*y  +  2xyi 

X2 

-r 

.-.  H.C.F.=  a*  +  y2. 

.   a?4  +  x*y  +  ary8  —  y4  _ 
X*  —  afy  _  ay8  —  y4 

(x2  +  y2)  (x*  +  a?y  -  y2) . 

(«■  +  y2)  (*■  -  *v  -  y%) 

_x*  +  xy  —  y2 

«*  -  ^y  -  y* 

11. 

o8-3a  +  2 
4 

a*  +    4a2—   5 
a8-   3a  +    2 
4a2  +    3a  -    7 
-3 

i 

4a3            -12a +  8 
4a8  +  3a2-    7a 
-3a2-    5a  +  8 
-3a2  +    3a 

-  8a  +  8 

-  8a  +  8 

a 

-12a2-    9a  +21 
-12a2 -20a  +32 

4 

ll)lla  -11 
a  -    1 

-3a 

.\H.C.F.  =  a-l. 

-8 

.   a3  +  4a2-5_(a 

-l)(a2+5a  +  5)__a2 

+  5a 

+  5 

"a8 -3a +  2        (a 

&-l)(a2  +  a-2)        a 

■  +  a 

-2 

12. 

x*  +  x*  —  x  —  1 
3 

3a;2  +  2a;-l 
3o?-6a;-9 

2C 

3 

3x*  +  3x*-$x-'S 
3a?  +  2a;2-    x 

x*-2x-S 

X2  +      X 

8)8a;  +  8 
a?  +  l 

ar-3 

-3a;-3 
-3a?-3 

.\  H.C.F.=  o;  + 

1. 

.     3a;2  +  2x-l 

_(*  +  l)(3a>-l)_3 

r-1 

a^ +««-*_  1      (i  +  l)(^_l)       a^-1 


82 


ALGEBRA. 


13.  a8-    a2 -2a +  2 

a8-3a2  +  2a 

2a*-4a  +  2 

2a?-6a  +  4 

2)2a-2 

a-1 

.-.  H.C.F.=  a-l. 
'.   a8-3a*  +  4a-2 

a8-a2-2a  +  2  " 


a3-3a2  +  4a-2 

a3-    a?-2a  +  2 

-2)-2a2  +  6a-4 

a2-3a  +  2 

a2—    a    . 

-2a? +  2 

-2a +  2 


x-2 


(a  -  Vjjx2  -  2x  +  2)  _  a*  -  2x  +  2 
(a  _!)(**_  2)  a* -2 


14. 


4a*-12aa  +  9a* 


8a8- 
(2  a 


27  a8 
3a)(2a?-3a) 


16. 


(2a-3a)(4a*  +  6aa+.9a2) 
,       2a-3a 
"4a2  +  6aa  +  9a2* 


a*-bi-2bc-ci 
aa  +  2ai  +  &2-c* 
_  q«-(6'  +  25c+c*) 

(a2  +  2aZ>  +  i2)-c2 
_  (a  +  &  +  c)(a— 5  — c) 

(a  +  i  +  c)(a  +  6  -c) 

a  — ft  — c 

a  +  6  — c 


15. 


15a2  +  a&-2&» 
9a*  +  3a&-2i2 

(5a  +  26)(3a-^ 

(3a  +  26)(3a-W 

5a  +  2& 

3a +  26 


17. 


x*-x2-2x  +  2 
2x*-x-\ 

(g-l)(z8  +  j8-2) 
=  (a-l)(2a8  +  2a  +  l) 

x*  +  x*-2 
=  2a*-f  2a  +  l 


18. 


ar"-2x2-    a  +  2 
a2-3a*  +  2a 


-3a  +  2 
-3a  +  2 


ar,-6x*  +  lla;-6 

a8 -2a*-      x  +  2 

-4)-4a*-f  12a-8 


x1-    3a  +  2 
H.C.F.  =  a*-3a  +  2. 
x9  -  6sa  +  11a  -  6  _  (a2  -  3s  +  2)(x  -  3) , 
a8  -  2**  -  x  +  2        (a2  -  3*  +  2){x  +  1) " 


x  +  1 


a-3 

a  +  1 


19.   6x8-17a8  +  lla;-2 
6a?8—    5a2  +      a; 


-12a2*  10a 
-  12a2  +  10a; 


6x*  -  23a;8  +  16a;  -3 

6x*-17x*  +  11a; -2 

-1)-    6x2+    5a; -1 


a-2 


6a2-    5a  +  l 
Qx3 -  23 a*  +  16a  - 3  _  (6a2  - 5a  +  l)(a - 3)      a-3 
'  Sa8-17aJ2  +  lla-2     (6a2- 5a+ l)(a-2)=Sa-2 


teachers'  edition. 


83 


20. 


-x*  —  x  +  l 


a*-2x*-x2-2x  +  l 
s»(g-l)-(s-l) 

W-D(x-l) 
(^  +  *  +  l)(as"-3a  +  l) 
_(g-l)(a'  +  a?  +  l)(g-l) 
=3(a>2-3a;  +  l)(:B2  +  a;  +  l) 

a*-3x  +  l 


21.  tt)q4-q8ft-a2ft2  +  a&3 
a*-a*b-ab*  +    ft3 


a5-a4ft 
a5-a4ft 


-  aft*  +  ft* 

-  a8ft2  +  a26» 


a* +  6* 


a8ft2-a2ft8-aft*  +  ft* 
a*b2-a2b*-ab*  +  & 

.  H.C.F.  =  a3-a2ft-aft2  +  ft3. 

a*ft  __  a»*  +  fts       (fli  +  y)  (as  __  a*h  _  ah*  +  y  ^  aa  +  52 


'  a4_  aSJ  _  a2&3  +  aft3  a(aS  _  a2b  _  ab2  +  JS) 


22. 


(a  +  &)2 


24. 


a2 -aft -2ft2 
_  (a  +  b)(a  +  b) 

(a-2ft)(a  +  ft) 
=  a  +  b 

a  -2b 

3  aft  (a2 -ft2) 

4(a2ft-aft2)2 

3aft(a4-ft)(a-&) 
4a2ft2(a-ft)(a-ft) 
3(a+ft) 

=  4aft(a-ft)' 

a2  +  2aft  +  ft2-c2 
a2  +  ab  —  ac 
(a2  +  2aft  +  62)-c2 

a2  -f  aft  —  ac 
(a  +  b  +  c){a  +  b-c) 
a(a  +  b  —  c) 
^a  +  b  +  c 
a 


25. 


26. 


27. 


<$x*-\\x2y  +  Zxy2 
§x2y-bxy2-§tf 

xjGx^-Uxy  +  Sy2) 

yijM-bxy-Qy2) 
=  ar(2ar-3y)(3g-y) 

y(2a-3y)(3a?  +  2y) 
a?(3a;  —  y) 

y(3s+2y)' 

o«  -  (ft  +  c  +  J)» 
(a_ft)a_(c  +  d)2 

(a  +  ft  +  c  +  a')(a  — ft  — < 


^) 


(a-5  +  c  +  d)(a- 

a  — 6  -f  c  +  d 
6a?-5s-6 
8z2-2a?-15 
_(3a?  +  2)(2g-3) 

(4»  +  6)(2s-3) 

3s +  2 
^40; +  5 


-b-c-d) 


84 


ALGEBRA. 


28.     s*  +  x  V  +  .V4 
'  (x-y){?*-y*) 
_(a?  +  xy  +  y*)(x*  -  xy  +  i/*) 
{x-y){x-y)(x*  +  xy  +y*) 
_  g*  —  xy  +  y* 

(*-yy  ' 


x6  +  y* 
x*  —  afy2  +  y* 
_(s2  +  .yW-a?y2  +  y*) 

a?*  —  afy*  +  y4 
=  a*+y2. 


'   (a8  -  #»)  (a2  -  afc  +  &*) 
_  (a+&)(q2-a&+&a)(q»+a&+&2) 

q  +  ft 


Exercise  LIII. 


,       3?-2a;  +  l        m        t 

1. — =*a  —  l. 

a  — 1 

2.  3a?  +    2a?+    l|«  +  4 


3a2  +  12a; 


3z-10 


-\Ox+    1 
-10a; -40 


,.^il=Lio+iL 

a;  +  4  <r+4 


3.   3a?  +    6a;  +    5 
3a^  +  12a; 


a;  +  4 


3a;-6 


-6a;  +    5 

-6a?-24 

+  29 

3a;2  +  6a;  +  5    ~        .       29 

=  da;  — bH -- 

a;  +  4  x+4 


5.  2a?  +5 

2a?-6a; 


5-3 


2a; +  6 


6x  +   5 

6a; -18 

+  23 

2*  +  5   >2a;  +  6+-23 


a;-3 


a;-3 


6.   10a2-17aa;  +  10a?|5q-3; 
10a2-   2ax  |2a-3a; 

-15aa;  +  10a? 
—  15oa;  +    3a? 

+  7a? 

lo^-nos+io*2  2a_Sx+JL*_. 

5a  — x  5a— x 


4.   a2 


aa;  +    a? 


a2  +    ax 


-2x 


-2aa;+    a;2 

-2oa;-2a? 

+  3a? 

a2  —  ax  +  x1  ~         3t* 
--        -a-2a;+- 


a  +  x 


a  +  a; 


7.   48  a? 
48  a?- 


+  16]4a;-l 


12a; 


112x4-3 


12a; +  16 

12a;-   3 

+  19 

4a;  — 1  4x— 1 


8.   2s2 -5s-    2 
2s2-8s 


teachers'  edition.  85 


s -4  9.  a2  +  &2ja-5 


2s  +  3  a2  -  ab         \a  +  b 
3s-    2  a&  +  62 

3s  -12  a&-&2 

+  10  +26* 


s  — 4  s— 4  a  — &  a  — 6 

10.  5s8-  s2        +  5|5s2  +  4s-l 
5s8+4s2—  s       |ag  — 1 
— 5s2+   s  +  5 
— 5  s2— 4s +  1 
+5s  +  4 
.    5s8-s2  +  5_/w   _j        5s +  4 


5sa  +  4s-l  5s2+4s-l 

Exercise  LIV. 


1.  l-*^2  4.   a-x  + 


a?  +  x* 


x  +  y  a  —  s 

_  s  +  y  -  (s  -  y)  _  a2  -  2ax  +  s2  +  (a2  +  Xs) 

x + y  a— s 

=  s+j/j- sj-^  _  2(o2  —  ax  +  x2) 

x +y  a— x 

^y  5.  5a-26-3a'-4iy 

5a  —  6o 

a._  _  25a2-40a&+12&2-(3o2-4y) 

2'  1+rTt  5a-66 


_22a2-40a5  +  16&2 
5a-66 


a  +  y 

=  xjLy^+j[s-1y) 

*  +  y 

=  s_+y_+s-^  fi    a  +  ^al+62 

a+y  a+6 

=   2x  ■  _a2  +  2a5  +  52-(a2  +  &2) 
*+?  a  +  6 

s 
3s2-(l+2s2)  7     7g     2-3a  +  4a2 

s  5  —  6a 

_3s2-l-2s2  ^  35a  -42a2  -  (2 -3a  +  4a2) 

s  5  — 6a 

_s*-l  _  38a -46a2 -2 

s    '  ""         5-6a 


86 


ALGEBRA. 


2a  14.  .  +  «-&=i? 

_6az-(5as-3)  s»  +  2s-15-  2x  +  15 

2a  "" 

_gg  +  3  jpj 


2a 


*  +  y 


x-3 


a?-3 


a-±*  +  l  15.  2a-b-™L 
a  —  b  a  +  b 

_a  +  b  +  (a-b)  _2a*  +  ab-b*-2ab 

a-b  <*  +  & 

__    2a  2a»-a&-6« 


a—b  a+b 

10.  a-ft     i  le-  3a?-10  +    41 

a  +  6 
_  a  -  &  -  (a  +  b) 

a  +  6 
_-26 
a  +  6 


s  +  4 
_3s*  +  2x-40  +  41 

17. 

»  +  4 
_3ar*  +  2a;  +  l 

»  +  4 

x*  +  a.  +  i+_L^ 
a  — i 

_ar»-l  +  2 

_2x*~(x*  +  2xy+f)  x-1 


*»+! 


_  g*  —  2sy  --  y»  «  — 1 

*  +  *  18.  **-Zx-3x(*-*) 

x-2 

12.  5a~12?  +  6a  +  3s  ^  s*-2s»-3s»+6s-9s+3s» 

x  —  2 
_  5a-  12a?  +  24a  +  12s  x*-2x*-3x 


4  x-2 

,29a 

4 


_g(s»-2ar»-3) 
s-2 


13.  a-l+_L-  1Q    ^.o^j.a-j       6a?» 


"^+\  19-  a2-2aa;+4a?»- 


,      ,      ,  .  a  +  2a? 

-  <*'  -  1  +  1  _as  +  8ar»-6ar» 

«  +  l  a  +  2» 

_    <*'  _a'  +  2a» 

a  +  1  "  a +  2*' 


TEACHERS     EDITION. 


87 


20.   x-g  +  y  +  o'-W+y' 
x  +  a 

_x2  —  a2  +  xy  +  ay  +  a*  —  ay  +  y* 

x  +  a 
_  x9  4-  xy  +  y2 
a;4-  a 


3a;-7    4s-9 


Exercise  LV. 


6  18 

L.C.D.  =  18. 

The  multipliers  are  3  and  1 
respectively. 

3s-7_9s-21. 
6  18      * 

4x-9     Ax-9 


18 


18 


4a  — 5c    3a  — 2c 

5ac    '      12a2c 

L.C.D.  =  60a2c. 

The  multipliers  are  12  a  and  5 
respectively. 

4a- 5c  _  48a2- 60 oc, 

5ac 
3a-2c 


12  a2  c 


60a2c 
15a -10c 
60a2c 


2    2g~4y    Sx-Sy 
5a?     '        10a; 

L.C.D.  =  10a* 

The  multipliers  are  2  and  x 
respectively. 

2x  —  Ay  =  4a?  — 8y  . 

5s*  10z* 

3s-8y  _3g»-8apy 

10a;  10a;2 


6 


1-a;     1-a* 

L.C.D.  =  l-a^. 

The  multipliers  are  l+a?and  1 

5     _5+5a;. 
l_a.      I-**1 

6 6_ 

l_ar»     i-aJ* 


5. 


1 


1 


(a-b){b-c)    (a-b)(a-c) 
L.C.D.  =  (a-6)(a-c)(6-c). 
The  multipliers  are  a  —  c  and  b  —  c. 

1 a  — c # 

(a_6)(6_c)     (a-6)(a-c)(6-c); 

1  __  b-c 

(a-b)(a-c)     (a-b)(a-c)(b-c) 


88  ALGEBRA. 


4a?              xy  7    8  x  4-2     2a?-l      3a;  4-2 

3{a  +  b)     6(a'-&2)'  '    x-2*    3x-6     5a;- 10' 

L.C.D.  -  6(a2-&»).  L.C.  D.  -  15(a;  -  2). 

The  multipliers  are  2(a—b)  The  multipliers    are    15,   5 

and  1.  and  3. 


4a*         8a?(a-fc). 
3(a  +  6)      6(a*-&2)' 
ay                 ay 

8a;  +  2_  30(4 x  +  1) 
a; -2        15(a?-2) 
2x-l      5(2a;-l). 

6(a»-62)     6(a*-&») 

3a;-6      15(a;-2)' 
3a; +  2  _3(3a;  +  2) 
5a; -10      15(a?-2) 

8    a~6m     1 

c  —  bn 

mx 

nx 

L.C.  D.  «=mnaj. 

The  multipliers  are  n,  mruc,  and  m. 

a  —  bm     an 

—  bmn. 

Tax 

mnx 

-      mnx  m 
mnx 
c  —  bn     cm 

—  bmn 

Exercise  LVI. 

1    Sx-2y     5a? -7y     8a?  +  2y 
5a;  10*  25 

L.C.D.  =  50  a;. 
The  multipliers  are  10,  5,  and  2  a?. 

30a;  —  20  y  =  first  numerator, 

25  a;  —  35  y  =  second  numerator, 

16a? +4ay =  third  numerator. 

16a?2  +  55a;  +  4a?y  —  55 y  =  sum  of  numerators. 

.-.  Sum  of  fractions  =  16**  +  55s  +  4sy-55y, 

50  a; 


teachers'  edition.  89 

4a? -7y2     3x-8y     5  -  2y 
3a?  6x  12 

L.C.D.  =  12x2. 

The  multipliers  are  4,  2  a;,  and  a?. 

16  x2                           —  28  y2  =  first  numerator, 
6x*               —  16  xy  =  second  numerator, 

5a?  — 2a?y =  third  numerator. 

27a?  —  2x2y  —  16xy  —  28y2  =  sum  of  numerators. 

.  Sum  of  fractions  =  27,^-2^-16^-28^ 

12a? 

4a*  +  562  (  3a  +  26  >  7-2o 


26*  56  9 

L.C.D.  =  9062. 

The  multipliers  are  45,  186,  and  106*. 
180a2  +  225 62  =  first  numerator, 

366s  +  54  a6  =  second  numerator, 

706^ -20o62  =  third  numerator. 

180a2  +  331 6*  +  54a6  -  20a62  =  sum  of  numerators. 

,.  Sum  offractions^180^  +  54^-20^+331^ 

90  62 

^  4x  +  5     3x-7        9 
3  5x         12a;2 

L.C.D.  =  60a?. 

The  multipliers  are  20  a?,  12  x,  and  5. 
80  a?  +  100  a?  =  first  numerator, 

—    36  a?  +  84  x  =  second  numerator, 

45  =  third  numerator. 

80a?  +    64a?  +  84  x  +  45  =  sum  of  numerators. 

...  Sum  of  fractions  =  80x»  +  64a?  +  84x  +  45, 

60  a? 

5    4x  —  3y     3x+7.y      5a;  -  2y      9x  +  2y 
7  14  21  42 

L.C.D.  =  42. 

The  multipliers  are  6,  3,  2,  and  1. 
24  x  —  18  y  =  first  numerator, 
9x  -f  21  y  =  second  numerator, 
— 10 x  -f    4y  =  third  numerator, 
9x  +    2y  =  fourth  numerator. 
32 x  +    9y  =  sum  of  numerators. 

.-.  Sum  of  fractions  -  32x  +  9V. 
42 


90  ALGEBRA. 


6.   3*y-4      5y»  +  7      6s* -11 
*V  ay8  ar*y 

3  a?  y2  —  4  xy  =  first  numerator, 

—  5  a?  y2  — 7a^  =  second  numerator, 

—  Qx2?/3 -flly2  =  third  numerator. 

—  Sa^y2  —  4ary  +  lly2  —  7a2  =  sum  of  numerators. 

. ..  Sum  of  fractions  «  11^-4^-8^-7*. 

a*y> 


7    g2-2gc  +  c2     y-2ftc  +  c? 
gV  frV 

L.C.D.-a»ftV. 

a262  —  2ab2c  +  62  c2  =  first  numerator, 

— of6f -f2a2frc         — g2^  =  second  numerator, 

—  2  g£>2  c+2  a2  6c+62  c2— g2  c2  =  sum  of  numerators. 

...  Sum  of  fractions  =  ** -2aVc  +  2a*bc -a>* 

a2b2<* 


8    5g8-2     3a2-a 


8a2  8 

L.C.D.  =  8a2. 

5  a*  —  2  =  first  numerator, 

g8  —  3  g4        =  second  numerator. 
6gs  —  3  g4  —  2  =  sum  of  numerators. 
6a3-3g4-2 


Sum  of  fractions  = 


8g2 


a  —  b      b  —  c     c  —  a     ab2  +  6c2  +  ca2 


cab  abc 

L.C.D.  =  g6c. 

g2  b  —  ab2  =  first  numerator, 

b2 c  —be2  =  second  numerator, 

gc2  —a2c=  third  numerator, 

ab2  +  bci  +  a2c=  fourth  numerator. 

g2  b  +  b2  c  +  gc2  =  sum  of  numerators. 

.-.  Sum  of  fractions  =  a' 'b  +  * 'e  +  "* 
abc 


teachers'  edition.  91 


10. 

I            1           1        2x  —  zy  —  2z 

2x*y     6y2z     2xz*      4a?2z2       taPyz 

L.C.D.  =  12sVz*. 

6yz*                                 =  first  numerator, 

—  2x2z                       =  second  numerator, 

—  6xy2            = third  numarator, 

6xy2  —  3  y2z  =  fourth  numerator, 

—  6  yz2                        +  3  y2z  =  fifth  numerator. 

—  2  x2  z                      =  sum  of  numerators. 

. 

0    2-                        "I 

Sum  ot  fractions  = -— -  =  —  — — • 

12zV2a         6y2« 

Exercise  LVII. 

1. 

j_+^_. 

a?-6     z  +  5 

L.C.D.=(a;-6)(a;  +  5). 

The  multipliers  are  jc  +  5  and  a?— 6  respectively. 

x  +  5  =  first  numerator, 

a?  — 6  =  second  numerator. 
2x  —  1  =  sum  of  numerators. 

Sum  of  fractions  =  -      x~ 


x*-x-20 

_1 1__. 

z-7     z-3* 

L.C.D.  =  (s-7)(3-3). 

The  multipliers  are  *  —  3  and  x  —  7  respectively. 

a;  —  3  =  first  numerator, 
—  x  +  7  =  second  numerator. 
4  =  sum  of  numerators. 
4 
,  Sum  of  fractions  = 


s2-10a:  +  21 


-x 

L.C.D  ^1-s2. 

The  multipliers  are  1— x  and  1  +  x  respectively. 
1  —  x  =  first  numerator, 

1  +  x  =  second  numerator. 

2  ^=  sum  of  numerators. 

o 
,  Sum  of  fractions  —  - — — • 
1  —  ar 


92  ALGEBRA. 


6. 


I 


l-x     l-x* 

L.C.D.=  l-x*. 

The  multipliers  are  \+x  and  1. 

1  +  x  —  first  numerator, 
—2        >=  second  numerator. 


— 1  +  x  =  sum  of  numerators. 

—  (1-*). 

.\  Sum  of  fractions  =  ~*    "*"*'  =  — • 

l-x"  1+a? 

c        1  x 

6.   + 


*-y  '  (*-y)" 

L.C.D.  =  (a-y)2. 

The  multipliers  are  a?  —  y  and  1. 

x  —  y  =  first  numerator,  I 

x-        =  second  numerator. 
2  x  —  y  =  sum  of  numerators.  \ 

Sum  of  fractions  =    x~*. 

(*-y)* 


1  1 


2a(a-f:e)     2a(a  — a) 

L.C.D.  =  2a(a  +  aO(a-x). 

The  multipliers  are  a— a;  and  a+c. 

a  —  x  =  first  numerator, 
o  +  a?  =  second  numerator. 
2  a        =  sum  of  numerators. 

,  Sum  of  fractions  =  -— ^- x  ........ 

2a(a-Kr)(a— aw     a2  — or 


(a  +  &)6     (a  — 6)a 

L.C.D.  =  a6(a2-62). 

The  multipliers  are  a(a  —  b)  and  J  (a  +  b) 

a8  —a2  6  =  first  numerator, 

—  a62  —  6s  =  second  numerator. 

az  —  a%b  —  db%  —  6s  =  sum  of  numerators. 
.  Sum  of  fractions  -  *-*h-&-». 


teachers'  edition.  93 


8. 


10. 


2x{x-\)     4x{x-2) 
L.C.D.  =  4a;(a?-l)(a:-2). 
The  multipliers  are  2  (a?— 2)  and  (a  — 1). 
10  a:  —  20  =  first  numerator, 
—3s  +   3  =  second  numerator. 
7x  — 17  =  sum  of  numerators. 

tj I  It 

.  Sum  of  fractions  = - . 

r  4s(xa-3a;  +  2) 

1  +x  1  — a; 


1  +  x  +  x*     1  — a?  +  x* 

L.C.D.  =  l+x*  +  a;4. 

The  multipliers  are  1— x  +  x*  and  1  +  x  +  xa. 

1  +  x8  —  first  numerator, 
— 1  +s*  =  second  numerator. 
2  3*  =  sum  of  numerators. 

.  Sum  of  fractions  —  • 


l  +  a*  +  x* 


2ax  —  Zby     2ox  +  35y 
2xy{x-y)     2xy(x  +  y) 
L.C.D.  =  2xy(x*-y*). 
The  multipliers  are  x+y  and  x—y. 

2 ax*  +  2axy  —  3  bxy  —  3 by9  =  first  numerator, 
—2ax*  +  2  gay  —  3 bxy  +  3 &y*  *=  second  numerator. 
4  oxy  —  6  fay  =  sum  of  numerators, 

or  2 ay (2a  —  3 6)  =  sum  of  numerators. 

•\  Sum  of  fractions  —  -^^ 

x*-y2 

Exercise  LVIII. 
1.-L.+    x    •    2a 


1  +  a     1  — a     1  — a* 

L.aD.-1-a".' 

The  multipliers  are  1  — a,  1+a,  and  1. 

1  —    a  «=  first  numerator, 

1  +    a  —  second  numerator, 

2a  =»  third  numerator. 

2  +  2a=*2(l  +  a)  =  sum  of  numerators. 

,  Sum  of  fractions  -      2^  +  a) ?— 

(l  +  a)(l-o)      1-a 


94  ALGEBRA. 


__1 1      ,     2  s 

1-x     1  +  x      14-a* 
L.C.D.*(l-x)(l  +  x)(l  +  x»). 

1  +x  +  x*  +    Xs  =  first  numerator, 
— 1  +  a?  —  x*  +    x8  =  second  numerator, 
2x         —2a8  =  third  numerator. 
4x  =  sum  of  numerators. 

4* 


\  Sum  of  fractions^ 


1-x* 


x  x*        _    X 


1-x     1-x     1+x* 
L.C.D.=  (l-x)(l  +  x*). 

x  +  X8         =  first  numerator, 

—  x*         —  x*  =  second  numerator, 

g    —x1 =  third  numerator. 

2x  —  2x*+x8  —  x*  =  sum  of  numerators. 
=  2x(l-x)+x»(l-x). 

,  Sum  of  fra*tions  =  (2*  +J<l-«),  *g±g. 
(1  +  3*)(1  -x)        l+s» 


?+_y_  + 


y     x+y     X*  +  xy 
L.C.D.  =  xy(x  +  y). 

x»    +  x*y  =  first  numerator, 

+  ajy*  =  second  numerator, 

•f  x*y =  third  numerator. 

x*  +  2x*y  +  xy*  =■  sum  of  numerators. 
=«x(x  +  y)2. 

...  Sum  of  fractions  =  x-^±^  =  ^±£ 
ajy(a?+y)        y 


x  —  2     x  —  3     x  — 4 

L.C.D.  =  (x-2)(x-3)(x-4). 

X8  —   8x2  +  19x-12  =  first  numerator, 
aJ—   8x*+20x  —  16  =  second  numerator, 
a*_   8xa  +  21x  —  18  =  third  numerator. 
3  a?  —  24  x*  +  60  a;  —  46  =  sum  of  numerators. 
3x8-24x»  +  60x-46 


,  Sum  of  fractions  = 


x.s_  9x*  +  26x-  24 


teachers'  edition.  95 


3  4a  5a2 


x  — a     (a—a)2     (x— a)s 
L.C.D.^x-a)8. 

3X2  —  6ax  +  3a2  =  first  numerator, 
4  ax  —  4  a2  =  second  numerator, 

—5a2  —  third  numerator. 

3s2  —  2ox  —  6  a2  =  sum  of  numerators. 

,  Sum  of  fractions  =33x2-2ox-6a2 


(x-af 


1  1 


x-1      x+2     (x  +  l)(x  +  2) 
L.C.D.  =  (x-  l)(x  +  l)(x  +  2). 

x2  +  3  x  +  2  =  first  numerator, 
—  x2  +  1  =  second  numerator, 

—  3x  +  3  —  third  numerator. 
6  =  sum  of  numerators. 

o 

,  Sum  of  fractions  » 


8. 


(x2-l)(x  +  2) 
a  —  b         ,         b  —  c 


(b  +  c)(c+a)     (c  +  a)(a  +  b)     (a  +  &)(6  +  c) 
L.C.D.  =  (6  +  c)(a+6)(c  +  a). 

a2  —  J2         —  first  numerator, 
+  6s  —  c2  =  second  numerator, 
—  q2  4-  c2  =  third  numerator. 

0  =  sum  of  numerators. 
.*.  Sum  of  fractions  —  0. 


9    x  —  a     x—  b  (a  —  b)1 

x—  b     x  —  d     (x  —  a)(x  —  b) 
L.C.D.  =  (x-a)(x-6). 

x2  —  2  ax  +  a2  «=  first  numerator, 

x2  —  2  6x  +  J2  —  second  numerator, 

-a2  +  2o6  —  62 —  third  numerator. 

2X2  —  2bx  +  2ab  —  2ax  =  sum  of  numerators. 
=  2(x-a)(x-6). 

.-.  Sum  of  fractions  -  2(s-a)(x-6)  _  2. 
(x-a)(x-6) 


96  ALGEBRA. 


11. 


12. 


13. 


10.  s  +  y        2j         afy-a8 

y      *  +  y    yO^-y*)' 

a*  —  sy*  +    ofy  —  y8  =  first  numerator, 
2ajy*  —  23*y         =  second  numerator, 

—  a* +    xhj  =  third  numerator. 

ay*  —  y*  =  sum  of  numerators. 

.-.  Sum  of  fractions  =  y'(*~#  =  -y— 
y(jc*-y2)     jr  +  y 


(6_c)(c_a)     (c-a)(a-b)     (a-b)(b-e) 
L.C.D.  =  (&-e)(c-a)(a-&). 

a*  —  6*         =  first  numerator, 
+  6*  —  c*  =  second  numerator, 
—  o*         +  c1  =  third  numerator. 

0  =  sum  of  numerators. 
•  \  Sum  of  fractions  =  0. 


q*  —  be       ,        62  —  oc        .       c*  +  qft 


(q  +  6)(q  +  c)     (&  +  a)(6  +  c)     (c+b)(c  +  a) 

L.C.D.  =  (a  +  6)(6  +  c)(a  +  c). 

q26  —  6'c  -f  a2c  —  6c*  =  first  numerator, 
aft*  +  ft*c  —  a*c—ae*  =  second  numerator, 
a*b  +  qc1  -f  6c*  4-  ob*  —  third  numerator. 
2q*o  +  2  aft*        —  sum  of  numerators. 
=  2aJ(a  +  6). 

Sum  of  fractions  ~ 2«K«+*) ^ 

(a  +  b)(b  +  c)(a  +  c)     (b  +  c)(a  +  c) 

a  x  a*+  x*  * 


a  —  x     a  +  2x     (a  —  x)(a  +  2x)  l 

L.C.T).  =  (a-x)(a  +  2x). 

a2  +  2  ax         —  first  numerator, 

—  ax  +  x*  =  second  numerator,  , 

—  a* —  a2  =  third  numerator.  I 

ax         =  sum  of  numerators.  | 

.•.  Sum  of  fractions  = — 

(q  — s)(q  +  2a:) 


14. 


16. 


teachers'  edition.  97 


6 


(a-6)(6-c)     (g-6)(g-c)     (g_c)(6-c) 
L.C.D.  =  (g-6)(g-c)(6-c). 
3  a  —  3  c  =  first  numerator, 

—  4  6  +  4  c  =  second  numerator, 
6o  —66  =  third  numerator. 

9  g  —  106    +  c  ==  sum  of  numerators. 

.-.  Sum  of  fractions 9a-10b+e    >t 

(g-6)(g-c)(6-c) 

15     x  —  2y        2x+y  2x 

'  x  {x  -  y )  ~~  y  (x  +  y)  ~"  x*  -  y2' 

L.C.D^a^-y2). 

x*y  —  xy2  —  2yi  =  first  numerator, 

—  2  a?  -f    afy  +  &y2  =  second  numerator, 
—  2x*y =  third  numerator. 

—  2a?  —  2y*  =  sum  of  numerators. 

=  -2(x  +  y)(x*-xy+y*). 

...  Sum  of  fractions  =  "2(g  +y)(a'"ay  +  y*). 
oy(a;+y)(aj-y) 

_      2(a?-sy  +  y2) 

ay(o;-y) 


a  — 6       (g  —  6)(a;  +  y) 


a;  (g  +  6  j     y  (g  +  6)         xy  (a  +  6) 
L.C.D.  =  ajy(g  +  6). 

ay —  ty  =  first  numerator, 

—  gar  +    6a;  =  second  numerator, 
—  gy  +  6y  —    ga;  +    6a;  =  third  numerator. 

—  2  ga;  +  2  6a;  =  sum  of  numerators. 

=  2a;(6-g). 

...  Sum  of  fractions  -  2x(b~«)  =  *!k=J$. 
xy(a  +  b)     y(g  +  6) 

17        3a;         x  +  2y         3y 
'  {x+yf     x*-y^{x-y? 
L.C.D.  =  (a;  +  y)2(a;-y)2. 
3a£  —  6a?y  +   3  ay2  =  first  numerator, 

—Xs  —  2x*y  +      xy%  +  2y*  =  second  numerator, 

+  3x2y  +   6xy2  +  Zy*  =  third  numerator. 
2a£  —  5x*y  +  10 xy3  +  5y*  =  sum  of  numerators. 

.-.  S«m  of  fractions  =  **-***+}<>*?  +  *?. 


98  ALGEBRA. 

18  a—c a—b 

•  (a  +  bY-<*     (a  +  cY-b* 
L.  C.  D.  -  (a  +  b  +  c)(a  +  b  -  c) (a  -  b  +  c). 
a*  —  c*  «=  first  numerator, 

—  a*  -f  oc  -f  5*  —  ab         =  second  numerator, 
ac  +  6*  —  o6  —  c*  ■»  sum  of  numerators. 
.-.  Sum  of  fractions  =  ac-ab  +  b*-<* 


(a  +  b  +  c)(a  +  b  —  c)(a  +  c—b) 


19. 


a  +  b  _  a  —  b  ab(x  —  y) 
ax  +  by  ax  — by  a V  —  &y 
L.C.D.  =  (ax  +  6y)(oa;  -  ty). 

a*x  +  ate  —  aby  —  b*y  =  first  numerator, 
—  a'a;  +  o&c  —  aby  +  6*y  —  second  numerator, 
+  qfcg  — o6y  =  third  numerator. 

+  Sabx—Saby         —  sum  of  numerators ; 
or,  3  ab  (a?  —  y)  =  sum  of  numerators. 

•.  Sum  of  fractions  =  3,°^*~y)- 
a*ar  —  by2 

Exercise  LIX. 


i.    x    |  a?~~y_  g     g-y 

x-y     y-x     x-y     x-y 
L.C.D.  =  x-y. 

a?  »=  first  numerator, 
y  —  a;  =  second  numerator, 
y        =  sum  of  numerators. 

.  \  Sum  of  fractions  =  — 2L. . 
x-y 

2    3  +  2a;     Sx-2     16x-x* 

'     2-s        2  +  s        s*-4 
^3  +  23?     3a:-2     16a?-ar« 
"  2-aj        2  +  x         4-s2  ' 
L.C.D.  =4 -a* 

6+    7aj  +  2xI  =  first  numerator, 
—  4+    8 a?  —  3  a?  =  second  numerator, 
—  16a?  +    x*  =  third  numerator. 
2     —  x  =  sum  of  numerators. 

n -I 

••.  Sum  of  fractions  =-  — 


4-a1      2  + a; 


teachers'  edition.  99 


3*  _£ X      ^       X*  X  X 


a?-l     x+1     1-x     a?-l     x  +  1     a?  — 1 
L.C.D.  =  a?-l. 

a?        =  first  numerator, 

a?  —  x  =  second  numerator, 

g*  rha;  =  third  numerator. 
3  a?        »=  sum  of  numerators. 

3a? 

,  Sum  of  fractions  =  - — -  • 
ar  — 1 


4. 


1 


3_3y*     2-2y     6y  +  6 

4  ,1,1 


3(l+y)(l-y)     2(1 -y)     6(1 +  y) 
L.C.D.  =  6(l+y)(l-y). 

8  =>  first  numerator, 
3y  H-    3  =  second  numerator, 
— y  +    1  =  third  numerator. 
2y  + 12  =  sum  of  numerators  ; 
or,  2(6  +  y)  =  sum  of  numerators. 

.*.  Sum  of  fractions  a  •—- — 2l_- 

3(1  -y*) 


(2-ro)(3-m)     (m-l)(m-3)     (m-l)(m-2) 
1  2,1 


(2-m)(3-m)     (l_w)(3-m)     (l-m)(2-ro) 
L.C.D.  =  (l-m)(2-m)(3-m). 

1  —    m  =  first  numerator, 
—  4  +  2  m  =  second  numerator, 

3—    m  =  third  numerator. 
0  ==  sum  of  numerators. 
.•.  Sum  of  fractions  =  0. 


6. 


(b  —  a)(x  +  a)     (a-b)(x  +  b) 
1  1 


(b-a)(x  +  a)     (b-a)(x  +  b) 
L.C.  D.  =  (b  -  a)(x  +  a)(x  +  6). 

x  +  b        —  first  numerator, 
—  a?        —a  =  second  numerator. 
6  —  o  =  sum  of  numerators. 
,  Sum  of  fractions  *  ~~a  — 


(b-a)(x  +  a)(x  +  b)     (ar  +  a)(x  +  b) 


100  ALGEBRA. 


a* +  6*       2a6*   ^  2a'&       a*  +  h1       2a&'    ,    2a'6 


L.C.I).  =  (a8- &»)(<*»  + 6s). 

a*  +2a4&*+2a'&*  +6*  =  first  numerator, 

— 2a*6*  — 2aJ*       =  second  numerator, 

+2a?b  -2a*b*  =  third  numerator. 


8. 


a«+2a*& 

—2abP+&  =  sum  of  numerators. 

Sum  of  fractions  = 

(a»-&»)(a8  +  &») 

b  —  a     a— 2b 

x  —  \      b +  x 

3x(a-b)_  b-a     a-26 
&*  — z2        a?  —  b      x  +  b 

3x(a- 

Xs- 

-b) 
b* 

L.  CD.  =  **-&». 

—  ab  —  ax  +    bx  ■ 

ab  —  ax  +  2bx  ■ 

Sax-Zbx 

+•    6s  =  first  numerator, 
-26*  =  second  numerator, 
=  third  numerator. 

■  6*  =  sum  of  numerators. 


Sum  of  fractions  =  -aaj 


x*  -b* 


S  +  2x     2-3a?     16g-ar»^3  +  2s     2-3.r     16S-S1 
2-a?        2  +  x         s*-4    =  2-a?        2  +  x         4  — ar* 
L.C.D.-4-x". 

6+    7  a?  -f  2  s*  =  first  numerator, 
—  4+    8  a?  —  3  e*  =  second  numerator, 
—  16  s  +    s*«°  third  numerator. 
2      —x  =  sum  of  numerators. 

2-x         1 


.  Sum  of  fractions  = 


4-ar*     2+a; 


10        3  7  4 -20s 3 7         4- 20a? 

'  1-2*     1  +  2*      4z*-l,Bal-2s     1+2*      l-4a*' 
L.C.D.  =  l-4ar*. 

3  +    6  a?  =  first  numerator, 

—  7  +  14  a?  =  second  numerator, 

4  —  20a;  =»  third  numerator. 

0  =  sum  of  numerators. 
•\  Sum  of  fractions  =  0. 


teachers'  edition.  101 


11.         fl  +  fr        ,  6  +  c  ,  c  +  a 


(J_c)(c_a)     (6-a)(a-c)     (a-6)(6-c) 
_  a  +  b  b  +  c  a  +  c 


(b-c){a~c)     (a-6)(a-c)     (a-6)(6-c) 
L.  C.  D.  =  (a  -  6)  (a  -  c)(b  -  c). 
—  a2  +  62         =  first  numerator, 
_  b2  +  c2  =  second  numerator, 
qa         —  c*  =  third  numerator. 

0  =  sum  of  numerators. 
.•.  Sum  of  fractions  =  0. 

12         <*-bc  b2  +  ac  <?  +  ab 

'  (a-6)(a-c)     (6  +  c)(6-a)     (c-a)(c  +  6) 
_       a2  — be  ac  +  b2  ab  +  c* 


(a-6)(a-c)     (6  +  c)(a-6)     (a-c)(6  +  c) 
L.C.D.=  (a-6)(a-c)(6+c). 
a*6— 62c+a2c— 6c*  =■  first  numerator, 

l?c—a2c        4-ac2— a6a  =  second  numerator, 

— a*b +bck—a&+alP=z  third  numerator. 

0  =  sum  of  numerators. 
.  \  Sum  of  fractions  =  0. 

i3.      y+2     i     z+x     ,     g+y 

(x-y)(x-z)     iy-x){y-z)     (z-x){z-y) 

^       y  +  z g  +  z  x  +  y 

(x-y)(x-z)     (x-y)(y-z)     (x-z){y-zj 
L.C.T>.  =  (x-y)(y-z){x-z). 

y2  —  z2=*  first  numerator, 
—  x2         -f  z2  =»  second  numerator, 
Xs  —  y2        =  third  numerator. 

0  =  sum  of  numerators. 
.\  Sum  of  fractions  =  0. 

14.  3  4  6 


(a_6)(6_c)     (6_a)(c-a)     (a-c)(c-6) 
3  4,6 


(a_6)(&_c)     (a-6)(a-c)     (a-c)(6-c) 
L.C.D.  =  (a-6)(a-c)(6-c). 
3  a  —  3  c  =  first  numerator, 

—   46  +  4c  =  second  numerator, 
6a—   66     '     =  third  numerator. 
9a  —  106  +    c  =  sum  of  numerators. 

,\  Sum  of  fractions  =-  ~ 


(a-6)(a-c)(6-c) 


102 


ALGEBRA. 


15. 


x(x-y)(x-z)     y(y-x){y-z)     xyz 

1 1 1_ 

x{x-y){x-z)     y{x-y){y-z)     xyz 
L.  C.  D.  ■»  xyz(x  —  y)(x  —  z)  (y  —  z). 

y*z  —yz*  =  first  numerator, 

— a^z+as*         =  second  numerator 
—xht+xyt—yh+sPz—xzi+yz*  =  third  numerator. 
—x*y+xy*  =  sum  of  numerators  ; 

or,  —  xy  {x  —  y)  =  sum  of  numerators. 


.•.  Sum  of  fractions  =  ■ 


-qy(*-y) 


xyz(x-y)(x-z)(y-z) 

1 


2(*-«)(y-2) 


Exercise  LX. 


1. 

a      C* 

5. 

8oW      15sy* 
45**y      24a86* 

Cancelling  common 

factor  x, 

Cancelling  8,  15,   a\  &»,  x, 

ac 

andy, 

bd 

=  -&. 
9  ox 

2. 

2*v3o5v3oc 
a        c         26 

A. 

9xY«          20aWc 

Cancelling  2a6c, 

10  aWc          18  sy'z 

=  9aa;. 

Cancelling  9xy*z,  10  cWc, 
and  2, 

3. 

3j>          2j> 
21>-2'<>-l 

=»  —  ax. 

«     3j>         »-l 
2(j>-1)       2^ 

7. 

3^yx5^x_12£ 
4xz»      6rry          2sy* 

Cancelling  p  and  j)  - 

-1, 

Cancelling  2,  6,  a?,  y8,  and  z, 

"i 

15  a? 
4z' 

4 

Sx*y       2x* 

15  aP     Sab* 

g 

9mVx5pVx24xV 

8**y      3a&* 

&jp<f      2xy      90  mn 

15a&»      2a* 

Cancelling  9,  5,  8,  mn,  j?1^, 

Cancelling  2»"  and  3 ab7, 

and  xy, 

4xy 

3mnxy 

5b 

W 

TEACHERS     EDITION. 


103 


25Pm2 


70nV  3pm 
14  nV  75p*m  4#n 
Cancelling  25  £2ra,  14n8g',  and 

5*ro2 


11. 


41>2 


10. 


'-62 


a-b       

a*  +  ab     a*—ab 

=    a-b     x(a  +  b)(a-b) 

a(a  +  b)  a(a  —  b) 

Cancelling  a  —  b  and  a  +  6, 

a-6 


a2  +  6*  .  a  —  b 
a2-b2^ a  +  b 
^       a2  +  6*  a  4-  6 

(a—b)(a  +  b)     a  —  b 
Cancelling  a  +  b, 

a2  +  62 


12. 


(a -ft)2 


3*  +  X  -  2      xf_ 


-13a: +  42 


x2  —  7x  x2  +  2x 

(s+2)(x-l)x(x-6)(x-7) 
x(x-7)  x(x+2) 

Cancelling  a?  —  7  and  x  +  2 

,(x-l)(—fl). 


13    g'-Hg-f  30     a»-3a? 
a^-Bx  +  S       a^-Sx 

(x-5)(x-6)     *(g-3) 
(x-3)(x-3)-    x(x-5) 

"**  — 3* 


14.  <*-*x(a  +  xf 
a3  +  x*     (a  — a;)2 

_  (a  -  x)(a2  +  ax  +  x2)      (a  +  a;)2 
(a  +  x)(a2  —  ax  +  x2)      (a  —  x)2 
(a  +  x)  (a2  +  ax  +  x2) 
(a  — x)(a2  — ax  +  X2) 


16    2a(x2-y2)2x  x» 

ex  (x-y)(x  +  y)2 

2a(x  +  y)2(x-y)\ 


*(.T-y)(x  +  y)2 


2ax*  (x  — y) 


104  ALGEBRA. 


16    aa+2q&xa6-26a  17     s»-4  xs»-25 
a3+461      a1-46»  '  x*  +  bx     3*+2x 

_a(a  +  26)         6(a-26)  _  (x+2)(s-2)     (g+5Xs-5) 

a* +46*      (a-26)(a+26)  s(ay+5)  x(x+2) 

ab  (x-2)(x-5) 

a*+461"  "  x* 

is.  a^  +  ay>/(*-y)1 

a;-y        a?*-y* 

_x(*  +  y)y      (s-y)Qg-y) 

*-y       («,  +  y,)(*  +  y)(a?-y) 
a; 


V  +  y* 


19. 


c8  +  d8      c  +  d 

_      (ro  +  n)  (ro  —  n)  c  +  d 

(c  +  rf)(c*  —  cd+  d*)         m  —  n 
m  +  n 
<?-cd  +  d* 


-4a  +  3      g»-9g  +  20      a* -  7a 


a1 -5a +  4     a* -10a +  21     a* -5a 
(a-3)(a-l)     (a-5)(a-4)     a(a-7) 
(a-4)(a-l)     (a-7)(a-3)     a(a-5) 

=  1. 

6»-76  +  6     6» +  106  +  24  .  6* +  66 
"   6*  +  36-4     6*-146+48T68-868 
(6-6)(6-l)     (6  +  6)(6  +  4)     P(6-8) 
(6  +  4) (6  - 1)     (6  -  8)(6  -  6)     6(6  +  6) 

=  6. 

22  s*-y»        x.a?y  -  2y8  x  x*-xy 

x2  —  3xy  +  2yi       x2  +  xy       {x—yf 
_  (s+y)(s-y)  x  y(g-2y)  x      a;(a?-y) 
(x-2y)(x-y)      x(x  +  y)      (*-y)(*-y) 

y 

as-y 


teachers'  edition.  105 


03    as-3q»&  +  3ay-6s     2ab-  2b*  „  a*  +  ab 

a*-6»             +  3         X  a-b 

_a3-3a86  +  3a6'-y  3            a*  +  ab 

a8-6*  2a&-26»      a-b 

_(a-b)(a-b)(a-b)  3        „a(a  +  b) 

(a  +  b)(a-b)  2b(a-b)       a-b 
^Sa 
2  b 

2±  (a+ftP-c*  .  <*-(a  +  bY 
a*-(b-c)*^c>-{a-bY 
_^ (a  +  b  +  c)(o  +  6 - c)(c -0  +  6)(c  +  a-b) 
(a  —  6  +  c)(o  +  6  -  c)(c  -  a  —  6)(c  +  a  +  6) 
_  c  —  a-f  6 


(x-&)»_a*     ^-(a-ft)* 
_  (a;  -  a  -f  6)  (a;  -  a  -  6)  (a  +  6  -  a)  (x  -  b  +  a) 
(3  —  b  +  a)(»  —  b  —  a)  (x  +  a  —  &)(s  —  a  +  6) 

a?  +  a  — 6 


(a  +  6)»-(c  +  aT     (a-c)»-(<*-6)» 
*   (a +  «)■-(&  +  <*)»-     (a -ft)1 -(<*-«)■ 
c (0  +  b  +  c  +  <?)(a  +  ft  —  c -  a*)(a -  6  +  d~ c)(a  -b-d  +  6) 
{a  +  c  +  b  +  d)(a  +  c  —  b  —  d)(a  —  c  +  d  —  ft)  (a  —  c-d  +  6) 
=  1. 


27     a?  —  2ary  +  y*  —  z*x  +  y  —  z 
x*  +  2xy+y*  —  z*     x  —  y  +  z 
_(x-y)i-z*yx  +  y-z 
(x  +  yf-z*     x-y  +  z 
^  (a  -  V  +  z)(x  -  y  -  z)(x  +  y  -  z) 
*°  (x  +  y  +  z)  (x  +  y  -  z)  (x  -  y  +  z j 

a;  +  y  +  2 


106  ALGEBRA. 

Exercise  LXI. 

3s     x-l 
2  3 


Multiply  both  terms  by  6, 
9a;  +  2a; -2 
13a? +  13 -2a -15 

__  11  x  —  2  _-.  ax+bx  +  cx—  be 

~lla;-2~ 


X 

—  a 

x 

(*- 

-»)(•-«> 

a?  +  a 
a?  — a 

x2  +  ax— x*  +  bx  +  cs- 

-6c 

(«■ 

ar  +  a 
-a)(s  +  a) 

-1+A 


\x     a/\x     a  J 


x-2  + 

x-6 


—'  l-x~a 


x  +  a 


Multiply  both  terms  by  *  -  6,  (  a'-^Ug+^N 


2a 

a  +  a? 


_g»-7a?  +  12 

"V-Ss  +  lS 

(a.-5)(a-3) ^-j 

a>-4 


3. 

3 
21  +  1      *' 

2a;-l 

+  5-1 
2     2 

Multiply  both  terms  of  sec- 
ond fraction  by  2, 

3 

x  +  1 

4x-2 
2x*  +  x-l 

3 

B  +  l 

2(2*- 1) 
(2*-l)(*+l) 

3 

2            1 

a?-y     a?-y» 
a;  y 


xy+y*     x*  +  xy 

x  +  y  —  x 


s*-ya 


y        *y(x  +  y) 
"a*-tfX    **-y» 


jc  +  1     x  +  l     a:+l  (a;+y)(x-y)» 


teachers'  edition.  107 


x+l+x— 1 

~    x  —  \     x  +  1 

x  +  1     a;—  1 

ar  — 1     x  +  1 

10. 

1 
x_ 

(a  +  1)2     (x-Xf 

1 

a^-l        x*-\ 

1 £_ 

(x  +  1?      (x-llP 

x  +  1 
x  +  1 

(ar  +  2x  +  l)  +  (x»- 
(x*  +  2x  +  l)-(x*- 

-2x  +  l) 
-2x  +  l) 

X  +  1  —X 

=  x+l 

2x*  +  2     x*  +  l 
4x           2x 

1 

-x  +  1. 

11. 

1 
1  1        * 

X 

1  ,               2ar 

l+x  +  -^- 

1— X 

1 

x  +  1 

1+l+% 

X  +  1  —X 

1-x 

x  +  1 

1 

1 

x  +  1 

i        X  —  X* 

1  +  x2 
^l+ar| 
"l  +  x* 

9.  1  +  . 


l+x  +  n 

1  — x 


e    .„  e+He+H 


=  1  + 


x(l-x) 
(l+x)(l-x)  +  2ar» 


(fir 

(l+x)(l-x)  +  2x3  /g»-2ax  +  x»\  /raa  +  2ox  +  xa\ 

1  +  g(l  ~ g)  V         ax         jy         ax  / 

"        "  *  /a8-xa\/a2-ar\ 

\    ax     J  \    ax     J 


l  +  x» 

1 1  +  X*  +  X  —  X* 

1  +x»  (q  —  x)(a  —  x)(a  +  x)(a  +  x) 

1  +  a?  (a  +  x)(a  -  x)(a  +  x)(a  -  x) 

'l+x*'  «1. 


108  ALGEBRA. 


g'  +  y1       2a?    (xy-&     x  +  y) 

13#  g'-y*    »+yt(^-y)>    g-y) 
»-y 

g»+y»       2a?    r aj_  +  g_+y  J 

g'-y*     g  +  y(     g~y     x-y\ 
=  g-y 

a?-y*     g+yjg-yj 

*-y 

g*  +  y»       2ay 

g-y 

g»  +  2gy +  y» 

g'-y* 
g-y 

_(g+y)(*+y)x   i    ^      *  +  y 

(g-y)(g  +  y)     g-y     x*-2xy  +  y* 
(g»-y»)(2g«-2a!y) 

14. 4<*-y>a 

_gy_ 
g  +  y 

_  (g  +  y)(g-y)2a?(a?-y)(a?  +  y) 
4(a?-y)(g-y)gy 

2y 

aft  oc 


jg    s* +(a  +  ft)a?  +  aft     a?  +  (a  +  c)x  +  ac 
b  —  c 
g*+(ft  +  c)a?  +  ftc 

(qfta;  +  aftc)  —  (aca;  -f  qftc) 
(a?  +  q)(g  + ft)(s+c) 

(a?  +  ft)(a;  +  c) 

ax(b  —  c)  (x  +  ft)  (x  +  c)  qg 

(a?  +  a)(g  +  ft)(g  +  c)(6  —  c)     g  +  a 


teachers'  edition.  109 


16.  - 

1 

x         1 
— Y  +  1-- 

X 

1 
'  +  1 

x+l 

1_ 

x+l 

x*  +  x 
x  +  l 

'X. 

a  +  b        b 
j~       b        a  +  b 

19. 

i+i+I 

ab     ac     be 

a     b 

a2-(6  +  c)J 
ab 

(a  +  b)*  +  b* 

b(a  +  b) 

a  +  b 

Multiply  the  terms  of  the  nu- 
merator by  abc,  and  factor 
the  denominator, 

~aT 

c  +  b  +  a 

_(a  +  b?  +  b*x   ab 
b(a  +  b)        a  +  b 

abc 

(a  +  b  +  c)(a-b-c) 
ab 

{(a  +  b)*  +  b*\a 

{a  +  b? 

_(a*  +  2ab  +  2b*)a 

_ c+b+a                ab 

abc    "  (a+b+c)(a-b-c) 

1 

(a  +  &)* 

c(a—b  —  c) 

1 


m 


2m-l 


20. 


2m-3  +  —  i  .         o 

18.  2  i   ,      3 


2m-i  1+rr; 


2m'-3m  +  l  3(1  — ag) 

™  l-s  +  3 


3 

"l-±x 


_(2m-l)(ro-l)  4-x 

2™-1  __3(4-aQ 

-»-l.  ""  7- 4a' 


110  ALGEBRA. 


Exercise  LXIL 

x*-9x8  +  7x*  +  9x-8  0    ot  +  6a-cs  +  2o5 


1. 


(x-8)(g*-x*-x  +  l)  ^16  +  }-l+4 

=a(x  +  8)(x*-xa-x  +  l)  ~16-J-1  +  1 

s-8  ^19} 

~*  +  8*  =15} 

Multiply  both  terms  by  4, 


3.  3«.  +  ^-^ 

c        b1 


■tt-l|- 


=  48  +  2-4  =  46. 

2  1  1 


(3*-l)*      2s»-4x  +  2     1-s* 


(x»-l)»     2(x-l)a     ar»-l 
L.C.D.=.2(x»-l)2. 

4  =  first  numerator, 
—Xs  —  2a:  —  1=  second  numerator, 
2s*  —2  =  third  numerator. 

x1  —  2s  +  1  =»  sum  of  numerators. 
.-.Sum  of  fractions-  **-2*  +  l  _        1 


2(x*-2x  +  l)(x  +  l)»      2(x  +  l)2 

Vb  +  1  x  +  1/      \x—l  x-lj 

/    x*     {x  +  l         l\/a*        x»-x 1_\ 

\x  +  l     x  +  1      x  +  l)      \x-l      x-1      s-lj 

_  x(x  +  1)       X  -  1 
X  +  l  X  — 1 


teachers'  edition.  Ill 


6.  (x-<*\*    (x-2a  +  b\ 
\x-b)      \x  +  a-2b) 


(a  +  b-2a\*    fa±b-^a±2b\ 
BS\a  +  b-2b)      \a  +  b  +  2a-4b) 

\a-b)      \Za-n) 
=  (-!)»-(- 1)  =  0. 

7    f    a  +  b  a-b  2b*   \a-b 

'  \2{a-b)     2(a  +  b)     a»-W  26 

L.C.D.  of  fractions  in  brackets  —  2(o*—  6*). 

a*  +  2  aft  +    6*  *=  first  numerator, 
—  a2  +  2  oft  —    6*  =  second  numerator, 

46*  =  third  numerator. 

4  ab  +  46*  =  sum  of  numerators  -, 
or,         4  6  (a  +  b)  =  sum  of  numerators. 

.\  Sum  of  fractions  in  brackets  =  — *a  +,  '■  —  -^—, -• 

2(a,-6»)     a- 6 

— ,x^  =  i. 

a-6       26 


Va^-y1     x*+y*)^\x-y     s  +  yy 

_  / s*+2 x*y*+y*     x*-2 afy* +y*\  #  (x*+2xy+y*     s*-2sy+y2\ 
\      a^-y4  a^-y4      y      \     a?-y2         ""iT^y2-"  / 

_  4aj*y*        4:cy 
x*  —  y*     x*—y* 

_4xyx*»-y' 
o^— y*       4ary 

„  *y 

a^+y8 


112  ALGEBRA. 

V  ya   J\*-y)    \  y8   A*  +  *y  +  W 

y2         s-y         y8         o^  +  ajy+y* 

y        y 

=  2. 

U8-W\     <*  +  &      y  Us  +  ^       A       a*+ab  +  b*J 
(a(a-b)\  (a2  +  ab  +  b*\      /a8-&8\  /q'-gft  +  yx 

=  U8-^A      <*  +  &  J      U8  +  &vU,  +  <**  +  &V 

=     a        a  — 6 
a  +  6     a  +  6 

=  2a-_6 
a  +  6 

....  a  +  jc  a3  +  x2 


-      a  — <c      -      a' 


a  +  x  a2  +  x2 

Multiply  both  terms  of  first  fraction  by  a+x,  and  both  terms  of 
the  second  by  a2  +  x2, 

a  +  x +  a  —  x     a2 +  X2 +  a2 —  x2 
a  +  x  — a  +  x     a2  +  a?  —  a2  +x* 
^2a     2X2     x 
2x     2a2     a 

=(-4)+3K)+4KVH) 

-(^-l+3l)  +  8(-l)  +  4 

X       X2 


teachers'  edition. 


113 


1- 


13. 


2xy 


1-'- 


1  + 


2ay 


1  + 


(»-yJ* 

g'  +  y1  '  /»  +  y' 


(^J 


=  1. 


14. 


15. 


x  +  2a     x  —  2a 
2b-x     2b  +  x' 
ab 


tab 
U*-x* 


a  +  b 


+  2a 


ab 
a  +  b 


2a 


~2b--?L 
a  +  b 

Sab  +  2a* 


~2b  +  ^- 
a  +  b 

2a*  +  ab 


tab 
tab  (a+b)* 


ab  +  2b*       Sab  +  2b*     b*(Sa*  +  Sab  +  U*) 
_a(3b  + 2a)      a(2a  +  b)  4ab(a  +  b)* 

b(a  +  2b)       b(Sa  +  2b)     b*(Sa  +  2b)(a  +  2b) 
L.C.D.  =  b*(3a  +  2b)(a  +  2b). 
Ga'b  +  13a*b*  +  6aP  =  first  numerator, 

—  2  a*b  —   5  a*b*  —  2  a&*  -» second  numerator, 

—  4 a*b  —   8 a262  —  4ag  » third  numerator. 
0  =  sum  of  numerators. 


0 
Sum  of  fractions  =  0. 

x  +  y—1 

x-y  +  1 

a  + 1       ab  +  a     « 
ab  +  1     ab  +  1 

a  +  1 
ab+1 

ab  +  a     * 
ab  + 1  + 

2a 

ab  +  1 
2 

ab  +  1 

—  a. 

114  ALGEBRA. 


16. 


1.1,1 


a(a-b)(a-c)     6(6  -  c)  (6  -  a)     c(c  -a)(c-  6) 
1  1,1 


a{a  -  b)(a  -  c)     6(6  -  c) (a  -  6)      c(a  -  c)(6  -  c) 

L.C.D.  =  a6c(a-6)(a-c)(6-c). 

6'c  —  6c*  —  first  numerator, 

—  a^c  +  ac3  =  second  numerator, 

+  ga6  —  a6*  =  third  numerator. 

6*c  —  6c*  —  a*c  +  ac2  +  a2b  —  ab*  =  sum  of  numerators. 

a         -.      ..  6*c  -  6c*  -  a*c  4-  ac2  +  a26  -  a6* 

.•.  Sum  of  fractions  =  — — ~ 

a6c(6*c  —  6c2  —  a*c  +  ac*  +  a*b  —  a6s) 

=  J-. 
abc 


17. 


a_l [ 5_i | c-1 
3a6c  a  6  c 


6c  +  ca  —  ab  1,1. 1 

a     6      c 

Multiply  both  terms  of  the  second  fraction  by  o6c, 

3o6c o6c  —  6c  +  o6c  —  ac  +  o6c  —  06 

6c  +  ca  —  ab  bc  +  ca  —  ab 

3  abc  3  a6c  —  6c  —  ac  —  ab 

be  +  ca  —  ab  be  +  ca  —  ab 

be  +  ac  +  ab 
bc  +  ac  —  ab 


m2  +  n* 


18. 


n 

-xro! 

-n» 

1 
w 

m2- 

1 
m 

-mn 

+  n8 
mn 

y        (m  +  n)(m-n) 

"  m  —  n     (m+  n)  (m*  —  mn  +  n*) 


^teachers'  edition.  115 


U-L- 

1Q    a     b  +  ef-,   ,  b2  +  <?-a'\ 

ml  ~^") 

a     b  +  c 

_(b  +  c  +  a)(2bc  +  b*  +<*-  a?) 

(b+e  —  a)2bc 
_(b  +  c  +  a){(b  +  cy-a*} 

(b  +  c—a)2bc 
__  (b  +  c  +  a)(b  +  c  +  a)(b  +  c  —  a) 

{b  +  c  —  a)2bc 
^(5-fc  +  a)2 
2bc 


20.  3a-[&+{2a-(&-c)}]  +  *  +  ^^l 

2c-f  1 

=  3a-[&  +  2a-&  +  c]  +  J+|^=i 

=  3  a  -  &  -  2  a  +  6  -  c  +  }  +  ?£=i 
2c+l 


1 

=  a- 

-•♦S3 

=  a- 

C  +  i  +  2(2c  +  1) 

—  a~ 

-c  +  J  +  c-J 

=  a. 

21. 

1 
a— a? 

1         X 

a  — y     (a— a;)2     (a 

y 

-y? 

1                          1 

(a-y){a-x)*     (a-x)(a-yf 
(a-x){a-yy-(a-y){a-xy+x(a-yf-y{a-xf 
= (a-a;)2(a--y)> 

s-y 


(a-x?{a-y)* 
«a(2a  — a?  — y). 


o(2o  -  x  -  y)(x  -y)^(g-  a;)*  (a  -  y)2 
{a-x)t{a-yf  x-y 


116  ALGEBRA. 


1 

xl     1 

X    I    ■                     - 

1+£±1 

3-x 

1 

*  +  3-^ 
3  +  1 

4 

4x  +  3  —  x 

4 

3(x  +  l) 

/c_5     <J_&r 

23    (x»-y»)(2^-2gy) 

v      y/     *+y 

(g  +  y)(s-y)(g-y)2s 

"  *(*-y)(*-y)(«+y) 
xy 

2 


L.  C.  D.  1st  expression  —  c8+6*.         L.  C.  D.  2d  expression  =■  <?— ft*. 

c»-2c26+2cft*-6,  =  lstnnm.  c*+2c6+   6s -1st  num. 

— c8 +6f  =  2d  num.  c*         +   6*  =  2d  num. 

— 2c*6+2c6*       =  sum  of  nums.    2c*+2c6+2o*  =  sum  of  nums. 
or,        —  2  c6  (c  —  6)  *»  sum  of  nums.  or,  2  (c*+co+62)  —  sum  of  nums. 

-2c6(c-6)  (c  +  6)(c-6) 

"  (c  +  6)  (c*  -  cb  +  b*) X  2  (<*  +  c6  +  &*) 

"  <<*  +  <jp  +  b* 

-bc(b-c)* 
"M  +  W  +  / 


25.        y       +       *       +     *+y 

(x-y){x-z)     (y-s)(y-«)     (*-*)(*-y) 

=  y x  x+y 

{x-y){x-z)     (x-y)(y-z)     (*-f)(y~s)' 
L.C.D.-(»-y)(*-*)(y-«). 

^         —  y«  =  first  numerator, 
—  jc*         +  xz         =  second  numerator, 

a*  — y* » third  numerator. 

rrz  —  yz «-  sum  of  numerators ; 
or,  z(x—y)=>  sum  of  numerators. 

.•.  Sum  of  fractions  —  -. v   ~~  V) 


(x  -y)(y  - *) (*  - *)    (* - *)(y  - *) 


teachers'  edition.  117 

26.  I + I J- 

a{a  —  b)(a  —  c)     b(b  —  a)  (b  —  c)     abc 

i i l. 

a(a  —  b)(a  —  c)     b(a  —  b)  (b  —  c)     abc 
L.C.D.  r  abc(a  -  b)(b  -  c)(a  -  c). 

—  be2  +  b*c  =  first  numerator, 

—  a?c  +  ac*  =  second  numerator, 

—  be9  —  a*b  +  ab2  —  b*c  +  a*c  —  ac*  =  third  numerator. 


—  a2b  +  ab*  =  sum  of  numerators. 

.-.  Sum  of  fractions ab(a-  b) 

abc(a  —  b)(b  —  c)  (a  —  c) 

1 

c(b  —  c){a  —  c) 


,-4  +  JL         i-!L+! 
27 1+1  x  ^ 


, 6_         (*-l)(*-2) 


*  +  l  «■-! 


s'-s-e       (x-1)  (as -2) 
x~l 

(s-l)(s-2)(s-l)  (g-3)(g  +  a 

>  +  l)(*-3)(*  +  2)  A  (x  +  1)(»-  1)(«-  1)(»-  2) 

1 

"  (*  +  !)»' 


Exercise  LXIII. 

1.5«-*±*-71.  2.  «-£n»-H 

3         3 

Multiply  by  3  ;  then 

3a;-  3  +  x  =  17, 
4*  =20, 
x  =  5. 


2 

Multiply  by  : 

2;  then 

10x-x-2  = 

-142, 

9a?  = 

=  144, 

x  = 

=  16. 

118 


ALGEBRA. 


■  +  Z  =  x  — 


4       "  2 

Multiply  by  4 ;  then 
5  -  2a;  +  8  =  4ar  -  12a?  + 16, 
6a?=3, 


5.   2s-5g-4-7-J-^ 
6  5 

Multiply  by  30 ;  then 
60  a; -25  a; +  20 

=  210-6  + 12a?, 
23  x  =184, 
a;  =  8. 


4. 


5a;     «r>*_9      3-g 
2        4       4        2 
Multiply  by  4  ;  then 
10a?-5a?  =  9-6  +  2a?, 
3a;  =  3, 


x  +  2_14     3  + 5a; 

2  9  4 

Multiply  by  36  ;  then 
18a; +  36  =  56 -27- 45  a?, 
63a?  =  -7, 


7    5a? +3     3-4a?     a;^31     9-5a? 
8  3  2*2  6 

Multiply  by  24  ;  then 

15a?  +  9  -  24  +  32a;  +  12a?  =  372  -  36  +  20a?, 
39a?  =  351, 
a?«9. 


Q    10a?  +  3     6a?-7     in,       1X    ,A    7a;  +  5     5a;-6     8-5a; 
8.  — __10(*-1).    10.  — — . 


3  2 

Multiply  by  6  ;  then 
20  a; +  6- 18a; +  21 

=  60  a; -60, 
58  a;  =  87, 
a;=l*. 


5a?-7      2a;  +  7 


=  3ar-14. 


Multiply  by  12 ;  then 
14a?+10-15a;  +  18  =  8-5a; 
4*  =-20, 
a;  =  ~5. 


11.  s  +  4     x~*  =  2  |  3s-l 


2  3 

Multiply  by  6 ;  then 
15a;-21-4a;-14  =  18a;-84, 
7a;  =  49, 
a? -7. 


3  5        "  '      15 

Multiply  by  15  ;  then 
5a; +  20- 3a; +12 

=  30+3a;-l, 
-  x  -  -  3, 
a?-3. 


12.  3a; +  5      2a;  +  7     iq_3jc^0 
7  3  5 

Multiply  by  105  ;  then 
45a-  +  75  _  70a;-  245  + 1050 -  63a;  - 0, 
-88  a;  =  -880, 
*  =  10. 


teachers'  edition.  119 

13.  }(3s-4)+£(5aH-3)  =  43-5a;.    14.   }(27-2a?)-f— ^(7*-54). 

Multiply  by  21 ;  then  Multiply  by  10  ;  then 

Pa; - 12  +  35s  +  21  135  -  10a?  =  45  -  7x  +  54, 

=  903  -105s,  -3*  =  -36, 

149*  =  894,  x  =  12. 
s  =  6. 


15.  5a;-{8aj-3[16-6a;-(4-5a;)]}  =  6, 
5a;-{8a;- 3  [16 -6a;- 4  + 5a;]}  =  6, 
5  x  -  {8  x  -  48  +  18  x  +  12  - 15  a;}  -  6, 
5a?  -  8a;  +  48  -  18a;  -  12  +  15a; -  6, 
~6a;  =  -30, 
a;  =  5. 


16.  5a;-3_9zi  =  5j     19(       4) 
7  3         2       6  v         ' 

Multiply  by  42 ;  then 

30a>  - 18  - 126  +  14a;  =  105  a;  +  133  a;  -  532, 
-194  a?  =  -388, 
a?  =  2. 


l7   2a?  +  7     9a?-8_a?-ll  lg    8a?-15     lla?-1^7a?  +  2 

'      7             11             2  '3                7       "     13 

Multiply  by  154  ;  then  Multiply  by  273  ;  then 

44a.  +  154  _  126a;  +  112  728a;  - 1365  -  429a;  +  39 

=  77a; -847,  =  147  a; +  42, 

-  159a;  =  - 1113,  152a;  - 1368, 

*-7.  a  =  9. 


1Q    7a;+9     3a;  +  1  __  9a;-  13     249 -9a; 
™'  —*  7  4  14~ 

Multiply  by  56 ;  then 

49a;  +  63-24a;-8  =  126a;-182-996  +  36a;, 
-137  a;  =  -1233, 
a?  =  9. 


120 


ALGEBRA. 


Exercise  LXIV. 


x    9*  +  20     4(*-3)     * 
'        36      "  5*-4      4 
Multiply  by  36  ;  then 

ox— 4 
144(x-3)    .gp 

5a;  — 4 
144* -432  =  100* -80, 
44  a;  =  352, 
*  =  8. 


«    9(2*-3)     11a? -1     9 a; +11 
14  3*  +  l  =       7 


Multiply  by  14  ;  then 
154  x  -14 


3*  +  l 


49. 


18*-27  +  - 
-  18a:  +  22, 
154* -14^ 

3x  +  l 
Divide  by  7, 
22a?-2_H 
3*  +  l 

22*-2  =  21x  +  7, 
x  =  9. 


10a; +  17     12*  +  2_5*-4 

18  13a;- 16        9 

Multiply  by  18  ;  then 
216  a; +  36 


10a;  + 17- 
=  10* -8, 
216  a; +  36 
13a;-  16 


13a; -16 


=  25, 


325  a; -400  =  216* +36, 

109*  =  436, 

*  =  4. 


6*+ 13      3a? +  5  ^2* 
15         5a; -25      5 
Multiply  by  15  ;  then 

8«  +  13-£*±??-8* 


5*-25 
13, 


45*  +  75_ 
5*-25  " 
45  a; +  75  =  65* -325, 
-20*  =  -400, 
*  =  20. 


5. 


18* -22  ^0^1  + 16* 

+  4S  + 


■*A- 


101-64* 


39-6*  24  l*  24 

Reduce  the  mixed  number  to  an  improper  fraction, 
18*-22    ,o-.l  +  16a?     53     101-64* 
12  24 


3(13-2*)  '  24 

Multiply  by  24  ;  then 

8(18* -22)     4ga,     1      16a;  =  106_  101  +  64a. 
13-2* 

144*  - 176     d 

13-2* 
144* -176  =  52 -8*. 
152*  =  228, 
*  =  1J. 


teachers'  edition. 


121 


6_5x       7-23* 


l  +  3s     10x-ll       1 
21  30  105* 


15         14(x-l) 
Multiply  by  210  ;  then 

84-70x-  105  ~  *0x*  =  10  +  30x-  70a;  +  77+2, 


x-1 
105-30x» 


.  30a;  +  5, 


x-1 

-  105  +  30  a?  =  30  x1  -  25  a;  -  5, 
25  a;-  100, 
x  =  4. 


7    9x+5     8a;-7^36a;+15    41, 
14       6X+2"      56         56* 
Multiply  by  56  ;  then 

36x  +  20+224*-196 

3a; +  1 
=  36  a; +  15 +  41, 
224  a; -196      ™ 
3x  +  l      =36' 
224  a; -196  =  108  a? +36, 
116  a;  =  232, 
a;  =  2. 


9    6a?  +  l      2x-4 
15         7x-16  = 


2x-l 


Multiply  by  15 ;  then 
ft^u.1      30  a; -60      ~       « 
6aJ  +  1-7^i6-  =  6aJ-3' 
30  a; -60 
7a; -16 
-  30x  +  60  =  - 
-2x  =  4, 
x  =  -2, 


28  a; +  64, 


6x+7     2x-2^_2x  +  l 

15         7x-6  5 

Multiply  by  15  ;  then 

6,  +  7-^^=6,  +  3, 
7a;  — 6 

30a; -30^      ^ 
7x-6 
-30x  +  30  =  -28x  +  24, 
-2x  =  -6, 
x  =  3. 


10. 


7x-6        x-5 


35         6a; -101     5 
Multiply  by  35 ;  then 

7x-6-35*-175  =  7x. 
6x-101 

Transpose,  and  clear  of  frac- 
tions, 
-35  a; +  175  =  36a; -606, 
-71  x  =  -781, 
x  =11. 


Exercise  LXV. 


1.  ax  +  bc  =  bx  +  ae% 
ax—bx  =  ac  —  bc1 
x(a  —  b)^c(a~  b), 
x-c. 


2a  —  ex  =  3  c—  56x, 
5bx  —  cx  =  3  c  — 2a, 
x(56-c)=3c-2a, 

3c-2a 

x  = • 

56-c 


122  ALGEBRA. 


3.  a*x  +  bx  —  c  =  b*x  +  cx—  dt 
a*x  —  b2x  +  bx  —  cx  =  c  —  dt 
x(a2  —  b*  +  b  —  c)  =  c  —  d, 

—         c~d 

a*  _  &»  +  b  -  c 


4.  —  aci  +  b*c  +  obex  =  abc  -f  cmx  —  ac*x  +  b*c  —  mc, 
obex  —  cmx  +  ac*x  =  abc  +  b*c  —  mc+ac*  —  6*c, 
x(abc  —  cm  +  ac*)  =  abc  —  mc  +  ac2, 
x  =  l. 


5.  (a  +  a;  +  6)(a  +  b  -  a)  « (a  +  x) (b  -  x)  -  abt 
—  x2  +  a2  +  2a&  +  62  =  ab  -f  6x  —  ax  —  x2  —  ab, 
ax  —  6x  =  -  a2  —  2ab  —  6*. 
ax  -  bx  =  -(a2  +  2ab  +  62), 

a-6 


6.  (a2  +  x)2  =  x2  +  4a2+a4,  q    affls+s8) ,  ax2 

a*  +  2a2x  +  x2  =  x2  +  4a2  +  a*l      *         te        -«*+y 

2a2x  =  4a2,  ...  - 

x  =  2  Divide  by  a ;  then 

62s+x»      ^  ,  i» 
— frx"^' 

7.  (^-.rXa2+x)  =  (a*+2ax-x2)f  Uxxltl^  **  **• 

a*  -x2  -  a*  +  2ax~  x2,  62x  +  x8  =  frcx2  +  x8, 

-  2  ax  =  0,  62x  =  fccx2, 

*-0.  x  =  0or_. 


ax  —  b  x-\-  ac 

■  +  a  = 


c  c 

Multiply  by  c ;  then 

ax  —  6  -f  ac  =  x  4-  ac. 

ax  —  x  =  6, 
•(a-l)-J, 

a  — 1  2a  +  6 


3a  -bx     1 
aa- _ — =_. 

2          2 

Multiply  by  2, 

2ax  — 3  a  +  6x  = 
2ax  +  6x  = 
x(2a  +  6)  = 

1, 

3a +  1 
3a +  1 

x  = 

3a+l 

teachers'  edition.  123 


11    a„     4as  — 26__  ab+x     V*—x     x—b     ab—x 

18a-4ax  +  26  =  3aj,  a86  -f  a2x  —  b*  +  bx 

-3s-  4ax  =  -18a~26,  -  b2x  -  68  -  a*b  +  a*x, 

*(3  +  4a)  =  2(9a  +  b),  -b*x  +  bx  =  -  a*b  -a86, 

_2(9a  +  6)  6x(6-l)  =  2a86, 

3  + 4a  '  Xr_  2a8 


6-1 


12. 


a—x     2x     a 


/-a-ax  +  *  =  2*-db,      17.  ax  -  ^±2  _  ^^H 


—  aa>  =  —  a6  +  a, 


x  x 


jP-aft  —  1.  arc2  — 6a;  — l  =  aa^  — a, 

_  6x  =  —  a  -f  1, 

1Q    3     ab—x2     ±x  —  ac  «  =  —-—. 

lo. = & 

c        bx  ex 

3  bx  —  abc  +  ex2  =  4  6a?  —  a6c, 

ex2  =  6x,  -^  __ 

■    b  18.  -^L+a  +  ^-O. 
a?  =  Oor-.  b-cx  c 

acaP  +  abc—aePx  +  abx—acx2 
=  0. 
14.  am-  b  -  y  +  -  =  0.  Divide  by  a, 

a6m'-6'm-amlC  +  6a;  =  0>  ^  +  6c-A  +  te-rf-0, 


6x  —  a7nar  =  b2m  —  a6m* 

6*771  —  a6m* 

a?  =  — 

6  — am 

—  bm. 


be 
c*-6 


15. 


19.  ^-fc  +  rf  +  L 
3as-26     oa;-a_aa;    2  *  * 

36  26        6     3'  .         o6-to  +  <fo  +  l, 

—  6ca?  —  ax  =  —  a6  +  1, 


6  ax— 4  6—3  aa;-f  3  a  =  6  oa;— 4  6, 
— 3aa:=—  3a, 


a6-l 


x=l.  be  +  d 

20.  «Ml±^)  =  ac  +  ^. 
ax  a 

ad2  +  ax*  =  acda?  +  ox*, 

aedx  =  ad2, 

d 

c 


124 


ALGEBRA. 


x-5       1 


s-3 

4(s-l)     6(*-l)  "  9 
Clear  of  fractions, 

9x-  27  =  6a;  -30+  4s  -4, 
_«  — 7, 
*=7. 


Exercise  LXVI. 

A  1 


1 


x-1 


2(x-3)    3(x-2)    (*-2X*-3) 

Clear  of  fractions, 

3*-6-2x  +  6  =  6x-6, 
-5x  =  -6, 
x=l*. 


x  +  _*_  _  («-2)(g  +  4) 

x  — 1  x+1 

Clear  of  fractions, 

x*-x+x*+x  =  xN-a^-lOx+S, 
10a;  =  8, 


g    ,     2(2g+3)     J> 5s+l 

9(7-*)  =  7-x    4(7-*)' 
Clear  of  fractions, 

252-36x-16x-24 

=  216 -45  a; -9, 
-7x=-21, 
*=3. 


7     _^6s+l     3(l+2x») 
a;  — 1       x  +  1         x*  —  l 
Clear  of  fractions, 

7x+7 = 6  **+*-6  *-l-3-6  a*, 
12*= -11, 


6.  -iL-.4  =  5(21  +  2*)-10. 
x  +  Z  3x  +  9 

Clear  of  fractions, 
51-12x-36 

=  105+ 10a; -30*- 90, 
8a;  =  0, 
*  =  0. 


7. 


*-7_^2x-15  1 

a;  +  7=*  2x-6      2(x  +  7) 
Clear  of  fractions, 

2x*-20x  +  42  =  2a?  -x-  105-s  +  3, 
-18x  =  -144, 
x-8. 


8. 


x  +  4  ,,  3x  +  8 
3x  +  5  *  2x  +  3 
Clear  of  fractions, 

12*«  +  66*  + 72  +  36  *,  +  114*+90  +  6x,  +  19s  +  15 
=  54x*  +  134x  +  240, 
-35* -63,        ^ 


*»-lf        \ 


teachers'  edition.  125 

9    132£+1     8aH-5  =  52  n    3x-l     4x-2_l 

3x+l        x-l  '   2x-l      3x-2     6* 

Clear  of  fractions,  Clear  of  fractions, 

132x*-131x-l+24x*+23x+5  54  x*-54x+ 12-48  x»+48x-12 

=  156x*-104x-52,  =  6x*-7x+2, 

-4x  =  -56,  x  =  2. 


x  =  14. 


1  6  12. 


3        x  +  l        x* 


10*   o 5  + o  =  5— To  '   x-l      X-l      1-3*' 

2x  — 3     x—  2     3x  +  2 

.„        3         x  +  l  _  —  x2 

Clear  of  fractions,  or>    x_\  "~  x_i  ""  a?— l' 

6x*-8a;-8  +  6aj2-5a;-6  *  ni         ,r     .. 

=  12a?  -  42*  +  36,  Clear  of  f™tlons. 

29x  =  50,  3aj  +  3-aa-2a?-l  =  -a!1, 

s=ljf           "  x  =  -2. 

13  g~4  _ ft  — 5  __  x  —  7 _ x  —  8 
x  —  5  x—  6  x  —  8  x  —  9 
Then 

(x-4)(x-6)     (x-5)(x-5)_(x-7)(x-9)     (x-8)(x-8) 
(x-5)(x-6)     (x-5)(x-6)     (x-8)(x-9)      (x-8)(x-9)' 

- 1  _  - 1 

(x-5)(x-6)     (x-8)(x-9)' 
Clear  of  fractions, 

-x2  +  17x-72  =  -x2  +  llx-30, 

6x  =  42, 

«-7. 

14.  (x-a)(x-b)  =  (x-a-b)2, 

tf-ax  —  bx  +  ab^x2-  2ax  —  2bx  +  a2  +  2ab  +  6\ 
ax  +  bx  =  a2  +  ab  +  b2, 

m     a2+ab  +  b2 
x  = _ 

a+b 


15.  (a-  b){x-  c)-(b  -  c)(x-a)-(c  -  a)(x  -  b)  =  0, 

ax  —  bx  —  ac  +  6c  —  bx  +  ex  4-  ab  —  ac  —  ex  +  ax  +  be  —  ab  =  0, 

ax—  bx  —  bx  +  ex  —  ex  +  ax  =  ac  —  be  —  ab  +  ac  —  be  +  ab, 

2ax  —  2bx  =  2ac  —  26c, 

2x(a-6)  =  2c(a-6), 

x  =  c. 


126  ALGEBRA. 


16.   ^-2+1  ,  **  +  x  +  l-2x  17.      4      |      7     -         37 

a;-l  a;  +  l         "  '  "   s  +  2     a;  +  3     aj*  +  5a;  +  6 

Clear  of  fractions,  Clear  of  fractions, 

a*  +  1  +  a*  -  1  -  2ar*-  2x,  4a;  +  12  +  lx  +  14  =  37, 
2a;  =  0,  11*  =  11, 

a;  =  0.  »  *  1. 

18.  (*+!)». *[6-(l-a0]-2, 
(a;+l)2  =  a;(6-l  +  a;)-2, 
a?  +  2a;+l  =  6a;-a;  +  a?-2, 
-3a;  =  -3, 
*-l. 

19.  ^-i^  +  ^  +  H— JL  +  5. 

a;  +  1  3a;+  2       x  +  1 

Reduce  the  complex  to  simple  fractions, 

75  -  x    ,    80a;  4-  21        23     .  K 
—  ■  +  5. 


20. 


3(a;  +  l)     5(3a;  +  2)     x  +  l 

Clear  of  fractions, 

1115a;  -  15s2  +  750  +  240a?  +  303a;  +  63 

=  1035  a;  +  690  +  225  x2  +  375  a;  +  150, 
8a;  =  27, 
*  =  3}. 

Sabc        a262        {2a-\-b)bix  =  3ca,  ,  &£ 
a  +  6     (a  +  6)8        a(a+6)2  a 

Clear  of  fractions, 
3  a46c+6  a862c+3  a268c+a8&2+2  a262a;+3  a&8a;+&4a; 

=  3  a4ca;+9  as6ez+9  a262ca;+3  aP'caj+a^aj+S  cWg+SoPoH-ftc, 
3  a4ca;  +  9  a3te  +  9  a262ca;  +  3  ab*cx  +  a36a;  +  a2b2x 

=  3  a46c  +  6  a*b2c  +  3  cWc  +  a862, 
aa;  (3  a8c  +  9  a2bc  +  9  ab2c  +  3  63c  +  a2b  +  ab2) 

=  a26  (3  a2c  +  6  a&c  +  3  b2c  +  aft), 
a;{3c(a  +  6)3  +  a&(a  +  &)}=a&{3c(a+&)2  +  a&}. 

a;  = -• 

a+b 

21    -±-+_? 29  =  ^—     22.'  5-J7-gV*-3aH4-5'> 

a>-8     2x-16     24     3x-24  V2    */     2  4 

Clear  of  fractions,  5_??_i_2  =  -  —  3a?~  4 +5  a; 

96 +  36 -29a; +  232  =  16,  2  2  4 

-  29  a;  =  -  348 ,  clear  of  fractions, 


x  =12. 


20-14a;+8  =  2a?-3a+4-5a?J 
-8  a;  =  -24, 
x  =  3. 


teachers'  edition.  127 

1 3  1  -  ar, 

5     x-1  3 

Multiply  both  terms  of  right  member  by  l—x\  then 

1        3  6-s 

5     x-1     3(1- a)' 

1         3     _    s-6 

5     a?-l      3(a;-l)' 

Clear  of  fractions, 

3aj-3-45  =  5s-30, 

-2a;  =  18, 
x  =  -  9. 


24.     x~i    +    «-f    ,1  + I - 

|(*-1)     f(x  +  l)  15(l-I^ 

Reduce  the  complex  to  simple  fractions, 
2aj-3  ,  2o-5  =  1  a? 


3a;-3     dx  +  5  15a?-15 

Clear  of  fractions, 

103s  -5s  -15  +  6a*  -21  a;  +  15  =  15a?-15  +  a* 
-26  a;  =  -15, 


Exercise  LXVII. 


1.  Find  the  number  whose  third  and  fourth  parts  added  to- 

gether make  14. 

Let 

x  =  the  number. 

Then 

-  =  one-third  of  the  number, 
3 

and 

-  =  one-fourth  of  the  number, 
4 

and 

-  +  -  =  sum  of  the  two  parts. 
3     4                               r 

But 

14  =  sum  of  the  two  parts. 

•" 

-.5  +  5-14.    Whence,  x  =  24. 
3     4 

128  ALGEBRA. 


2.  Find  the  number  whose  third  part  exceeds  its  fourth  part 
by  14. 


Let 

x  =  the  number. 

Then 

-  =  one- third  of  the  number, 
3 

and 

-  =  one-fourth  of  the  numbe 
4 

and 

X 

3 

-  7  =  the  excess. 
4 

But 

14  =  the  excess. 

.^-^H.    Whence,  x  =168. 
3     4 


3.  The  half,  fourth,  and  fifth  of  a  certain  number  are  together 
equal  to  76 ;  find  the  number. 

Let  x  —  the  number. 

Then  -  =  one-half  of  the  number, 

2 

and  -  =  one-fourth  of  the  number, 

4 

-  =  one-fifth  of  the  number, 
5 

-  +  -  +  -*=  sum  of  the  parts. 
2     4     5 

But  76  —  sum  of  the  parte. 

...*  +  !+  5  -  7&    Whence,  x  -  80. 


4.  Find  the  number  whose  double  exceeds  its  half  by  12. 

Let     '       x  —  the  number. 

Then  -  =  one-half  the  number, 

2 

and  2x  =  double  the  number, 

2a;  — -  =  the  excess. 
2 

But  12  =  the  excess. 

.-.  2x  - 1  =  12.    Whence,  x  -  8, 
2 


teachers'  edition.  129 

5.  Divide  60  into  two  such  parts  that  a  seventh  of  one  part 
may  be  equal  to  an  eighth  of  the  other. 

Let  x  =  one  part, 

and  60  —  x  =  the  other  part. 

Then  -  =  one-seventh  of  one  part, 

7 

and  — ^-^  =  one-eighth  of  the  other  part, 

o 

60  -  x  _  x 

"'*       8  7 

Whence,  re =28, 

and  60-x=32. 


6.  Divide  50  into  two  such  parts  that  a  fourth  of  one  part 
increased  by  flve-sixths  of  the  other  part  may  be  equal  to  40. 

Let  x  =  the  smaller  part. 

Then      50  —  x  —  the  larger  part, 

7  +  J(50—»)  —  J  of  one  part  increased  by  4  of  the  other. 
4 

But  40  =  J  of  one  part  increased  by  J  of  the  other. 

■'.f  + 1(50-*) -40. 
4 

Whence,         x  —  2$, 

and  50-a;  =  47f 


7.  Divide  100  into  two  such  parts  that  a  fourth  of  one  part 
diminished  by  a  third  of  the  other  part  may  be  equal  to  11. 

Let  x  =  one  part. 

Then    100  -  x  « the  other. 

jP^  =  J  of  one  part  diminished  by  J  of  the  other. 

But  11  =  J  of  one  part  diminished  by  J  of  the  other. 

•     X  1Q0  ~  X  r-  11 

'  "4  3 

Whence,         x  =76, 
and  100-a;  =  24. 


130  ALGEBRA. 


8.  The  sum  of  the  fourth,  fifth,  and  sixth  parts  of  a  certain 
number  exceeds  the  half  of  the  number  by  112.  What  is  the 
number  ? 

Let  x  =  the  number. 

Then  -  =  one-half  of  the  number, 

2 

-  —  one-fourth  of  the  number, 
4 

\  —  one-fifth  of  the  number, 
5 

-  =  one-sixth  of  the  number. 
6 


...?  +  fE  +  ?  =  ii2  +  ?. 
4     5     6  2 

Whence,         a?  =  960. 


9.  The  sum  of  two  numbers  is  5760,  and  their  difference  is 
equal  to  one-third  of  the  greater.    What  are  the  numbers  ? 

Let  x  =»  the  greater  number. 

Then  5760  —  x  —  the  smaller  number, 
a? -(5760- a;)  =  |- 

.\  3a;  - 17,280  +  3x  =  x. 
Whence,        x  =  3456, 
and        5760-  x  =2304. 

10.  Divide  45  into  two  such  parts  that  the  first  part  divided 
by  2  shall  be  equal  to  the  second  part  multiplied  by  2. 

Let  x  =  first  number. 

Then      45  —  a:  =  second  number, 

-  =  first  divided  by  2, 

2  J 

90  —  2  a;  =  second  multiplied  by  2. 

Then  -  =  90-2ar. 

2 

.-.  x  =  180- 4a;. 

Whence,         x  =  36, 

and  45  —  x  =  9. 


TEACHERS*    EDITION.  131 


11.  Find  a  number  such  that  the  sum  of  its  fifth  and  its 
seventh  parts  shall  exceed  the  difference  of  its  fourth  and  its 
seventh  parts  by  99. 

Let  x  =  the  number. 

Then  -  =  one- fifth  of  the  number, 

5 

-  =  one-fourth  of  the  number, 
4 

^  =  one-seventh  of  the  number, 

?  +  ^  =»  sum  of  i  and  \  of  the  number, 
5     7 

?  _  ?  =  difference  between  J  and  \  of  the  number. 

(-+-]—(-—  -J138  the  excess  of  the  sum  of  its  fourth  and 
*        '/      \        '/  seventh  parts  over  the  difference  of  its 

fourth  and  seventh  parts. 
But  99  =  this  excess. 

-H)-(h)-w- 

Whence,     x  «=  420. 


12.  In  a  mixture  of  wine  and  water,  the  wine  was  25  gallons 
more  than  half  of  the  mixture,  and  the  water  5  gallons  less  than 
one-third  of  the  mixture.  How  many  gallons  were  there  of 
each? 

Let  x  =  number  of  gallons  in  mixture. 

.  Then     -  +  25  =  number  of  gallons  of  wine, 
-  —  5  =  number  of  gallons  of  water, 
-  +  25  +  -  —  5  =  number  of  gallons  in  mixture. 

It  O 

.•  *  +  25  +  £- 
2  3 

Whence,        x  =  120, 

and  ^  +  25  =  85,   £-5  =  35. 

2  3 


132  ALGEBRA. 


13.  In  a  certain  weight  of  gunpowder  the  saltpetre  was  6 
pounds  more  than  half  of  the  weight,  the  sulphur  5  pounds  less 
than  the  third,  and  the  charcoal  3  pounds  less  than  the  fourth  of 
the  weight.    How  many  pounds  were  there  of  each? 

Let  x  =  number  of  pounds  in  mixture. 

Then   -  +  6  =  number  of  pounds  of  saltpetre, 

-  —  5  =  number  of  pounds  of  sulphur, 

and         --3  =  number  of  pounds  of  charcoal. 
4 

2  3 

Whence,  z  =  24,   |+6  =  18,   |-5  =  3,  |  —  3  =  3. 

14.  Divide  46  into  two  parts  such  that  if  one  part  be  divided 
by  7,  and  the  other  by  3,  the  sum  of  the  quotients  shall  be  10. 


Let  x  =  first  part. 

Then  46  —  x  =  second  part, 

and^46-*-- 


3         7 
Whence,     x  =  18,     and  46  -  x  «  28. 

15.  A  house  and  garden  cost  $  850,  and  five  times  the  price  of 
the  house  was  equal  to  twelve  times  the  price  of  the  garden. 
What  is  the  price  of  each? 

Let  x  =  number  of  dollars  the  house  cost, 

and      850  —  x  =  number  of  dollars  the  garden  cost. 
Then        5&  =  five  times  cost  of  house, 
10,200— 12 &  =  twelve  times  cost  of  garden. 

.-.5  x  =10,200  -12  x. 
Whence,     x  =  600,     and  850  -  x  =  250. 

16.  A  man  leaves  the  half  of  his  property  to  his  wife,  a  sixth 
to  each  of  his  two  children,  a  twelfth  to  his  brother,  and  the 
remainder,  amounting  to  #600,  to  his  sister.  What  was  the 
amount  of  his  property? 


teachers'  edition.  13S 

Let  x  —  number  of  dollars  the  property  amounted  to 

Then  -  =  number  of  dollars  left  to  wife, 

2 

-  =  number  of  dollars  left  to  each  child, 
6 

^-  =  number  of  dollars  left  to  brother. 
12 

*  +  *  +  *  +  JL+600  =  number  of  dollars  in  all. 

2     6     6      12 

But  x  =  number  of  dollars  in  all. 

...!  +  §  +  §  +  JL  +  600  =  *. 
2     6     6     12 
Whence,  x  =  7200. 

17.  The  sum  of  two  numbers  is  a  and  their  difference  is  b ; 
find  the  numbers. 

Let  x  =  the  smaller  number. 

Then    x  +  b  =  the  larger  number, 

2  x  +  b  =  the  sum  of  the  numbers. 
But  a  =  the  sum  of  the  numbers. 

.•.  2x  +  b  =  a. 
Whence,     rr  =  ^,    andz  +  &  =  ^±i. 


18.  Find  two  numbers  of  which  the  sum  is  70,  such  that  the 
first  divided  by  the  second  gives  2  as  a  quotient  and  1  as  a 
remainder. 

Let  x  =  first  number, 

and        70  —  x  =  second  number. 

Then  -^1  =  2. 

70 -a? 
Whence,     x  =  47,     and  70  -  x  =  23. 

19.  Find  two  numbers  of  which  the  difference  is  25,  such  that 
the  second  divided  by  the  first  gives  4  as  a  quotient  and  4  as  a 
remainder. 

Let  x  =  smaller  number. 

Then  x  +  25  =  larger  number, 

x  +  25  4 

=  4  +  - 

x  x 

Whence,     x  =7,    and  x  +  25  =  32. 


134  ALGEBRA. 

20.  Divide  the  number  208  into  two  parts  such  that  the  sum 
of  the  fourth  of  the  greater  and  the  third  of  the  smaller  is  less 
by  4  than  four  times  the  difference  of  the  two  parts. 

Let  x  =  the  greater  part. 

Then    208  —  x  =  the  smaller  part, 

-  + — ^  =  sum  of  \  the  greater  and  £  the  small""". 

4  3 

x  —  (208  —  x)  =-  difference  of  the  parts. 

...£  +  20^-?  +  4  =  4(iE_208+a!). 
4  o 

Whence,         x  =  112,        and  208  -  x  =  96. 

21.  Find  four  consecutive  numbers  whose  sum  is  82. 

Let  x  =  first  number. 

Then        x  -f  1  =  second  number, 

x+  2  =  third  number, 

x  +  3  =  fourth  number. 
Then        x  +  x  +  l  +  x  +  2  +  x+3*=  sum  of  the  numbers. 
But  82  =  sum  of  the  numbers. 

.-.  x  +  x  +  1  +  x  +  2  +  x  +  3  =  82. 
Whence,  s=19,  s  +  l  =  20,  s  +  2  =  21,  s  +  3  =  22. 

22.  A  is  72  years  old,  and  B's  age  is  two-thirds  of  A's.  How 
long  is  it  since  A  was  five  times  as  old  as  B? 

Let "  x  =  number  of  years  since  A's  age  was  five  times 

that  of  B. 
J  of  72  =  48  =  B's  age  at  present, 
72  —  x  =  A's  age  x  years  since, 
48  —  x  =  B's  age  x  years  since. 
Then      72  -  x  =  5  (48  -  a). 
Whence,         a;  =  42. 

23.  A  mother  is  70  years  old,  her  daughter  is  half  that  age. 
How  long  is  it  since  the  mother  was  three  and  one-third  times  as 
old  as  the  daughter? 

Let  x  =  number  of  years  since. 

Then      70  —  x  =  mother's  age  x  years  since, 
35  —  x  =  daughter's  age  x  years  since. 
.-.  70-s  =  3J(35-3). 
Whence,         x  =  20. 

24.  A  father  is  three  times  as  old  as  the  son ;  four  years  ago 
the  father  was  four  times  as  old  as  the  son  then  was.  What  is 
the  age  of  each? 


teachers'  edition.  135 

Let  x  —  number  of  years  in  son's  age. 

Then  3x  =  number  of  years  in  father's  age, 

x  —  4  =  number  of  years  in  son's  age  4  years  since, 
3  a;  —  4  =  number  of  years  in  father's  age  4  years  since. 
.-.  3s- 4  =  4a; -16.    Whence,  x  =  12,  and  3a;  =  36. 

25.  A  is  twice  as  old  as  B,  and  seven  years  ago  their  united 
ages  amounted  to  as  many  years  as  now  represent  the  age  of  A. 
Find  the  ages  of  A  and  B. 

Let  x  =  number  of  years  in  B's  age. 

Then  2x  =  number  of  years  in  A's  age, 

x  —  7  =  number  of  years  in  B's  age  7  years  since, 
2x  —  7  =  number  of  years  in  A's  age  7  years  since. 
.\  a;- 7+ 2a;- 7  =  2a?.     Whence,  a;  =14,  and  2a;  =  28. 

26.  The  sum  of  the  ages  of  a  father  and  son  is  half  what  it 
will  be  In  25  years ;  the  difference  is  one-third  what  the  sum  will 
be  in  20  years.     What  is  the  age  of  each? 

Let  x  —  number  of  years  in  father's  age. 

Then      50  —  x  =  number  of  years  in  son's  age, 

x  —  (50  —  x)  —  difference  of  their  ages. 
But  (x  +  20)+(50-*)+20  _  difference  of  their  ageg 

.  „     CA  ,  _     x  +  20  +  60-  x  +  20 

a     .     X  ~~  tJ\J    "T    X   —  ' 

Whence,  x  =  40,  and  50  -  x  =  10. 

27.  A  can  do  a  piece  of  work  In  5  days,  B  in  6  days,  and  C  in 
7\  days ;  in  what  time  will  they  do  it,  all  working  together? 

Let  x  —  number  of  days  required  for  A,  B,  and  C, 

together. 

Then  -  =  part  all  can  do  in  one  day. 

But  -  =  part  A  can  do  in  one  day, 

5 

-  =  part  B  can  do  in  one  day, 

2 

—  =  part  C  can  do  in  one  day. 
15 

112 
Then-  +  -  +  —  =  what  all  can  do  in  one  day. 

But  -  =  what  all  can  do  in  one  day. 

a? 

.•.i  +  I  +  A  =  l.    Whence,  x  =  2. 
5     6      15     x 


136  ALGEBRA. 


28.  A  can  do  a  piece  of  work  in  2\  days,  B  in  3}  days,  and  C 
in  3 J  days ;  in  what  time  will  they  do  it,  all  working  together? 

Lot  x  ■---  number  of  days  required  for  A,  B,  and  C, 

together. 

Then  -  =  part  they  can  do  in  one  day. 

Now  —  =  part  A  can  do  in  one  day, 

—-  =  part  B  can  do  in  one  day, 

-—  =  part  C  can  do  in  one  day. 

Then  ~-  +  —  +  --  =  part  all  can  do  in  one  day. 

2y      3£     3J 

But  -  =  part  all  can  do  in  one  day. 

x     2^3J^3| 
Whence,  x  =  l^y. 

29.  Two  men  who  can  separately  do  a  piece  of  work  in  15 
days  and  16  days,  can,  with  the  help  of  another,  do  it  in  6  days. 
How  long  would  it  take  the  third  man  to  do  it  alone? 

Let  x  =  number  of  days  required  for  third  man. 

-  +  —  4-  — -  =  part  all  can  do  in  one  day. 
x     15      16     r  J 

But  -  =  part  all  can  do  in  one  day. 

.-.  i  +  —  +  -i  =  i-      Whence,  x  =  26J. 
x     15      16      6  '  * 

30.  A  can  do  half  as  much  work  as  B,  B  can  do  half  as  much 
as  C,  and  together  they  can  complete  a  piece  of  work  in  24  days. 
In  what  time  can  each  alone  complete  the  work? 

Let  x  •-=  number  of  days  C  works. 

Then  2x  =  number  of  days  B  works, 

4  x  =  number  of  days  A  works. 

Then  -  +  —  -\ =  part  all  can  do  in  one  day. 

x      2x     4a: 

But  —  =  part  all  can  do  in  one  day. 

.-.  -  +  —  +  i  =  — .     Whence,  s=  42,  2x  =  84,  and  4x -  168. 

x     2x     4tx     21  ' 


teachers'  edition.  137 

31.  A  does  $  of  a  piece  of  work  in  10  days,  when  B  comes  to 
help  him,  and  they  finish  the  work  in  3  days  more.  How  long 
would  it  have  taken  B  alone  to  do  the  whole  work? 

Let  x  =  number  of  days  required  for  B. 

Then  -  =  part  B  can  do  in  one  day, 

— -  =  part  A  can  do  in  one  day, 

-  —  part  left  to  be  finished, 

9     r 

14       4 

-  of  -  or  —  =»  part  both  can  do  in  one  day. 

3      9      27     r  J 

But        -r  +  -  =  part  both  can  do  in  one  day. 
18     x 

...±+I  =  ± 

18     a     27  # 

Whence,         x  =  10}. 


32.  A  and  B  together  con  reap  a  field  in  12  hours,  A  and  C  in 
16  hours,  and  A  by  himself  in  20  hours.  In  what  time  can  B  and  C 
together  reap  it?    In  what  time  can  A,  B,  and  C  together  reap  it? 

—  =  part  A  and  B  can  do  together  in  one  hour, 

and  —  =  part  A  can  do  in  one  hour; 

20     r 

.: or  —  =  part  B  can  do  in  one  hour, 

12     20      30     r 

—  =  part  A  and  C  can  do  together  in  one  hour. 
16 

.-. or  —  =  part  C  can  do  in  one  hour. 

16     20      80     v 

Let  -  =  part  A,  B,  and  G  can  do  together  in  one  hour. 

Then         '    I-JL  +  2.  +  JL 
x     20     30     80 

Whence,         x  =*  10$}.    B  and  C  together  in  21T\  hours. 


138  ALGEBRA. 


33.  A  and  B  together  can  do  a  piece  of  work  in  12  days,  A  and 
C  in  15  days,  B  and  C  in  20  days.  In  what  time  can  they  do  it, 
all  working  together? 

Let  x  —  number  of  days  required  working  together. 

—  =  part  A  and  B  do  in  one  day, 

—  «  part  A  and  C  do  in  one  day, 

—  =  part  B  and  C  do  in  one  day. 

Then  —  +  —  +  —  =  part-all  do  in  two  days. 

2 
But  -  =  part  all  do  in  two  days. 

...Ll  +  i  +  JL 

x     12     15     20 


Whence,         x  =  10. 


34.  A  tank  can  be  filled  by  two  pipes  in  24  minutes  and  30 
minutes  respectively,  and  emptied  by  a  third  in  20  minutes.  In 
what  time  will  it  be  filled  if  all  three  are  running  together? 

Let  x  =  number  of  minutes  required  for  all  running 

together, 

-  =»  part  filled  by  all  in  one  minute, 


—  =  part  filled  by  first  in  one  minute, 


24 

—  =  part  filled  by  second  in  one  minute, 

30     l  J 

—  =  part  emptied  by  third  in  one  minute, 

—  + =  part  filled  by  all  in  one  minute. 

24     30     20     *  J 

But  _  =  part  filled  by  all  in  one  minute. 

.  i  =  j,  J J_ 

"  x     24     30     20* 

Whence,         x  =  40. 


teachers'  edition.  139 

35.  A  tank  can  be  filled  in  15  minutes  by  two  pipes,  A  and  B, 
running  together.  After  A  has  been  running  by  itself  for  5  min- 
utes, B  is  also  turned  on,  and  the  tank  is  filled  in  13  minutes 
more.    In  what  time  may  it  be  filled  by  each  pipe  separately? 

Let  x  =  number  of  minutes  required  for  A. 

Then  -  =  part  filled  by  A  in  one  minute, 

18 
and  —  =  part  filled  by  A  in  eighteen  minutes, 

x 

=  part  filled  by  B  in  one  minute, 

15     x 

13     13 

=>  part  filled  by  B  in  thirteen  minutes. 

15      x 

.^  18  +  13_13_1 

x      15      x 

Whence,        x  =  37J. 

Therefore,  it  can  be  filled  by  A  in  37£  minutes,  and  by  B  in 

25  minutes. 

36.  A  cistern  could  be  filled  by  two  pipes  in  6  hours  and  8 
hours  respectively,  and  could  be  emptied  by  a  third  in  12  hours. 
In  what  time  would  the  cistern  be  filled  if  the  pipes  were  all  run- 
ning together? 

Let  x  =  number  of  hours  required  for  all  running 

together, 

-  =  part  all  can  fill  in  one  hour, 
x 

-  =  part  filled  by  first  pipe  in  one  hour, 
6 

%  -  =  part  filled  by  second  pipe  in  one  hour, 

8 

— -  =  part  emptied  by  third  pipe  in  one  hour. 

La 

Then-+ — — -  =part  filled  by  all  in  one  hour. 

6    8     12     r  J 

But  -  =  part  filled  by  all  in  one  hour. 

.   M      1__J_ 
"  x     6     8     12 
Whence,         x  =  4$ . 


140  ALGEBRA. 

37.  A  tank  can  be  filled  by  three  pipes  in  1  hour  and  20  min- 
utes, 3  hours  and  20  minutes,  and  5  hours,  respectively.  In 
what  time  will  the  tank  be  filled  when  all  three  pipes  are  running 
together? 

Let  x  =  number  of  minutes  required  for  all  to  nil  it, 

—  =  part  first  will  fill  in  one  minute, 
80     r 

=  part  second  will  fill  in  one  minute, 

200     r 

=  part  third  will  fill  in  one  minute, 

300     r 

-  =  part  all  will  fill  in  one  minute. 
x 

x     80     200     300 
Whence,         x  =  48. 


38.  If  three  pipes  can  fill  a  cistern  in  a,  b,  and  c  minutes, 
respectively,  in  what  time  will  it  be  filled  by  all  three  running 
together? 

Let  x  =  number  of  minutes  required  for  all. 

Then  -  —  part  first  fills  in  one  minute, 

I  • 

-  =  part  second  fills  in  one  minute, 
o 

-  =  part  third  fills  in  one  minute, 

-  +  r  +  -  =  part  all  fill  in  one  minute. 
a     b     c 

But  -  =  part  all  fill  in  one  minute. 

x     a     b     c 

abc 
x  =  - 


Whence, 


ab  +  ac  +  be 


39.  The  capacity  of  a  cistern  is  755 \  gallons.  The  cistern 
has  three  pipes,  of  which  the  first  lets  in  12  gallons  in  3 J  min- 
utes, the  second  15 }  gallons  in  2 \  minutes,  the  third  17  gallons 
in  3  minutes.  In  what  time  will  the  cistern  be  filled  by  the  three 
pipes  running  together? 


teachers'  edition.  141 

Let  x  =  number  of  minutes  required  for  all. 

7551 
Then    — I  =  number  of  gallons  let  in  per  minute  by  all, 

12 

—  =  number  of  gallons  let  in  per  minute  by  first, 

-£&  =  number  of  gallons  let  in  per  minute  by  second, 

17 

— -  =  number  of  gallons  let  in  per  minute  by  third, 

12     15i     77 

— -  +  -— j*  4-  -—  =  number  of  gallons  let  in  per  minute  by  all. 

7551 
But       — i  ==  number  of  gallons  let  in  per  minute  by  all. 

.   755}  =  12      15j     17 
"     x        3J      2J       3 ' 
Whence,     x=48}. 


40.  A  sets  out  and  travels  at  the  rate  of  7  miles  in  5  hours. 
Eight  hours  afterwards,  B  sets  out  from  the  same  place,  and 
travels  in  the  same  direction  at  the  rate  of  5  miles  in  3  hours. 
In  how  many  hours  will  B  overtake  A? 

Let  x  =  number  of  hours  A  is  travelling. 

Then    x  —  8  =  number  of  hours  B  is  travelling, 

1|  =  number  of  miles  per  hour  A  is  travelling, 
if  =  number  of  miles  per  hour  B  is  travelling, 
If  x  =  number  of  miles  A  travels, 
1§  (x  —  8)  =  number  of  miles  B  travels. 


.•.lf*-lf(*-8). 
ice,     a:  =50,  x- 8  =  42. 


Whence, 


41.  A  person  walks  to  the  top  of  a  mountain  at  the  rate  of  2  J 
miles  an  hour,  and  down  the  same  way  at  the  rate  of  3J  miles  an 
hoar,  and  is  out  5  hours.  How  far  is  it  to  the  top  of  the  moun- 
tain? 

Let  x  =  number  of  hours  required  to  go  up, 

and  5  —  x  =  number  of  hours  required  to  go  down. 

Then      2£x  =  distance  up  the  mountain, 
and  3J(5  —  x)  =  distance  down  the  mountain. 

.-.21* -31(5-*). 
Whence,     x  =  3,     and  2Jjc  =  7. 


142  ALGEBRA. 

42.  A  person  has  a  hours  at  his  disposal.  How  far  may  he 
ride  in  a  coach  which  travels  b  miles  an  hour,  so  as  to  return 
home  in  time,  walking  back  at  the  rate  of  c  miles  an  hour? 

Let  x  =  number  of  miles  he  may  go. 

Then  ^  =  number  of  hours  he  is  riding, 

b 

and  -  =  number  of  hours  he  is  walking. 

•  •  T  +  -  =  a. 
b     c 

Whence,     x  = 

43.  The  distance  between  London  and  Edinburgh  is  360  miles. 
One  traveller  starts  from  Edinburgh  and  travels  at  the  rate  of  10 
miles  an  hour ;  another  starts  at  the  same  time  from  London, 
and  travels  at  the  rate  of  8  miles  an  hour.  How  far  from  Lon- 
don will  they  meet? 

Let  x  =  number  of  hours  both  travel. 

Then      10  a;  =  number  of  miles  first  travels, 
and  8x  =  number  of  miles  second  travels. 

10  a;  +  Sx  =  number  of  miles  both  travel. 
.-.  18x  =  360. 
Whence,     x  =  20,  and  8  x  =  160. 

44.  Two  persons  set  out  from  the  same  place  in  opposite 
directions.  The  rate  of  one  of  them  per  hour  is  a  mile  less  than 
double  that  of  the  other,  and  in  4  hours  they  are  32  miles  apart 
Determine  their  rates. 

Let  x  =  rate  of  second  in  miles. 

Then  2x  —  1  =  rate  of  first  in  miles, 
and        3  x  —  1  =  number  of  miles  apart  in  one  hour. 
12a?  —  4  =  number  of  miles  apart  in  four  hours. 
.-.  12rr-4  =  32. 
Whence,     a;  =  3,  and  2a;— 1  =  5. 

45.  In  going  a  certain  distance,  a  train  travelling  35  miles  an 
hour  takes  2  hours  less  than  one  travelling  25  miles  an  hour. 
Determine  the  distance. 

Let  x  =  number  of  miles. 

Then        —  =  number  of  hours  first  was  travelling, 

and  —  =  number  of  hours  second  was  travelling. 

.%  £  +  2  =  £.    Whence,  x  =  175. 
35  2o 


TEACHERS     EDITION. 


143 


46.  At  what  time  are  the  hands  of  a  watch  together : 
I.    Between  3  and  4? 
II.    Between  6  and  7? 
III.    Between  9  and  10? 

I.   Let  CH  and  CM  denote  the  positions  of  the  hour  and  minute 

hands  at  3  o'clock,  and  CB  the  position  of  both  hands  when  together. 

Then  arc  HB  =  *  of  arc  MHB. 

Then    x  =  number  of  minute-spaces  in 

arc  MB, 
Then  —  =  number  of  minute-spaces  in 
12         arc  HB, 
and         15  =  number  of  minute-spaces  in 
arc  MH. 
Now  arc  MB  =  arc  MH  +  arc  HB. 


That  is,        x  =  15  +  ~ 


Whence, 


'16*. 


II.   Let  CM  and  CH  denote  the  positions  of  hour  and  minute 
hands  at  6  o'clock,  CB  the  position  of  both  when  together. 
Then  arc  HB  =  *  of  arc  MHB. 
Let       a;  — number  of  minute-spaces  in 

arc  MHB. 
Then  —  =  number  of  minute-spaces  in 
12        arc#£, 
and         30  =  number  of  minute-spaces  in 
arc  MH 
Now  arc  MHB  =  arc  MH+  arc  HB. 


That  is, 
Whence, 


3  =  30  + 
x  =  32*. 


12 


III.   Let  BC  and  BA  denote  the  positions  of  the  hour  and  minute 

hands  at  9  o'clock,  and  BD  the  position  of  both  hands  when  together. 

Then  CD  =  *  of  arc  AECD. 

Let       x  =  number  of  minute-spaces  in 

arc  AECD. 
Then  —  =  number  of  minute-spaces  in 
12         arcCA 
and  45  =  number  of  minute-spaces  in 

arc  AEC 
Now  arc  AECD  =  arc  AEC  +  arc  CD. 


That  is, 
Whence, 


x  =  45  +  - 
3  =  49* 


12 


144 


ALGEBRA. 


47.  At  what  time  are  the  hands  of  a  watch  at  right  angles: 
I.    Between  3  and  4? 
II.    Between  4  and  5? 
III.    Between  7  and  8? 

I.   Let  CB  and  CD  denote  the  positions  of  the  hour  and  minute 
hands  when  at  right  angles. 

Let       x  =  number  of  minute-spaces  in 
arc  MHBD. 


Then  —  =  number  of  minute-spaces  in 
12         arcJra, 
15  =  number  of  minute-spaces  in 

arc  IflT, 
15  =  number  of  minute-spaces  in 
arc  BD. 
Now  arc  MHBD  =  arcs  MH+HB+BD. 


That  is, 
Whence, 


x  =  15+^-  +  15. 
x  =  S2^. 


II.   Let  CE  and  DE  denote  the  positions  of  the  hour  and  minute 

hands  when  at  right  angles. 

Let       x  =  number  of  minute-spaces  in 

arc  ABCD. 
Then  4:=  number  of  minute-spaces  in 
12        arc  BQ 
and  20  =  number  of  minute-spaces  in 

arc  AB, 
also         15  =  number  of  minute-spaces  in 
arc  CD. 
Now  arc  ABCD  =  arcs  BC+AB+CD. 


That  is, 
Whence, 


3=  20 +  -£-+15. 


x  =  38^. 


12 


Let 
Then 
and 


x  *=  number  of  minute-spaces  in 
arc  AB. 


—  =  number  of  minute-spaces  in 
12         arcCA 

20  =  number  of  minute-spaces  in 
arc  ABC, 

also  15  =  number  of  minute-spaces  in 

arc  BCD. 

Now  arc  AB  =  arcs  CD  +  AC-  BD. 


That  is, 
Whence, 


x  =  ^-+20-15. 
12 


teachers'  edition. 


145 


III.  Let  2?Cand  DC  denote  the  positions  of  the  hour  and 
hands  when  at  right  angles. 

Let      x  =  number  of  minute-spaces  in 

arc  MHBD. 
Then  —  =  number  of  minute-spaces  in 
12         arc  HB, 
and         35  =  number  of  minute-spaces  in 

arc  MAH, 
also  15  =  number  of  minute-spaces  in 

arc  BD. 
Now  arc  MHBD  =  arcs  MAH+HB+BD. 

That  is,  x  -  35  +  —  +  15. 

12 
Whence,  x  =  54^. 

Let      x  =  number  of  minute-spaces  in 

arc  MB. 
Then  —  =  number  of  minute-spaces  in 
12         arcffl), 
and         35  =  number  of  minute-spaces  in 

arc  MBH% 
also         15  =  number  of  minute-spaces  in 
arc  BHD. 
Now  arc  MB 

=  arcs  HBM+  HD  -  BHD. 

That  is,    x  =  35  +  -£  -  15. 


minute 


Whence,  x  =  21^, 


12 


48.  At  what  time  are  the  hands  of  a  watch  opposite  to  each 
other: 

I.    Between  1  and  2? 

II.    Between  4  and  5? 

III.    Between  8  and  9? 

I.  Let  CB  and  CD  denote  the  positions  of  the  hour  and  minute 
hands  when  opposite. 

Let      x  =  number  of  minute-spaces  in 

arc  MHBD. 
Then  --  =  number  of  minute-spaces  in 
12         arc  HB, 
and  5  =  number  of  minute-spaces  in 

arc  MS, 
also         30  =  number  of  minute-spaces  in 
arc  BAD. 
Now  arc  MHBD 

-  arcs  MH+  HB  +  BAD. 


That  is, 
Whence, 


.5  +  ^  +  30. 
■38* 


146 


ALGEBRA. 


II.  Let  CB  and  CD  denote  the  positions  of  the  hour  and  minute 
hands  when  opposite. 

Let       x  =  number  of  minute-spaces  in 

arc  MHBD. 
Then  —  =  number  of  minute-spaces  in 
12         arc  HB, 
and  20  =  number  of  minute  spaces  in 

arc  MH, 
also  30  =  number  of  minute-spaces  in 

arc  BAD. 
Now  arc  MHBD 

=  arcs  MH+  HB  +  BAD. 

That  is,       x  =  20  +  -|  +  30. 

Whence,      x  =  54^. 

III.  Let  CB  and  CD  denote  the  positions  of  the  hour  and  minute 
hands  when  opposite. 

Let       x  =  number  of  minute-spaces  in 

arc  MD. 
Then  —  =  number  of  minute-spaces  in 
12         arc  HB, 
and  40  «=  number  of  minute-spaces  in 

arc  MDH% 
also  30  =  number  of  minute-spaces  in 

arc  DHB. 
Now  arc  MD  -  arcs  MDH+HB-DHB. 


That  is, 
Whence, 


x  =  40+^--30. 
12 


and  I 


49.  It  is  between  2  and  3  o'clock ;  but  a  person  looking  at  his 
watch,  and  mistaking  the  hour-hand  for  the  minute-hand,  fancies 
that  the  time  of  day  is  55  minutes  earlier  than  it  really  is.  What 
is  the  true  time? 

Let  CB  and  CD  denote  the  positions  of  the  hour  and  minute  hands 
id  CE  the  1  o'clock  point. 

Let       a;  »number  of  minute-spaces  in 

arc  MED. 
Then  —  =  number  of  minute-spaces  in 
12         arc  HB, 
and  10  =  number  of  minute-spaces  in 

arc  MEDH,     - 
also  5  =  number  of  minute-spaces  in 

arc  DHB. 
Now  arc  MED 

=  arcs  MEDH+  HB  -  DHB. 

Thatis,x=10  +  ^--5. 


teachers'  edition.  147 

50.  A  hare  takes  6  leaps  to  a  dog's  5,  and  7  of  the  dog's  leaps 
are  equivalent  to  9  of  the  hare's.  The  hare  has  a  start  of  50  of 
her  own  leaps.  How  many  leaps  will  the  hare  take  before  she  is 
caught? 

Let  6  a;  =  number  of  leaps  taken  by  the  hare. 

Then        5  a;  =  number  of  leaps  taken  by  the  dog. 
Also  let      a  =  number  of  feet  in  one  leap  of  the  hare. 
Then        -~  =  number  of  feet  in  one  leap  of  the  dog. 

...  f—\5x  =  (50  +  6a?) a, 

MM  =  50a+6aa;, 

7 

45aa;  =  350a  +  42oaj. 
Divide  by  a,   3  a;  =350. 
Whence,  x  = 116$, 

6s  =  700. 


51.  A  greyhound  makes  3  leaps  while  a  hare  makes  4 ;  but  2 
of  the  greyhound's  leaps  are  equivalent  to  3  of  the  hare's.  The 
hare  has  a  start  of  50  of  the  greyhound's  leaps.  How  many  leaps 
does  each  take  before  the  hare  is  caught? 

Let  3a;  =  number  of  leaps  taken  by  the  greyhound. 

Then        4  a;  —  number  of  leaps  taken  by  the  hare. 
Also  let      a  =  number  of  feet  in  one  leap  of  the  hare. 

o  - 

Then        —  =  number  of  feet  in  one  leap  of  the  greyhound. 

A 

That  is,   3  a;  X  -£■  =  the  whole  distance. 

150a 

But v  4  ax  =  the  whole  distance. 

2 

9  ax     150a  .  A„„ 

2  2 

Divide  by  a,    i*  =  1^  +  43;, 

9a;  =  150  +  8a;. 
Whence,  x  =  150, 

3a;  =  450, 
4  a;  =  600. 


148  ALGEBRA. 


52.  A  greyhound  makes  two  leaps  while  a  hare  makes  3 ;  but 
1  leap  of  the  greyhound  is  equivalent  to  2  of  the  hare's.  The 
hare  has  a  start  of  80  of  her  own  leaps.  How  many  leaps  will 
the  hare  take  before  she  is  caught? 

Let  2x  =  number  of  leaps  taken  by  the  greyhound. 

Then    «    3x  =  number  of  leaps  taken  by  the  Hare. 
Also  let      a  =  number  of  feet  in  one  leap  of  the  hare. 
Then        2  a  =  number  of  feet  in  one  leap  of  the  greyhound. 
That  is,  2x  X  2  a  =  whole  distance. 

But  (80  +  'Sx)  a  =  whole  distance. 

.-.  (80  +  3a:)a  =  4aa:. 
Divide  by  a,    80  +  Sx  =  4x.     Whence,  x  =  80,  and  3  a;  =  240. 

53.  A  rectangle  whose  length  is  5  feet  more  than  its  breadth 
would  have  its  area  increased  by  22  feet  if  its  length  and  breadth 
were  each  made  a  foot  more.    Find  its  dimensions. 

Let  x  —  number  of  feet  in  breadth. 

Then    x  +  5  =  number  of  feet  in  length. 
x(x  +  5)  —  number  of  square  feet  in  area, 

x  +  1  =  number  of  feet  in  breadth  + 1, 

x  +  6  =  number  of  feet  in  length  + 1. 
.-.  {x  +  l)(a:  +  6)  -  x(x  +  5)  =  22. 
Whence,  x  =  8,  and  x  +  5  =  13. 

54.  A  rectangle  has  its  length  and  breadth  respectively  5  feet 
longer  and  3  feet  shorter  than  the  side  of  the  equivalent  square. 
Find  its  area. 

Le^  x  —  3  =  number  of  feet  in  breadth, 

and  x  +  5  =  number  of  feet  in  length. 

Then  (x— 3)  (a; +  5}  =  number  of  feet  in  area. 
But  ar  =  number  of  feet  in  area. 

.-.  a*  =  x*  +  2x  -  15.     Whence,  x  -  7£,  and  x*  =  56J. 

55.  The  length  of  a  rectangle  is  an  inch  less  than  double  its 
breadth ;  and  when  a  strip  3  inches  wide  is  cut  off  all  round, 
the  area  is  diminished  by  210  inches.  Find  the  size  of  the  rect- 
angle at  first. 

Let  x  =  number  of  inches  in  breadth. 

Then  2x  —  1  =  number  of  inches  in  length, 

and  6a;+12a:  —  6— 36  =  number  of  inches  in  area  cut  off. 
But  210  =  number  of  inches  in  area  cut  off. 

.-.  6<c+12<c-6-36  =  210.     Whence,  x -  14,  and  2«-l  =  27. 


teachers'  edition.  149 

56.  The  length  of  a  floor  exceeds  the  breadth  by  4  feet ;  if 
each  dimension  were  increased  by  1  foot,  the  area  of  the  room 
would  be  increased  by  27  square  feet.     Find  its  dimensions. 

Let  x  =  number  of  feet  in  breadth. 

Then  x  +  4  =  number  of  feet  in  length, 

and  a?  +  4  x  =  number  of  feet  in  area, 

x  +  1  =  number  of  feet  in  breadth  +  1  foot, 
x  +  5  =  number  of  feet  in  length  + 1  foot, 
x*  +  6 a?  +  5  =  number  of  feet  in  area  after  addition. 
But  x2  +  4a;  H-  27  =  number  of  feet  in  area  after  addition. 
.-.  s2  +  6x  +  5  =  z2  +  4a;  +  27.    Whence,  a;  =  11,  and  a; +  4  =  15. 

57.  A  mass  of  tin  and  lead  weighing  180  pounds  loses  21 
pounds  when  weighed  in  water ;  aud  it  is  known  that  87  pounds 
of  tin  lose  5  pounds,  and  23  pounds  of  lead  lose  2  pounds,  when 
weighed  in  water.  How  many  pounds  of  tin  and  of  lead  in  the 
mass? 

Let  x  =  number  of  pounds  of  tin. 

Then         180  —  x  =  number  of  pounds  of  lead, 

—  «=  number  of  pounds  x  pounds  of  tin  lose  in 
'37  water, 

—  (180  —  a;)  =  number  of  pounds  180  — a;  pounds  of  lead 
23  lose  in  water. 

But  21  =  number  of  pounds  tin  and  lead  lose  in  water. 

...*L?  + JL(i80-aO  =  21. 
37      23 v  ' 

Whence,  x  =  111,  and  180  -  x  =  69. 

58.  If  19  pounds  of  gold  lose  1  pound,  and  10  pounds  of  silver 
lose  1  pouud,  when  weighed  in  water,  find  the  amount  of  each 
in  a  mass  of  gold  and  silver  weighing  106  pounds  in  air  and  99 
pounds  in  water. 

Let  x  =  number  of  pounds  of  gold. 

Then        106  —  x  =  number  of  pounds  of  silver, 

-^-  =  number  of  pounds  the  gold  loses  in  water, 

=  number  of  pounds  the  silver  loses  in  water. 

10  l 

1 ^~  =  number  of  pounds  both  lose  in  water. 


But  7  =  number  of  pounds  both  lose  in  water. 

.    x      106-a;_7 

"19  10 

Whence,  x  =  76,  and  106  -  x  =  30. 


150  ALGEBRA. 


59.  Fifteen  sovereigns  should  weigh  77  pennyweights ;  but  a 
parcel  of  light  sovereigns,  having  been  weighed  and  counted, 
was  found  to  contain  9  more  than  was  supposed  from  the  weight; 
and  it  appeared  that  21  of  these  coins  weighed  the  same  as  20 
true  sovereigns.    How  many  were  there  all  together  ? 

Let  x  =  number  in  parcel, 

77 

—  *=  number  pennyweights  a  good  sovereign  weighs, 

15 

x  —  9  =  number  good  sovereigns  that  weigh  same  as  bad, 

\~   '  =  number  pennyweights  the  good  coins  weigh, 
15 

on     77      44 

—  X  —  or  —-  =*  number  pennyweights  a  bad  coin  weighs. 

2t\      15        9 

.  44s_77(s-9) 
"9  15 

Whence,     x  =  189. 

60.  There  are  two  silver  cups,  and  one  cover  for  both.  The 
first  weighs  12  ounces,  and  with  the  cover  weighs  twice  as  much 
as  the  other  without  it ;  but  the  second  with  the  cover  weighs 
one-third  more  than  the  first  without  it.  Find  the  weight  of  the 
cover. 

Let  x  =  weight  of  cover  in  ounces, 

12  +  x  =  weight  of  first  cover  and  cup  in  ounces, 

2(16  —  x)  =  double  the  weight  of  the  second  cup  in  ounces. 
But    12  +  x  =  double  the  weight  of  the  second  cup  in  ounces. 

.-.  12  +  x=2(16-x). 
Whence,     x  —  6  j. 

61.  A  man  wishes  to  enclose  a  circular  piece  of  ground  with 
palisades,  and  finds  that  if  he  sets  them  a  foot  apart  he  will  have 
too  few  by  150 ;  but  if  he  sets  them  a  yard  apart  he  will  have  too 
many  by  70.    What  is  the  circuit  of  the  piece  of  ground? 

Let  x  =  number  of  feet  in  circuit  of  field. 

Then  a? —150  =  number  of  palisades  he  had. 
But    -  +  70  =  number  of  palisades  he  had. 

.-.  <c-150  =  |  +  70. 
Whence,     x  =  330. 


teachers'  edition.  151 

62.  A  horse  was  sold  at  a  loss  for  $  200 ;  but  if  it  had  been 
sold  for  $250,  the  gain  would  have  been  three-fourths  of  the  loss 
when  sold  for  $  200.    Find  the  value  of  the  horse. 

Let  x  =  number  of  dollars  the  horse  is  worth. 

Then     250 — x  =  number  of  dollars  made  if  sold  for  $  250, 
re -200  =  number  of  dollars  lost  if  sold  for  $200. 
.-.  250-a;  =  t(x-200). 
Whence,         x  =  228f 


63.  A  and  B  shoot  by  turns  at  a  target.  A  puts  7  bullets  out 
of  12,  and  B  9  out  of  12,  into  the  centre.  Between  them  they 
put  in  32  bullets.    How  many  shots  did  each  fire? 

Let  x  =  number  of  shots  each  fired, 

-~  =  number  of  centres  made  by  A, 

—  =  number  of  centres  made  by  B. 
12  J 

But  32  =  number  of  centres  made  by  both. 

...  ^  +  £?  =  32. 
12      12 

Whence,         x  =  24. 


64.  A  boy  buys  a  number  of  apples  at  the  rate  of  5  for  2  pence. 
He  sells  half  of  them  at  2  a  penny  and  the  rest  at  3  a  penny,  and 
clears  a  penny  by  the  transaction.     How  many  does  he  buy? 

Let  x  =  number  bought. 

Then  — -  =  number  of  pence  paid, 

5 
_     -I       x 
and      -  X  -  or  -  =  selling  price  of  one-half. 

£t  &  4 

But  -  X  -  or  -  =  selling  price  of  the  other  half. 
2     3      6 


■G+5)- 


2x     * 
.  1. 


5 
Whence,        x  =  60. 


152  ALGEBRA. 


66.  A  person  bought  a  piece  of  land  for  $6750,  of  which  he 
kept  $  for  himself.  At  the  cost  of  $250  he  made  a  road  which 
took  tV  °f  tne  remainder,  and  then  sold  the  rest  at  12}  cents  a 
square  yard  more  than  double  the  price  it  cost  him,  thus  clearing 
his  outlay  and  $500  besides.  How  much  land  did  he  buy,  and 
what  was  the  cost-price  per  yard? 

Let  x  =  number  of  yards. 

Then  — -  =-  number  of  yards  kept, 

y 

—  =  number  of  yards  used  for  road, 

-  =  number  of  yards  sold. 

.-.  6750  + -x-=  7500. 
8      2 

Whence,         x  =  12,000, 
and  $6750.00  -s-  x  =  $0.56J. 

66.  A  boy  who  runs  at  the  rate  of  12  yards  per  second  starts 
20  yards  behind  another  whose  rate  is  10}  yards  per  second. 
How  soon  will  the  first  boy  be  10  yards  ahead  of  the  second? 

Let  x  =  number  of  seconds  they  are  running. 

Then  12  a;  =  number  of  yards  first  boy  runs, 

01  r 
and  — -  =  number  of  yards  second  boy  runs. 


12s-(l0  +  ^\  =  20, 

12jc_20  +  21* 

2 
24x-20-21z  =  40, 
3a:  =  60, 
z  =  20. 


67.  A  merchant  adds  yearly  to  his  capital  one-third  of  it,  but 
takes  from  it,  at  the  end  of  each  year,  $5000  for  expenses.  At 
the  end  of  the  third  year,  after  deducting  the  last  $5000,  he  has 
twice  his  original  capital.     How  much  had  he  at  first? 


teachers'  edition.  153 

Let  x  =  number  of  dollars  he  had  at  first. 

4-rr 

Then     —  —  5000  =  number  of  dollars  he  had  at  the  end  of 
3  the  first  year, 

or  g~~     ' =  number  of  dollars  he  had  at  the  end  of 

3  the  first  year, 

4/4s-15,000\     5(XX)  =  number  of  dollar8  he  had  at  the  end  of 
H         *         J  foe  second  year, 

or       — ^~~Z — ! =  numDer  of  dollars  he  had  at  the  end  of 

9  the  second  year, 

4/16s-105,000\  _5000==  number  of  dollarB  he  had  at  the  end  of 
3\  9  /  the  third  year, 

or        — x~       ' =  number  of  dollars  he  had  at  the  end  of 

27  the  third  year. 

But  2x  =  number  of  dollars  he  had  at  the  end  of 

the  third  year. 

64  s  -555,000     „ 

.*.  : =  Ax. 

27 
Whence,  x  =  55,500. 


68.  A  shepherd -lost  a  number  of  sheep  equal  to  one-fourth  of 
his  flock  and  one-fourth  of  a  sheep ;  then,  he  lost  a  number  equal 
to  one-third  of  what  he  had  left  and  one-third  of  a  sheep ;  Anally, 
he  lost  a  number  equal  to  one-half  of  what  now  remained  and 
one-half  a  sheep,  after  which  he  had  but  25  sheep  left.  How 
many  had  he  at  first? 

Let  x  =  number  of  sheep  he  had  at  first. 

q  r 1 

Then  — - —  =  number  of  sheep  he  had  left  after  first  loss, 
'^/T    '  =  number  of  sheep  he  lost  the  second  time, 
~    =  number  of  sheep  he  had  left  after  second  loss, 

r  4-  1 

— - —  =  number  of  sheep  he  lost  the  third  time, 
4 

x~~    =  number  of  sheep  he  had  left  after  third  loss. 
But  25  =  number  of  sheep  he  had  left  after  third  loss. 

4 
Whence,     x  =  103. 


154  ALGEBRA. 


69.  A  trader  maintained  himself  for  three  years  at  an  expense 
of  $  250  a  year ;  and  each  year  increased  that  part  of  his  stock 
which  was  not  so  expended  by  one-third  of  it.  At  the  end  of 
the  third  year  his  original  stock  was  doubled.  What  was  his 
original  stock? 

Let  x  =  number  of  dollars  in  stock  at  first. 

Then      ^(s-250) 

or  a"~ =  number  of  dollars  in  stock  at  the  end  of 

3  first  year, 

|/4«-1000_2BO\ 

or  — 2Lz =  number  of  dollars  in  stock  at  the  end  of 

second  year, 

-(" 

3\         9 


4 /16  a:  -  7000  _  ^X  =  number  of  dollar8  in  gtock  at  the  end  of 


third  year. 

But  2x  —  number  of  dollars  in  stock  at  the  end  of 

third  year. 

■  •■f(16*-7000-250)-2s. 
Whence,  x  =  3700. 


70.  A  cask  contains  12  gallons  of  wine  and  18  gallons  of 
water ;  another  cask  contains  9  gallons  of  wine  and  3  gallons 
of  water.  How  many  gallons  must  be  drawn  from  each  cask  to 
produce  a  mixture  containing  7  gallons  of  wine  and  7  gallons  of 
water? 

Let  x  =  number  of  gallons  drawn  from  1st  cask, 

14  —  x  =  number  of  gallons  drawn  from  2d  cask, 

2 

-  =  proportion  of  wine  to  water  in  1st  cask, 
o 

-  =  proportion  of  wine  to  water  in  2d  cask. 

.-.  ^  +  !(14-*)  =  7. 
Whence,  x  =  10, 

and  14  —  x  =  4. 


teachers'  edition.  155 

71.  The  members  of  a  club  subscribe  each  as  many  dollars  as 
there  are  members.  If  there  had  been  12  more  members,  the 
subscription  from  each  would  have  been  $  10  less,  to  amount  to 
the  same  sum.     How  many  members  were  there? 

Let  x  =  number  of  members  of  the  club. 

Then  x  =  number  of  dollars  each  subscribed, 

x  +  12  =  number  of  members  +  12, 
and        x  — 10  =  number  of  dollars  each  would  have  subscribed 
in  second  case. 
But  x2  =  number  of  dollars  all  subscribed. 

.-.  (x  +  12)(x-10)  =  x*. 
Whence,  x  =  60. 

72.  A  number  of  troops  being  formed  into  a  solid  square,  it 
was  found  there  were  60  men  over ;  but  when  formed  in  a  col- 
umn with  5  men  more  in  front  than  before  and  three  men  less  in 
depth,  there  was  lacking  one  man  to  complete  it.  Find  the 
number  of  troops. 

Let  x  =  number  of  men  on  one  side. 

Then  x2  +  60  =  number  of  men  in  the  square, 
x  +  5  =  number  of  men  on  a  side  +  5, 
x  —  3  =  number  of  men  on  a  side  —  3, 
and  (x  +  5)(x  —  3)  —  1  =  number  of  men  in  the  square. 

.-.  (x  +  5)(x-S)-l  =  x*  +  60. 
Whence,  x  =  38, 

and  x2  +  60  =  1504. 

73.  An  officer  can  form  the  men  of  his  regiment  into  a  hollow 
square  12  deep.  The  number  of  men  in  the  regiment  is  1296. 
Find  the  number  of  men  in  the  front  of  the  hollow  square. 

Let  x  =  number  of  men  in  front. 

Then       12  x  =  number  of  men  in  twelve  lines, 
and  24  x  =  number  of  men  in  twelve  lines  front  and  rear. 

12  (x  —  24)  =  number  of  men  on  a  side, 
12  (x  —  24)  x  2  =  number  of  men  on  both  sides. 
Then  24 x  + 12  (x  -  24)  X  2  =  whole  number  of  men. 
But  1296  =  whole  number  of  men. 

.\  24x  +  12(x-24)x2  =  1296. 
Whence,  x  =  39. 


156  ALGEBRA. 


74.  A  person  starts  from  P  and  walks  towards  Q  at  the  rate 
of  8  miles  an  hour ;  20  minutes  later  another  person  starts  from 
Q  and  walks  towards  F  at  the  rate  of  four  miles  an  hour.  The 
distance  from  P  to  Q  is  20  miles.  How  far  from  P  will  they 
meet? 

Let  x  —  number  of  miles  first  travels. 

Then  20  —  x  =  number  of  miles  second  travels, 

-  =s  number  of  hours  first  travels. 
3 

— ^—  =■■  number  of  hours  second  travels. 
4 

^.   x^M-x     1     whence,  x  =  9f . 
3         4  3  »7 

75.  A  person  engaged  to  work  a  days  on  these  conditions : 
for  each  day  he  worked  he  was  to  receive  b  cents,  and  for  each 
day  he  was  idle  he  was  to  forfeit  c  cents.  At  the  end  of  a  days 
he  received  d  cents.     How  many  days  was  he  idle? 

Let  x  =  number  of  days  he  was  idle. 

Then    a  —  x  =  number  of  days  he  worked, 
and  ex  =  number  of  cents  he  forfeited, 

b  (a  —  x)  =  number  of  cents  he  received, 
(rib  —  bx)  —  cx  =  whole  amount. 
But  d  =  whole  amount. 

.  \  (rib  —  bx)  —  cx  =  d. 

Whence,     x  =  ab~d 


6  +  c 


76.  A  banker  has  two  kinds  of  coins :  it  takes  a  pieces  of  the 
first  to  make  a  dollar,  and  b  pieces  of  the  second  to  make  a  dol- 
lar. A  person  wishes  to  obtain  c  pieces  for  a  dollar.  How  many 
pieces  of  each  kind  must  the  banker  give  him? 

Let  x  =  number  of  pieces  of  first  kind. 

Then    c  —  x  =  number  of  pieces  of  second  kind, 

-  =  the  part  of  a  dollar  in  one  piece  of  first, 
a 

-  =  the  part  of  a  dollar  in  one  piece  of  second. 
b 

.   x     c  —  x  =  -. 
a        b 

Whence,     i^4   and  c-x  =  S(e  ~a). 
b  —a  b  —  a 


TEACHERS     EDITION. 


157 


Exercise  LXVIII. 


3 


(1) 

(2) 


2a;+3y  =  7 
4a;-5y  =  3 

Multiply  (1)  by  2, 

4z  +  6y  =  14 

(2)  is  4a;-5y  =   3 

Subtract,  lly  =  ll 

.-.  y  =  l 

Substitute  value  of  y  in  (2), 

4a;-5  =  3. 

.\x=*V. 

a;-2y  =  4 
2a;-    y  =  5 
Multiply  (1)  by  2, 

2a;-4y  =  8 
(2)  is  2s-    y  =  5 

Subtract,  -  3y  =  3 

.-.y  =  -l. 
Substitute  value  of  y  in  (2), 
x  +  2  =  4. 
.-.  a;  =  2. 

7a;  +  2y  =  30 
-3a;  +    y=    2 
(l)is  7a;  +  2y  =  30 

(2)  by  2,    -6a;  +  2y  =    4 
Subtract,    13  a;  = 26 

.-.  x  =  2. 
Substitute  value  of  a;  in  (1), 
14  +  2y  -  30, 
2y  =  16. 
.vy  =  8. 

3a;-5y  =  51     (1) 
2a;  +  7y  =  3       (2) 
Multiply  (1)  by  2,  and  (2)  by  3, 
6a;-10y  =  102 
6a;  +  21y«     9 
Subtract,         -31y=    93 

.-.y 3. 

Substitute  value  of  y  in  (1), 
Sx  +  15  =  51, 
3a;  =  30. 
.*•£*»  12. 


8 


5.  5a;+4y  =  5S     (1) 
3a;  +  7y  =  67    (2) 

Multiply  (1)  by  3,  and  (2)  by  5, 
15a;+12y  =   174 
15a?  +  35y  =   335 
Subtract,         -23y=-U>l 

.-.  y  =  7. 
Substitute  value  of  y  in  (1), 
5a; +  28  =  58. 
.-.  a;  =  6. 

6.  3a;  +  2y  =  39    (1) 
3y-2a;=13     (2) 

Multiply  (1)  by  3,  and  (2)  by  2, 

9a; +  6y  =  117 
-4a;  +  6y=    26 

Subtract,    13  a;  =91 

.\*-7. 

Substitute  value  of  x  in  (1), 
21  +  2y  =  39,  2y  =  18. 
.-.y  =  9. 

7.  3a;-4y  =  -5  (1) 
4x-5y  =  l       (2) 

Multiply  (1)  by  4  and  (2)  by  3, 
N12a;-16y  =  -20 
12s -15y-       3 
Subtract,  -  y  =  -  23 

.-.y  =  23. 
Substitute  value  of  y  in  (1), 
3a; -92  =  -5,  3a;  =  87. 
.-.  a;  =  29. 

8.  lla;  +  3y  =  100  (1) 

4a;-7y  =     4  (2) 
Multiply  (1)  by  4  and  (2)  by  11, 
44a;  +  12y  =  400 
44a;-77y  =  44 
Subtract,  89y  =  356 

.-.  y  =  4. 

Substitute  value  of  y  in  (1), 
11a; +  12  =100,  11  x  =  88. 
.-.  x  =  8. 


158 


ALGEBRA. 


*  +  49y  = 
49*+      y  = 


093(1) 
357(2) 


Add,  50a;  +  50y  =  1050(3) 
Divide  by  50,  *  +  y  =  21  (4) 
Subtract  (4)  from  (1), 

48y  =  672. 

.-.  y  =  14. 
Subtract  (4)  from  (2), 

48*  =  336. 

.-.  *  =  7. 


12.  2x-7y=   8      (1) 

-9*  +  4y=19      (2) 
Multiply  (1)  by  4  and  (2)  by  7, 
8x-28y  =    32    (3) 
-63*  +  28y  =  133    (4) 
Add,- 55  x  =165 

.\  x  =  -3. 
Substitute  value  of  x  in  (2), 
27  +  4y=19, 
4y  =  -8. 


10.  17*  +    3y=    57(1) 

-3*  +  16y  =    23(2) 
Multiply(l)  by  3  and  (2)  by  1 7, 
51*+     9y  =  171 
-51*  +  272y  =  391 
Add,  281y  =  562 

.-.y  =  2. 
Substitute  value  of  y  in  (2), 
3a; +  32  =  23, 
-3s  =  -9. 
.-.  *  =  3. 


13.  69y-17*  =  103    (1) 

14*-13y  =  -41  (2) 

Multiply  (1)  by  14,  and  (2) 

by  17, 

-238  a;  +  966  y  =  1442(3) 
238*-221y  =  -697(4) 
Add  745y  =   745 

.-.y  =  i. 

Substitute  value  of  y  in  (2), 
14s -13  =  -41, 
14s  =  -28. 
.-.  *  =  -2. 


11.  12*  +    7y  =  176  (1) 

3y-19*  =  3      (2) 

Multiply  (1)  by  3  and  (2)  by  7, 

36*  +  21y  =  528  (3) 

-133*  +  21y=   21 

Subt.,   169*  =507 

.\  *  =  3. 
Substitute  value  of  *  in  (2), 
3y-57  =  3, 
3y  =  60. 
.-.y  =  20. 


14.  17*  +  30y  =  59      (1) 

19*+28y  =  77     (2) 
Multiply  (1)  by  14,  and  (2) 
by  15, 

238*  +  420y=    826(3) 
285*  +  420y=  1155(4) 
-47*  =-329 

.-.  *=7. 
Substitute  value  of  *  in  (2), 
133  +  28y  =  77, 
28y  =  -56. 

.-.y  =  -2. 


TEACHERS     EDITION. 


159 


Exercise  LXIX. 


3a?-4y  =  2 
7a?-9y  =  7 

Transpose  — 4y  in  (1), 

3x=2  +  4y. 
Divide  by  coefficient  of  a?, 

3 
Substitute  value  of  a?  in  (2),. 


i^y°>- 


■  7. 


Simplify, 
14+28y-27y  =  21. 

.•■y-7. 

Substitute  value  of  y  in  (1), 
3a;- 28  =  2, 
3*  =  30. 
.-.a;  =  10. 


(1)  3.  3  a; +  2y  =  32        (1) 

(2)  20a;-3y  =  l  (2) 

Transpose  2y  in  (1), 

3x  =  32-3y. 
Divide  by  coefficient  of  a?, 

x=™-2y. 

3 
Substitute  value  of  x  in  (2), 

20(M^_3y  =  l, 

3  * 

Simplify, 

640-40y-9y  =  3, 

-49y  =  -637. 
.-.y  =  13. 
Substitute  value  of  y  in  (1), 
3a? +26  =  32. 
.:x  =  2. 


7a;-5y  =  24        (1) 
4a;-3y  =  ll         (2) 

Transpose  by  in  (1), 

7a;  =  24  +  5y. 
Divide  by  coefficient  of  x, 

Substitute  value  of  x  in  (2), 

4(M±S«)-s,-ii. 

Simplify, 

96  +  20y-21y  =  77, 
-y  =  _19. 
.-.y  =  19. 
Substitute  value  of  y  in  (1), 
7* -95  =  24,    ' 
7a?  =  119. 
.-.a?  =17. 


4.  lla?-7y  =  37        (1] 

8a?+9y  =  41         (2; 

Transpose  7y  in  (1), 

llx  =  37+7y. 
Divide  by  coefficient  of  x, 

x-37  +  7y 
11 
Substitute  value  of  x  in  (2), 

8(^i^)4  9y  =  41. 

Simplify, 
296  +  56y  +  99y  =  451, 
155y  =  155. 
.-.J-l. 
Substitute  value  of  y  in  (2), 
83  +  9  =  41. 
.-.  a?  =-4. 


160 


ALGEBRA. 


5.  7*+   5y  =  60        (1) 

13a:-lly  =  10        (2) 

Transpose  5y  in  (1), 

7&  =  60-5y. 
Divide  by  coefficient  of  x, 

60  -5y 

Substitute  value  of  x  in  (2), 
IS^-^-lly-lQ, 

Simplify, 
780-65y-77y  =  70, 
780-142y  =  70, 
142y=710. 
...y  =  5. 

Substitute  value  of  y  in  (1), 
7x  +  25  =  60, 
7a;  =-36. 
.-.*=»  5. 


7.  10*+    9y  =  290      (1) 

12z-lly=130      (2) 

Transpose  9y  in  (1), 

10s  =  290-9y. 
Divide  by  coefficient  of  ar, 

j,_290-9.y 
10 
Substitute  value  of  x  in  (2), 
12/290-9y\     lly  =  130 

Simplify, 
3480 -108y-110y  =  1300, 
218y  =  2180. 
.-.y  =  10. 

Substitute  value  of  y  in  (1), 
10a? +  90  =  290, 
10*  =  200. 
.-.  x  =  20. 


6.  6s-7y  =  42        (1) 

7z-6y  =  75        (2) 

Transpose  7y  in  (1), 

6x  =  42  +  7y. 
Divide  by  coefficient  of  x, 

^12  +  7^. 
6 
Substitute  value  of  x  in  (2), 

Simplify, 

294  +  49y-36y  =  450, 
13y  =  156. 
.-.y=12. 

Substitute  value  of  y  in  (1), 
6* -84  =  42. 
.\*-21. 


8.  3s-4y  =  18        (1) 

3s  +  2y  =  0  (2) 

Transpose  4y  in  (1), 

3*  =  18+  4y. 
Divide  by  coefficient  of  xt 

o 
Substitute  value  of  x  in  (2), 

Simplify, 

54  +  12y  +  6y  =  0, 

18y  =  -54. 

.-.y  =  -3. 

Substitute  value  of  y  in  (2), 
3s-6  =  0. 
.\x-2. 


TEACHERS     EDITION. 


161 


9.                 9a;-5y  =  52 

(1) 

11.               9y-7z  =  13        (1) 

8y-3z  =  8 

(2) 

15a?-7y  =  9          (2) 

Transpose  5y  in  (1), 

Transpose  —  7  a;  in  (1), 

9a:  =  52  +  5y. 

9y=13  +  7a;. 

Divide  by  coefficient  of 

*. 

Divide  by  coefficient  of  y, 

•      52  4 

x  =  — - 
S 

,..      13+ 7a: 
y~       9      * 

Substitute  value  of  x  in 

(2). 

Substitute  value  of  y  in  (2), 

8y    3(5-H^)  =  8- 

15*     7(13  +  7*W 

Simplify, 

Simplify, 

72y-156-15y  =  72, 

135  a; -91 -49a;  =  81, 

57y  -  228. 

86  a;  =  172. 

.-.y-4. 

.-.a;  =  2. 

Substitute  value  of  y  in 

(1). 

Substitute  value  of  x  in  (1), 

9a?  -20  =52, 

9y-14  =  13. 

9*  =72. 

.-.y-3. 

.-.a;  =  8. 

10.                5a?-3y  =  4 

(1) 

12.               5a;-2y  =  51         (1) 

12y-7a;  =  10 

(2) 

19a;-3y  =  180      (2) 

Transpose  —  3y  in  (1), 

Transpose  2y  in  (1), 

5a;  =  4  +  3y. 

5a;  =  51  +  2y. 

Divide  by  coefficient  of 

X, 

Divide  by  coefficient  of  x, 

*         5 

*y 

-     51  +  2y 
—6 

Substitute  value  of  x  in 

(2), 

Substitute  value  of  x  in  (2), 

12y       7(*+»jrt_10. 

19pLt2jrt_8y-l80. 

Simplify, 

Simplify, 

60y-28-21y  =  50, 

969  +  38y-15y  =  900, 

39y  =  78. 

23y  =  -69. 

.-.y  =  2. 

.-.y  =  -3. 

Substitute  value  of  y  in 

(1), 

Substitute  value  of  y  in  (1), 

5s- 6  =  4, 

5* +  6-51, 

5a?  =  10. 

5a;  =  45. 

.-.a:  -2. 

.-.  x  =  9. 

162 


ALGEBRA. 


13.  4s  +    9y=106       (1) 

8x  +  17y=198       (2) 

Transpose  9y  in  (1), 

4s=106-9y. 
Divide  by  coefficient  of  x, 

«_"»-»y. 

Substitute  value  of  x  in  (2), 
8/106-9y\  +  17y=198 

Simplify, 

212 -18y  +  17y  =  198. 
.-.y  =  14. 

Substitute  value  of  y  in  (1), 
43  +  126=106, 
4s  =  -20. 
.•.  x  =  —  5. 


14.  8x  +  3y=3  (1) 

12s  +  9y  =  3  (2) 

Transpose  3y  in  (1), 

8x  =  3-3y. 
Divide  by  coefficient  of  x, 

8 
Substitute  value  of  x  in  (2), 

12(^)+9y=3- 

Simplify, 

9-9y  +  18y  =  6, 
9y  =  -3. 

Substitute  value  of  y  in  (1), 
8s-l  =  3, 
8x  =  4. 


Exercise  LXX. 


1.  a;  +  15y  -  53  (1) 

3*+      y=-27  (2) 

Transpose  15  y  in  (1)  and  y 
in  (5), 

x  =  53-15y(3) 
3x  =  27-y      (4) 

Divide  (4)  by  3, 

x  = 


27  -y 


Equate  values  of  a, 

27-y 


3 


(5) 
(6) 


2.  4*+    9y  =  51  (1) 

8x-I3y  =  9  (2) 

Transpose  9y  in  (1),  and—  13j/ 
in  (fe), 

4x  =  51-9y  (3) 
8x  =  9  +  13y  (4) 

Divide  (3)  by  4  and  (4)  by  8, 
,-61-9y(5) 


53  -  15y  = 

Reduce, 

159 -45y  =  27-y, 
44y  =  132. 

.-.y-3. 

Substitute  value  of  y  in  (2), 

3*  +  3  =  27, 
3x=-24. 

.\  x  =  8. 


._9+13y 


(6) 


Equate  values  of  x, 

51-9y_9  +  13y  m 
4  8        W 

Reduce, 

102-18y  =  9  +  13y. 

.-.y  =  3. 
Substitute  value  of  y  in  (1), 
4x  +  27-51. 
.-.  x-6. 


TEACHERS'    EDITION. 


163 


3.            4a;  +  3y  =  48              (1) 

5.         5.*-    7y«33              (1) 

5y-3a:  =  22              (2) 

lla;+12y=100             (2) 

Transpose  3y  in   (1)  and  5y 

Transpose  —7y  in  (1)  and  12y 

ln(2)'      4a;  =  48-3y     (3) 
3z  =  5y-22     (4) 

in  (2)»      5x  =  33  +  7y     (3) 
llx  =  100-12y(4) 

Divide  (3)  by  4  and  (4)  by  3, 

Divide  (3)  by  5  and  (4)  by  1 1, 

,  =  4^, 

g-33  +  *y, 

a^-y-22. 

,-100-i2y.. 

3 

11 

Equate  values  of  x, 

Equate  values  of  x, 

48-3y_5y-22 

33  +  7y     100 -12y 

4                3 

5                 11 

Reduce, 

Reduce, 

144-9y  =  20y-88, 
-  29y  =  -  232. 

.-.  y  =  8. 

363  +  77y  =  500-60y, 

147y  =  147. 
.-.y  =  i.V.    -''' 

Substitute  value  of  y  in  (1), 

Substitute  value  of  y  in  (1), 

4*  +  24  -  48, 

5a;- 7  =  33, 

4a?  =  24. 

5a;  =  40. 

.-.a;  =  6. 

.-.  *-8. 

L            2a:  +  3y  =  43              (1) 

6.           5a;  +  7y  =  43              (1) 

10a:-    y  =  7                (2) 

lla:  +  9y  =  G9              (2) 

Transpose  3y  in  (1)  and  y  in 

Transpose  9y  in  (2)  and  7y 

(2)'           2a;  =  43-3y     (3) 

lnW.      5x  =  43-7y     (3) 
lla;  =  69-9y     (4) 

10a?  =  7  +  y         (4) 

Divide  (3)  by  2  and  (4)  by  10, 

Divide  (3)  by  5  and  (4)  by  11, 

-     43 -3y 

..H=U 

«_*±*. 

*-69-9.y. 

10 

n 

Equate  values  of  a;, 

Equate  values  of  x, 

43 -3y      7+v 

43-7.y     G9-9y 

2              10 

5               11 

Reduce, 

Reduce, 

215-15y  =  7+y, 
-  16y  =  -  208. 

473-77y  =  345-45y, 

-  32y  =  -  128. 

.-.y^l3. 

.-.  y  =  4. 

Substitute  value  of  y  in  (1), 

Substitute  value  of  y  in  (1), 

2x  +  39  -  43. 

5x  + 28  =  43. 

.-.  x  -2. 

.\*  =  3. 

164 


ALGEBRA. 


7.  8x-21y  =  33  (1) 

6x  +  35y=177  (2) 

Transpose  21  y  in  (1)  and  35y 
in  (2),       8x  =  33  +  21y  (3) 
6*=177-35y  (4) 
Divide  (3)  by  8,  and  (4)  by  6, 
33  +  21y 
X~        8       ' 
x     m -35y 
6 
Equate  values  of  x, 

33  +  21y_177-35y 
8  6 

Reduce, 

99  +  63y=708-140y, 
203y  =  609. 
-■-J -3. 
Substitute  value  of  y  in  (1), 
801-63  =  33, 
8a?  =  96. 
.\x  =  12. 


2ly  +  20s=165  (1) 

77y-30a;  =  295  (2) 

Transpose  20  s  in  (1)  and  30* 

in(2)'  21  y  =  165  -20a?  (3) 

77y  =  295  +  30a;  (4) 

Divide(3)by  21  and(4)by77, 

=  165  -  20a: 


Vs 


21 
295  +  30a; 

77 


Equate  values  of  x, 

165 -20s^  295  +  30a; 
21  77 

Reduce, 

1815-  220  a;  =  885  +  90x, 

-310a;  =  -930. 

.-.a;  =  3. 

Substitute  value  of  ar  in  (1), 

21y  +  60  =  165, 

21y  =  105. 


8.  3y-7a;  =  4 

2y +  5a;=22 
•  Transpose  7  a;  in  (1)  and 

in<2)>      3y  =  4  +  7s 
2y  =  22-5a; 
Divide  (3)  by  3  and  (4)  by 

y         3 
22 -5a; 


5a; 

(3) 
(4) 

0 


8 


Equate  values  of  y, 

4+ 7a;  ^22 -5a 
3  2 

Reduce, 

8  +  14a;  =  66 -15a;, 
29x  =  58. 
.\  a;  =  2. 
Substitute  value  of  x  in  (1), 
3y-14  =  4. 
y,.y  =  6. 


10.     llx-10y  =  14 
5a;  +    7y  =  41 
Transpose  — 10  y  in  (1)  and  7y 

in(2)'  lla;  =  10y  +  14  (3) 

5a;  =  41-7y     (4) 

Divide  (3)  by  11  and  (4)  by  5, 

,-10y  +  14, 

5 
Equate  values  of  a;, 
10y  +  14_^41-7y 
11  5 

Reduce, 

50y+70  =  451-77y, 

127y  =  381. 

...y  =  3. 

Substitute  value  of  y  in  (1), 

lla;-30=14. 

.-.a;  =  4. 


TEACHERS      EDITION. 


165 


11.        7y-3a;=139  (1) 

2a;  +  5y  =  91  (2) 

Transpose  7y  in  (1)  and  5y 

™  (2),     3x  =  >7y-i$9   (3) 

2a;  =  91-5y     (4) 

Divide  (3)  by  3  and  (4)  by  2, 

„     7.y  - 139 

x  =  — * > 

3 
„.     91  -5y 

Equate  values  of  a;, 
ly  -  139  _  91  -  by 
3  2 

Reduce, 

14y-278  =  273-15y, 
29y  -  551. 
.-.y  =  19. 
Substitute  value  of  y  in  (4), 
2s  =  91 -95, 
2s  =  -4. 
.\  x  =  -2. 


13. 


24*  +    7y  =  27 
8a;-33y  =  115 


(1) 
(2) 


12.     17a;  +  12y  =  59 
19a:  -4y    =153 


s 


Transpose  7y  in  (1)  and  33 y 

m  W  24  a;  =  27  -7y     (3) 
Sx  =  115  +  33y  (4) 

Divide  (3)  by  24  and  (4)  by  8, 

24 
115  +  33v 
8 
Equate  values  of  x, 

27-7y_115  +  33y 
24  8 

Reduce, 

27  -  7y  -  345  +  99y, 
-  106y  =  318. 

Substitute  value  of  y  in  (3), 
24a:  =  27  +  21, 
24  a;  =  48. 
.-.  x  =  2. 


Transpose  12y  in  (1)  and  4y      +* 
in(2),  I7a;  =  59_i2y   (3) 
19ar=153  +  4y   (4) 
Divide  (3)  bv  17  and  (4)  by  19, 
",  =  59-12y| 

17 
x_163+4£ 
19 
Equate  values  of  x, 

153  +  4y  ^  59  -  12y 
19  17 

Reduce, 
2«01+68y=1121-228y, 
296y  =  - 1480. 
.-.  y  =  -5. 
Substitute  value  of  y  in  (1), 
17a:-60  =  59. 
.\»-7. 


a;  =  3y-19     (1) 
y  =  3a?-23     (2) 

Transpose  3  x  and  y  in  (2), 

3a;=23+y        (3) 
Divide  (3)  by  3, 

*=23±2. 
3 
Equate  values  of  x, 

3y-19  =  23±i/. 
y  3 

Reduce, 

9y-57  =  23  +  y, 

8y  =  80. 

.-.y  =  10. 

Substitute  value  of  y  in  (1), 
a;  =  30-19. 

.VJC-11. 


166                                                 ALGEBRA. 

Exercise 

LXXI. 

1.               z(y  +  7)  =  v(*+l) 
2x  +  20  =  3y  +  l 

$ 

3                     2    _     3 

a:  +  3     y-2 

(1) 

Simplify  (1), 

5(*  +  3)  =  3(y-2)+2(2) 

xy  +  7x  =  sy  +  y. 

Simplify  (1), 

Transpose  and  combine, 

2y-4  =  3a:  +  9. 

7x-y  =  0 

(3) 

Transpose  and  combine, 

Transpose  and  combine  (2), 

2y-3x  =  13 

(3) 

2a:-3y  =  -19 

(4) 

Simplify  (2), 

Multiply  (3)  by  3, 

5a+15  =  3y-6  +  2. 

21x-3y  =       0 

Transpose  and  combine, 

(4)  is         2s-3y  =  -19 

5x-3y  =  -19 

(4) 

Subt.,     19  x          =19 
.-.  x  =  l. 

Multiply  (3)  by  3  and  (4)  by  2, 

6y-   9a:  =     39 

Substitute  value  of  x  in  (3), 

-6y  +  10x 38 

7-y  =  0. 

Add,              x=l. 

Substitute  value  of  x  in 

(3), 

2y  -  3  =  13. 

.-.y-8. 

2.       2a;-2^l3_4  =  0 
5 

(1) 

3y  +  ^-9  =  0 

(2) 

4.  ^r:i_2Mj  =  o 
5           10 

(1) 

Simplify  (1), 

£  +  3tJ-3 
6        4 

(2) 

10x-y  +  3-20  =  0. 

Transpose  and  combine, 

Simplify  (1), 

10x-y  =  17 

(3) 

2x-8-y-2  =  0, 

Simplify  (2), 

2x-y  =  10 

(3) 

9y  +  x-2-27  =  0. 

Simplify  (2), 

Transpose  and  combine, 

2x  +  3y-6  =  36, 

x  +  9y  =  29 

(4) 

2x  +  3y  =  42 

« 

Multiply  (3)  by  9, 

Subtract  (4)  from  (3), 

90a:  -  9y  =  153 

2s-    y=    -10 

(4)  is            ar  +  9y=    29 

2x  +  3y=      42 

Add,       91a:           =182 

-4y 32 

.-.  x  =  2. 

.-.y-8. 

Substitute  value  of  x  in  (4), 

Substitute  value  of  y  in 

(3), 

2  +  9y  =  29. 

2a -8 -10. 

.-.y-3. 

.-.  z-9. 

teachers'  edition.  167 


(*  +  l)(y  +  2)-(*  +  2)(y  +  l)--l  (1) 

3(a;+3)-4(y  +  4)  =  -8  (2) 

Simplify,  (1),       xy  +y  +  2x  +  2  —  xy  —  2y  —  x  —  2  =  —  1. 

Combine,  x  —  y  =  —  1  (3) 

Simplify  (2),  3x  +  9  -  4y  -  16  =  -  8. 

Transpose  and  unite,  3  a  —  4y  =  —  1  (4) 

Multiply  (3)  by  3,  3s-3y  =  -3 

Subtract, 

Substitute  value  of  y  in  (3), 


Simplify  (1), 

Transpose  and  combine, 

Simplify  (2), 

Transpose  and  combine, 

Multiply  (3)  by  3, 

Multiply  (4)  by  8, 

Add, 

Substitute  value  of  y  in  (3), 


-y-    2 

.-.y  — 2. 

X+2  =  -\. 

.-.  s  =  -3. 

e    x  —  2     10-a;     y-10 
'      5            3              4 

(1) 

2y  +  4     2x  +  y     x  + 13 
3               8              4 

(2) 

12*  -  24  -  200  +  20x  -  15y  -  150. 

32s  -  15y  =  74 

(3) 

16y  +  32  -  6a?  -  3y  =  6x  +  78. 

-12a>  +  13y  =  46 

(4) 

96a;-   45y=>222 

-96a  +  104y  =  368 

59y  =  590 

.-.y-10. 

32a: -150  =74, 

32a;  =224. 

.\a>-7. 

7     x  +  1     y  +  2     2(x-y) 
'        3            4              5 

(1) 

4            3           * 

(2) 

20a;  +  20 -  15y  -  30  =  24a;  -  24y. 

-4a;  +  9y=10 

(3) 

3a; - 9  -  4y  +  12  =  24y  -  12x. 
15a;-28y  =  -3 

W 

60a;-135y  =  -150 

60a;-112y  =  -    12 

Simplify  (1), 

Transpose  and  combine, 

Simplify  (2). 

Transpose  and  combine, 

Multiply  (3)  by  15, 

Multiply  (4)  by  4, 

Subtract,  -23y  =  -138 

.-.  y  =  6. 
Substitute  value  of  y  in  (3),  —  4  a;  +  54  =  10, 

*  -4a;  =  -44. 

.\*-ll. 


168  ALGEBRA. 


8. 

3x- 


2^  +  5x-3^  =  x  +  1  (1) 


5  3 

2s-3y+4s-3y==y  +  1  (2) 

o  A 

Simplify  (1),  9x-6y +  25x-15y  =  15x+ 15. 

Transpose  and  combine,  19  x  —  21  y  =  15  (3) 

Simplify  (2),  4x-6y  +  12x-9y  =  6y  4- 6. 

Transpose  and  combine,  16  a:  —  21  y  =  6  (4) 

Subtract  (3)  from  (4),  19x-21y=15 

-3x  =-9 

.\  x  =  3. 
Substitute  value  of  x  in  (4),  48  —  21  y  =  6, 

-21y  =  -42. 

.-.y  =  2. 

9. 

2x-y  +  3     x-2y+3_d  ,.* 

3                     4  W 

3x-4y  +  3  {  4x-2y-9_1  ^ 

Simplify  (1),  8 x  -  4y  +  12  -  3 *  +  6y  --  9  =  48. 

Transpose  and  combine,  5x  +  2y  =  45  (3) 

Simplify  (2),  9x  -  12y  +  9  +  16x  -  8y-  36  =  48. 

Transpose  and  combine,  25  x  —  20  y  =  75  (4) 

Divide  (4)  by  5,  5x  -  4y  =     15 

(3)  is  5x  +  2y  -     45 
Subtract,  -6y  =  -30 

.-.  y  =  5. 

Substitute  value  of  y  in  (3),  5x  + 10  =  45, 

5x  =  35. 

.-.  x  =  7. 

10.  li^  =  ily+4A    (l) 

4Jx  =  Jy-21A     (2) 
Simplify  (1),  18  x  -  16y  =  53  (3) 

Simplify  (2),  54 x-   4y  =  -259  (4) 

Multiply  (3)  by  3,  54x-48y-     159 

(4)  is  54x-   4y  =  -259 

Subtract,  -  44y  -     418 

.-.y  =  -9J. 

Substitute  value  of  y  in  (3),  18  x  +  152  =  53, 

18x--99. 
.-.x  =  -5J. 


teachers'  edition.  169 


11.      — — = * (1) 

a;  +  2y  +  3         4a;-5y+6 ' 

3  19         (2) 


6a;-5y  +  4     3a;  +  2y+l 

Simplify  (1),  55a;-59y  =  -87  (3) 

Simplify  (2),  - 105  x  + 101  y  -  73  (4) 

Transpose  59 y  in  (3)  and  101  y  in  (4),  and  divide  by  55  and  105 

respectively, 


x  = 

59y  -  87 
55      ' 

x  = 

^101y-73 
105 

59y- 
55 

87 

_101y-73 
105 

Equate  values  of  x, 

Simplify,  1239y  -  1827  -  11 1 1  y  -  803, 

128y  =  1024. 

.-.y-8. 

Substitute  value  of  y  in  (3),  55  a?  —  472  =  —  87, 

55  a;  =  385. 
.*.  x=*  7. 


12.                *±2«1§ 
y-x      8 

(1) 

9*     2*±ii-ioo     • 

7 

(2) 

Simplify  (1), 
Transpose  and  combine, 
Simplify  (2), 
Transpose  and  combine, 

8x  +  8y  =  15y-15a;. 
23a;-7y  =  0 
63  a; -3y- 44  =  700. 
63a;-3y  =  741 

(3) 
(4) 

Multiply  (3)  by  3, 
Multiply  (4)  by  7, 
Subtract, 

69a>-21y=.     0 
441x-21y  =     5208 
-372  a;            =-5208 
.-.a;  =14. 

Substitute  value  of  x  in  (3), 

322-7y  =  0, 
-7y--322. 
.-.  y=46. 

170  ALGEBRA. 


13. 

3a?~5y  +  3  =  2x+y         (i) 

Z  o 

Z__x-2y=ix     y  (2) 

4  2     3 

Simplify  (1),  15a;- 25  y  +  30  =  4a;  +  2y. 

Transpose  and  combine,  1 1  x  —  27  y  =  —  30  (3) 

Simplify  (2),  96  -  3a:  +  6y  =  6a;  +  4y. 

Transpose  and  combine,  —  9a;  +  2y  —  —  96  (4) 

Multiply  (3)  by  9,  99x  -  243y  -  -   270 

Multiply  (4)  by  - 11,  99a;-   22y=     1056 

Subtract,  -  221  y  =  - 1326 

.-.  y  =  6. 
Substitute  value  of  y  in(4),  -  9x  + 12  =  -  96. 

.\x=12. 

14    4a?-3y-7^3s     2y     5     m 
5  10      15     6     W 

3        2      20  15       6     10 w 

Simplify  (1),  24a;  -  18y  -  42  =  9a;  -  4y  -  25. 

Transpose  and  combine,  15a;—  14y  =  17  (3) 

Simplify  (2),  20y- 20  +  30a;-9y-60=:4y-4a;  +  10a;+6. 

Transpose  and  combine,  24a;  +  7y  =  86  (4) 

Multiply  (4)  by  2,  48  x  +  14y  =  172  (5) 

(3)  is  15a;-14,y=    17 

Add  (3)  and  (5),  63  a;  =189 

.-.  x  =  3. 
Substitute  value  of  a;  in  (3),  45  — 14  y  =  17. 

.-.y  =  2. 
16.  •ri-lgi  (!) 

I+V2=3  (2)      ! 

Simplify  (1),  2ar-8  =  y  +  2.  I 

Transpose  and  combine,  2x  —  y  =  10  (3) 

Simplify  (2),  2x  +  3y  -  '6  =  36. 

Transpose  and  combine,  2a;  +  3y  =  42  (4) 

2x-    y  =  10  (3) 

Subtract,  4y  =  32 

.-.y  =  8. 
Substitute  value  of  y  in  (4),  2a;  +  24  =  42. 

.-.  s«9. 


I 


TEACHERS     EDITION. 


171 


16.       3*  +  12.v-9 

(i) 

1&       3x-0.25y  =  28        (1) 

11 

0.12<r+0.7y  =  2.54     (2) 

l-3x_ll-3j 

'(2) 

Multiply  (1)  by  0  04, 

7                5 

0.12a; -0.01y=  1.12     (3) 

Simplify  (1)  and  (2), 

0.12x+0.7  v  =  2.54'   (2) 

3a;  +  12y  =  99 

(3) 

Subtract,  —  0.71y  =  —  1.42 

15a;-21y  =  -72 

(4) 

.-.y  =  2. 

Divide  (3)  by  3  and  (4)  by 

15, 

Substitute  value  of  y  in  (1), 

x-"-12 

■V 

3a: -0.5  =28. 

3 

.-.  x  =  9.5. 

*--72  +  ! 

21  y 

15 

Equate  values  of  x, 

99-12.7^-72+1 

21  y. 

3                   15 

Simplify, 

495-60y  =  -72  +  J 

21  y. 

-81y  =  -567. 

.-.y-7. 

Substitute  value  of  y  in  (3), 

3s  +  84  =  99. 

19.           7(*-l)-3(y+8)(l) 

.-.  x  =  5. 

4a;  +  2     5y  +  9    ((>) 
9     "       2        W 
Simplify  (1)  and  (2), 

17.    5a-i(5y  +  2)  =  32 

(1) 

7a;-7  =  3y  +  24, 

3y  +  J(*  +  2)  =  9 

(2) 

7a:-3y  =  31           (3) 

Simplify  (1)  and  (2), 

8*  +  4-45y +  81, 

20aj-5y  =  130 

(3) 

8a:-45y  =  77          (4) 

a?  +  9y  =  25 

(4) 

Multiply  (3)  by  8  and  (4)  by  7, 

Multiply  (4)  by  20, 

56a-    24y  =  248 

20a:  +  180y  =  500 

(5) 

56a;-315y  =  539 

20a;  -     5y  =  130 

(3) 

Subtract,     291y  =  -291 
.-.y  — 1. 

Subtract,     185y  =  370 

.:y  =  2. 
Substitute  value  of  y  in  (3), 

Substitute  value  of  y  in  (3), 

20  a?- 10  =*  130. 

7* +  3-31. 

t  **  •?  —  I  • 

.\a?-4. 

172 


ALGEBRA. 


20.    7s  +  J(2y  +  4)=16 
3y-J(a?  +  2)  =  8 
Simplify  (1), 

35  s  +  2y  +  4  =  80. 
Transpose  and  combine, 

35a?  +  2y  =  76 
Simplify  (2), 

12y-a;-2  =  32. 
Transpose  and  combine, 
12y-z=»34 
.  Multiply  (4)  by  35, 

-  35  s  +  420y  =  1190 

35a?  +      2y=      76 

422y  = 1266 

.-.y-3. 

Substitute  value  of  y  in  (3), 
35a?  +  6  =  76, 
35  a;  =70. 
•  \  a?=  2. 


(1) 
(2) 


(3) 


(4) 


(3) 


21.    ^L=_^+3s  =  4y-2     (1) 

5s  +  6y_3a;-2y  , 

6  4  *  W 

Simplify  (1), 

5a?  -  6y  +  39a;  =  52y-26. 
Transpose  and  combine, 

44a?-58y  =  -26      (3) 
Simplify  (2), 
10a?+12y-9a;+6y  =  24y-24. 
Transpose  and  combine, 

a;_6y  =  -24      (4) 
Multiply  (4)  by  44, 

44a;  -  264y  =  -  1056 
44a?-  58y  =  -  26  (3) 

206y  =  1030 

.-.y  =  5. 

Substitute  value  of  y  in  (4), 
x  _  30  -  -  24. 
.-.a?  =-6. 


22. 

5s-3      3a 
2 
2x+y 


-19 


2 
9a;- 


«4 


_3jL 


(1) 


3(y  +  3)     4a;  +  5y  ,9) 
4  16       K"' 


2  8 

Simplify  (1),  15 x  -  9  -  9x  +  57  -  24  -  6y  +  2ar. 

Simplify  (2),  16a; +  8y-18a?  +  14  =  12y +  36-4a?-5y. 

Transpose  and  combine  (1), 
Transpose  and  combine  (2), 
Divide  (1)  by  2, 


Substitute  value  of  y  in  (4), 


4x+6y  =  -24 

(3) 

2a;  +    y=      22 

(4) 

2a;  +  3y  =  -12 

2y  =  -34 

.-.y  =  -17. 

2a; -17 -22, 

2a?  =  39. 

.-.  a;=19J. 

TEACHERS*    EDITION.  173 

23. 

3y  +  ll  =  i*l=^L±^  +  3l-4a    (1) 

(a;  +  7)(y-2)  +  3  =  2z£-(y-l)(x  +  1)  (2) 

Simplify  (1),  Zxy - 3 y2  +  12y  +  11a—  lly  +  44 

=  4a;2-zy-3ya  +  31a;-31y  +  124-4a?  +  4.Ty-lG.T 
Transpose  and  combine,     32y  —  4a;  =  80 
Divide  by  4,  8y-a;  =  20 

Simplify  (2),    xy  +  7y  —  2x  — 14  +  3  =  2xy-xy— y  +  x  +  1  (6) 

Transpose  and  combine,       8y  —  3a;=12 
Subtract  (5),  8y-    a;  =  20 

-2a;  =  -8 
.\a>  =  4. 
Substitute  value  of  a;  in  (5),    8y  —  4  =  20, 

8y  =  24. 

.-.y-s. 

24. 

6a;  +  9  3a?  +  5y  = «,       3a?  +  4      ,,v 

4  4a;  — 6                     2 

8y  +  7  6a;-3y      ,      4y -9         (2) 

10  2y  -  8                  5             w 

Multiply  (l)by4,  6a?  +  9  +  6*  +  *0V  =13  +  6a;  +  8. 

Transpose  and  combine.  — -*  =  12. 

^  2a;-3 

Divide  both  sides  by  (2),  3*  +  5 /  =  6. 

2a;  —  3 
Multiply  by  2a;  -  3,  3a;  +  5y  =  12a;- 18. 

Transpose  and  combine,  —  9  a;  +  5y  =  — 18  (3) 

Multiply  (2)  by  10  8 y  +  7  +  S0x-]5y  =  40  +  8y  -  18. 

y-4 

Transpose  and  combine,  — — *  =  15. 

y-4 

Divide  both  sides  by  15,  ^~a  ~ l' 

Multiply  by  y  —  4,  z. x  —  y  =  y  —  4. 

Transpose  and  combine,  2ar— 2y=  —  4. 

Divide  by  2,  x -y  =  -  2  (4) 

Multiply  (3)  by  1  and  (4)  by  9,  -  9  x  +  5y  =  -  18 

9a;-9y  =  -18 
Add,  _4y  =  -36 

.-.  y  =  9. 
Substitute  value  of  y  in  (4),  x  —  9  =  —  2. 

.-.  a;  =  7. 


174  ALGEBRA. 


26.  s-2JLz*  =  20-^2*     (1) 

23 -a;  'J  v  J 

Multiply  (1)  by  2,  2x  -  -£=-^  -  40 -  59  +  2x. 

23  — a; 

Ay    O  «. 

Transpose  and  combine,  — * =  19, 

Multiply  by  23  -  xt 

Transpose  and  combine,  4y  +  17  a?  =  437  (3) 

Multiply  both  sides  of  (2)  by  3, 
Transpose  and  combine, 


Multiply  by  x  -  18, 

Transpose  and  combine,  3y  —  17  a;  =  —  297  (4) 

Add  (3), 

Substitute  value  of  y  in  (3), 


23-a; 

4y-2*  = 

=  437 -19a;. 

4y +  17a;  = 

=  437 

"♦Si*- 

=  90-73  +  3y. 

3y-9_ 
a; -18 

*i7. 

3y-9  = 

.  17a;  -306. 

3y-17a;  = 
4y  +  17a;  = 

=  -297 
=     437 

7y 

.      140 
20. 

80  + 17a;  = 
17a;  = 

437, 
357. 

.*.  x  = 

21. 

Exercise  LXXII. 


1.                      x  \-y  =  a 

0) 

x-y=>b 

(2) 

Add,               2a;        =  a  +  b 

. x~a+h 
2 

Subtract  (2)  from  (1), 

2y  =»  a  —  b. 

.:y-aZb- 

(2) 


2.  ax  +  6y  =  c 

px  +  qy  =  r 

Multiply  (1)  by  p  and  (2)  by  a, 
apx  +  6py  =  cp  (3) 
apx  •+  agy  =  ar       (4) 

Subt.,      y(bp  —  aq)  =  cp  —  ar. 

6p  —  aq 

Multiply  (1)  by  q  and  (2)  by  b, 
aqx  +  bqy  =  eg 
fcpa;  +  bqy  =  br 

Subt.,     (ag  —  bp)  x  =cq  —  br 

.    «.      cq  —  br 

ag  — op 


teachers'  edition. 


175 


3. 


mx  +  ny 

px  +  qy. 

Multiply  (1)  by  p 

mpx  +  npy 

mpx  +  mqy 


(1) 
-*  (2) 

and  (2)  by  m, 
=  ap  (3) 

=  mb         (4) 


4. 


ax+by  =  e 
ax  4-  cy—  d 


(1) 
(2) 


•y  =  ; 


Sub. ,  (np — mq)  y  --=  ap—mb 
ap  —  mb 
np—mq 
and  (2)  by  n, 
=  aq 
=  nb 


Multiply  (1)  by  q 

mqx  +  nqy  -- 

npx  +  nqy  = 
Sub.,  (mq  —  np)a;  =  o^  —  nb 

■  ■■  ;»-«?-"*. 
mq  — np 


Multiply  (1)  by  m', 
Multiply  (2)  by  m, 
Subtract, 


Multiply  (1)  by  n', 
Multiply  (2)  by  n, 
Add, 


Multiply  (1)  by  d, 
Multiply  (2)  by  a, 
Subtract, 


Multiply  (1)  by/, 
Multiply  (2)  by  b, 
Subtract, 


Subt.,       (b—c)y=  t  —  d 

b  —  c 
Multiply  (1)  by  c  and  (2)  by  bt 
acx  +  bey  =»  ce 
abx  +  bey  =  bd 
Subt.,  (ac  —  ab)x  =  ce-bd 
•   x  —  M  —  bd 
a(c-b) 


6.       maj  —  wy  =  r 
m'a;  +  n'y  =  r1 
mm'x  —  m'ny  =  mfr 

mm9x  +  m  n'y  =  m/ 

(m'n  +  m  n')y  =  mr'  —  ra'r 


•y- 


7?17^  ■ 


m'n  +  mn' 
mnfx  —  nn'y  =  n9r 
mrnx  +  7in;y  =  n/ 
(mw'  +  m'n)  a:  =  n'r  +  n/ 
.    ~       w;r  +  nr1 


mnf  +  m'n 


6. 


ax  +  by  =  c 

dx+fy  =  <? 
adx  -f  6dy  =  erf 
acfe  +  g^  =  ac2 
bdy  —  afy^cd  —  ac2 

-   v-c(<*-«0 
"y       bd-af 

a  fa  +  tyy  =  cf 
bdx  +  bfy  =  6c» 


(a/-  6c?) a;  = 


•  cf-bc* 
af-bd 


(1) 
(2) 
(3) 
(4) 


(1) 
(2) 


176 


ALGEBRA. 


7. 


a     o 


a) 

(2) 


6     a 

Simplify  (1),  bx  +  ay  =  abc     (3) 
Simplify  (2),  ax  +  bey  =  0         (4) 
Multiply  (3)  by  a  and  (4)  by  6, 
abx  +  c?y  —  a*bc 
abx  +  IPcy  —  0 

Subt.,        a%y  —  b*cy  =  a2bc 
(a*  —  6*c)y  =  a26c, 


a'-62c 

Multiply  (3)  by  6c  and  (4)  by  a, 
6*cx  +  a6cy  =  a6V 
a's  +  a6cy  =  0 

Subt.,         b?cx  —  a2x  —  ab2c2 

(6»c-a*)a;  =  a6V. 

:x-     a6a°2 


9. 


(1) 

(2) 


b +  y      3a  +  x 
as  +  26y  =  d 
Simplify  (1), 

3  a*  +  ax  =  6*  +  by. 
Transpose  and  combine, 

ax-   6y  =  6*-3a*      (3) 

(2)  is     ax  +  2by  =  a* 

Subt.,       -  3 by  =  b%-  3a*-  </ 

"y  36 

Multiply  (3)  by  2, 

2aaj-26y  =  26*-6a* 

(2)  is     ax  +  2by=*d 

Add,  3a*  =  26*-6a*  +  d 

.  26»-6a»  +  rf 

•  •  x  —  ^ 

3a 


6*c- 


abx  +  cay  = 
ax—  cy  = 

Simplify  (2), 

a6aa;  —  6cay  * 
Multiply  (1)  by  6, 

a6*sc  +  6c<fo  = 
(3)  is  abdx  —  bedy  ■- 
Add,     (a62  +  a6a>  = 


(1)   10. 

■(2) 


bd 
d-6(3) 


26      (4^ 
d-bh] 


=  6+o* 
b  +  d 


or,a;  =  —  ■ 


Multiply  (1)  by  d, 

a6o*a;  +  cd2y  = 
(3)  is  a6aa;  —  6coy  = 
Add,     (cd2  +  bcd)y-- 

•••yj 

or,  y- 


o6(6  +  d) 
1_ 
a6' 


2d 
d-b 


=  b  +  d 

b  +  d  t 
~cd(b  +  dj 

"cd 


x 
a  +  6 

x 


1 
a  +  6 


+  _2 : 

a+6     a— 6     a- 

Add  (1)  and  (2), 


(1) 
(2) 


1 


a+6     a+6    a-6 

Simplify, 

2s(a-6)  =  2a, 

x(a  —  6)  =  a. 

a 

.-.  x  = -• 

a  — 6 

Subtract  (1)  from  (2), 

_2£_  =  _1 L 

a—b     a—b     a  +  ft 

Simplify, 

2y(a  +  6)  =  26, 
y(a  +  6)  =  6. 

•••y  =  aTV 


teachers'  edition.  177 

11. 

a(a  —  x)  =  b(x  +  y  —  a)  (1) 

a<y-ft-*)-%-6)  (2) 

Simplify  (1),  a2  -  as  =  bx  +  6y  -  afc.  (3) 

Simplify  (2),  ay  —  ab  —  ax  =  by  —  b2.  (4) 

Transpose  a2  and  for  in  (3),         ax  +  bx  =*  a2  +  ab —  by  (6) 

Transpose  ay  —  ab  in  (4),  aa;  —  ay  —  ab  —  by  +  6*.  (6) 

Divide  (5)  by  (a  +  b)  and  (6)  by  o,        x  =  ^^-fy, 

a+  6 

__ay  —  ab  —  by  -f  6* 

a 

Equate  values  of  x,  *+«»-fr  .  ay-ot-iy  +  y 

a  +  6  a 

Simplify,  a8  +  a26  —  a&y  =»  aty  —  a2b  —  62y  -f  J3, 

aty  +  a6y  —  U*y  «  os  +  2a26  —  6s. 
...  y  =  a  +  6. 
Substitute  value  of  y  in  (5),        ax+bx=*a2 +  ab  —  ab  —  b2. 

.*.  jc  =  a  — 6.     . 


12. 

^L±l     a  (1) 

ar-y-1 

ar  +  y-1 

Simplify  (1),  x—  y  +  \  =  ax—  ay  —  a. 

Simplify  (2),  x  +  y  +  1  =  bx  +  by  —  b. 

Trans,  and  combine,  (a  —  1)  x — (a — 1)  y  =  a  + 1  (3) 

(6-l)a>  +  (ft-l)y-&  +  l  (4) 

Multiply  (3)  by  6-1  and  (4)  by  (a-1), 

(a-l)(6-l)*-(a-l)(6-l)y-(a  +  l)(ft-l)  (5) 

(o-l)(6-l)a  +  (a-l)(6-l)y-(o-l)(6  +  l)  (6) 

Add,  2{a-l)(b-l)x  =  2(ab- 1) 

aJ-1 


(2) 


(a-l)(ft-l) 
Subtract  (5)  from  (6),    2 (a - 1)(6  -  l)y  =  2(a - 6). 

...  y «z^ 

y     (a-l)(*-l) 


178  ALGEBRA. 


13.       o*  =  ty  +  ^±£        (i) 

{a-b)x  =  (a  +  b)y  (2) 

Simplify  (1),  2ox-2&y  =  a»  +  6*  (3) 

Simplify  (2),  ax  -  bx  -  ay  -  by  =  0  (4) 

m  (3).  .-£±*±«Sr. 

In  (4),  -1*rfL 

Equate  valaes  of*,         °!+*±2&  _  2JLt|jf. 

Simplify, 

a8  +  ab*  +  2a&y  -  a*b  -  ft8  -  2b2y  =  2a*y  +  2a6y. 
Transpose  and  combine,     2a'y  +  26*y  =  a8  —  a*6  +  a62  —  6s, 

a-6 

Substitute  value  of  y  in  (1),  ax  *  ^^  +  £±£ 


14. 


ax  +  by  ~c* 

a) 

a 

y     a+  x 

(2) 

Simplify  (2), 
Add  (1)  and  (3), 

ax  —  by  =  -  o*  4-  0* 

ax  +  by  =  <? 

ax  —  by  =  —  a*  +  b* 

(3) 

2aa;  =  c2-a,  +  6* 

Subtract  (3)  from  (1), 

c^-a8*©* 

•'•—         2a       * 

aa?  +  6y  =  c* 

oar  —  by  =  ~a*  +  o8 

2&y  =  c8  +  aa-68 
*  *  y             ox 

teachers'  edition.  179 

16. 

-4  +  -*.  =2a  (1) 

a  +  6     a-b  ' 

x  —  y^  x  +  y 

2a6      a2  +  62 

Clear  (1)  of  fractions,     ax  —  bx  +  ay  +  Ay  =  2  a8  —  2  a62  (3) 

Clear  (2)  of  fractions,  a2* +62sc — a2y — 6^  =  2  a6z  +  2  aby  (4) 

2a8  — 2a62  — as  +  6a? 


(2) 


In  (3),  y- 

In(4),  y- 

Hence, 


a+6 
a'g—  2a6a;  +  b2x 
a2  +  2a6  +  62 
2as  -  2a62  —  ax  +  bx     a?x  -  2abx  +  b2x 


a  +  b  a2  +  2a6  +  62 

2a4  -  2a262  -  a*x  +  2a86  -  206s  +  b*x  =  a2*-  2abx  +  62x. 
Transpose  and  combine,         2a?x  —  2  abx  =  2  a4  —  2  a262  4-  2  a86  -  2  a68. 
Divide  by  2  a,  asc  —  bx  =  a8  —  a62  +  a26  —  6s 

a8  -  ao2  +  a*b  -  ft8 
a  —  b 
or,  a  =  a2  +  2a6  +  62. 
Substitute  value  of  a;  in  (3), 

o8  +  2a26  +  a62-a26-2a62-68+ay  +  6y  =  2a8-2a62. 
Transpose  and  combine,  ay  -f  by  =  a8  —  a26  —  062  +  6s. 

a8-a26-a62+68 


a  +  6 
or,  y  =  a*  —  2ab  +  62. 


16. 


bx 

-6(5  = 

-ay- 

-ac 

(1) 

X 

-y- 

=  a  — 

6 

(2) 

Transpose  (1), 

bx- 

-«y  = 

.(&- 

-a)c 

Multiply  (2)  by  a, 

ax- 

-«y  = 

.(«- 

-b)a 

(3) 

Subtract, 

(6- 

a)a?  = 

■  c(b 

-a)+a(6- 

-a) 

.•.  a?  = 

=  c  +  a. 

bx- 

-<*y  = 

=  (b~ 

■a)c 

(1) 

Multiply  (2)  by  6, 

bx  - 

-*y- 

.(a- 

-6)6 

(4) 

Subtract, 

(b- 

a)y  = 
-.  y  = 

=  c(6 

=  c  + 

-a)  + 
b. 

6(6- 

-a) 

180  ALGEBRA. 


17. 


(1) 


y-b 

a{x  -  a)  +  6(y  -  6)  +  abc  =  0  (2) 

Simplify  (1),  x  —  cy  =  a  —  bc  (3) 

(2)  is  ax  +  by  =  a*  +  6*  —  a6e  (4) 

Multiply  (3)  by  a,  ax  —  acy  =  a*  —  abc 

Subtract,  by  +  acy  =  6* 

...y— £_ 
b  +  ac 

Multiply  (3)  by  b  and  (4)  by  c,    6s  —  icy  =  ab  —  c6* 

aca;  +  bey  =  a*c  +  6*c  —  a6c* 
Add,  bx  +  acx  =  ab  +  a*c  —  abc1 

abc1 

,\  x  =  a r 

ac+b 

18. 

(a  +  b)x -(a -  b)y  =  4a6  (I) 

(a  -  6)x  +  (a  +  6)y  -  2a*  -  2b1  (2) 
Multiply  (1)  by  (a -6), 

(a»  _  b*)  x  -  (a  -  b)1  y  -  4  a*6  -  4  a6*  (3) 

Multiply  (2)  by  (a  +  6), 

(a*-6*)x  +  (a  +  6)*y  =  2as-2a6*  +  2a*6-26*(4) 

Subtract  (3)  from  (4),  (2a*  +  26*)y  =  2a*  -  2a*6  +  2a6* - 26s. 

.-.  y  =  a-6. 

Multiply  (1)  by  (a  +  6)  and  (2)  by  (a  -  6), 

(a  +  6)*x  -  (a*  -  6*)y  -  4a*6  +  4a6* 

(a  -  ft^a,  +  (gi  _  y) y  =  2as  -  2a*6  -  2a6»  +  26s 

Add,  (2a*  +  26*)*  =  2a1  +  2a*6  +  2a6*  +  26* 

.\  *  =  a  +  6. 

19. 

(»  +  o)(y  +  6)-(x  -  a)(y  -  6)  -  2(a  -  6)*  m 

Simplify  (1)  and  (2),      *~  y  +  2(a-A)  =  0  ® 

ay  4-  6a;  +  ay  4-  a6  —  ay  +  ay  4-  6a?  —  ab  =  2(a  —  6)*  (3) 

x-y  +  2a-26  =  0  (4) 

Transpose  and  combine,         2  ay  +  26a?  =  2  a*  —  4  a6  +  26* 

x  — y  =  26  —  2a 
Divide  (5)  by  2,  ay  +  6x  =  a*  — 2a6  +  62 

Multiply  (6)  by  a,  —  ay  +  ax  =  2a6  —  2a* 

Add,  (6  +  a)x=6*-a* 

.  \  x  =  6  —  a. 
Substitute  value  of  x  in  (6),      6  —  a  —  y  =  26  —  2a. 

.-.  y  =  a-6. 


TEACHERS     EDITION. 


181 


20. 

(a  +  ft)fa>  +  y)-(a-6)(*-y). 

(a-b)(x  +  y)  +  {a+b){x-y)~- 
Simplify  (1),  2  bx  +  2  ay  ■- 

Simplify  (2),  2ax-2by- 

Multiply  (3)  by  a,  2abx  +  2a*y- 

Multiply  (4)  by  6,  2abx-2b2y- 

Subtract,  (2a2+2b*)y: 


Multiply  (3)  by  6, 
Multiply  (4)  by  - 
Subtract, 


a, 


62 
a* 
b\ 
a3 
b> 


•.  y  = 

2b*x+2aby  = 
—  2a2a?  +  2aby  -- 


(2a3 +  26*)*  = 


a8 -6s 

a8 -ft8 
2(a*  +  J8)' 
a86 
-ab* 


a86  +  a&* 
a6  (a  +  6) 
2(a*  +  68)' 


(3) 
(4) 


Exercise  LXXIII. 


l  +  5-io 

x     y 

4      3      on 

-  +  -  =*  20 

x     y 
Multiply  (1)  by  4, 

4       8       Art 

-  +  -  =  40 
or     y 

l  +  ?  =  20 

*   y 

5-20 

y 
•••y=i- 

Multiply  (1)  by  3, 

?  +  ?-     30 

x     y 

(2)  by  2,        ?  +  ?=     40 


(2)  is 
Subtract, 


Subtract,  —  - 


=  -10 


or,  10a>  =  5. 


(i) 

(2) 
(3) 


(5) 


2. 


1     2 

_  +  _  _  a  (1) 

x     y 

M-5  (2) 

x     y  v 

Multiply  (1)  by  3, 

?  +  5  =  3a         (3) 
x     y  w 

3  4  . 
-  +  -  =  6 
*     y 


(2)i8 
Subtract, 


2_ 

y 

.-.  y  = 
Multiply  (1)  by  2, 


3a-6 

2 
Za-b 


(2)is 


2     4     o 
-  +  -  =  2a 
a;     y 

»  +  *-» 
a?     y 

1 


Subtract,     -        =b  —  2a. 
x 

.  _  1 


b-2a 


182 


ALGEBRA. 


3.  2__5  =  4.  (1) 

z     3y     27  W 

-1  +  1=11  (2). 

ix     y     72  W 
(1)i.          ?_5.  =  A 
K)             x     Zy     27 

8X(2)i8      M-£  0) 


Subtract, 

29      29 

3y     27 

,-.y  =  9. 

Substitute  value  of  y  in  (1), 

2       5       4 

*      27     27' 

2      9 

a?     27 

.-.  *  =  6. 

!  +  ?-4 

(1) 

*   y 

?_2_4 

(2) 

*    y 

Multiply  (1)  by  3, 

5  +  5-12 

a?    y 

(2)  is 

3_2«4 
x     y 

Subtract, 

5.8 

y 

.-.  y=l. 

(l)is 

1  +  2  =  4 

.»    y 

(2)  is 

!_2  =  4 

x     y 

Add, 


=  8 


1-5  =  6        (2) 

*   y 

Multiply  (1)  by  (4), 

*  y 

(2)by3,      ^-15  =  18 

*  y 

Subtract,  —  i=   2 

y 
.-.y  =  -J. 

Substitute  value  of  y  in  (1), 

-  +  8  =  5. 
x 

.•.*  =  -!. 


6.  ?  +  *  =  ??      (1) 

x     y      b 

*  +  ?  =  ^       (2) 
x     y      a 

Multiply  (1)  by  b  and  (2)  by  a, 

<*>     P     „. 
1-  —  =  ac 

*    y 


Subtract, 


I-  —  =  be 

x     y 


,y=  — 


ac  —  bc 


Multiply  (1)  by  a  and  (2)  by  k 
a?     ah     a*c 


Subtract, 


x       y       b 
6*     ab_¥c 
x      y       a 

a* -J2     a?c-Pc 
x              ab 
ab(a  +  b) 

ciat  +  ab  +  V*) 

teachers'  edition.  183 


7. 

A+3    5 

ax     by 

A_JL  =  3 
ax     by 

(i) 

(2) 

Multiply  (1)  by  5, 

10  +  16  =  25 
ax     by 

Multiply  (2)  by  2, 

l°-±  =   6 
ax     by 

Subtract, 

W-19 

i 

Multiply  (1)  by  2, 

*+i.-io 

Multiply  (2)  by  3, 

15  _  «.-   9 

aa;     by 

Subtract 

1?-19. 

1 
.  •.  x  -=  — 

a 

a 

—  +  —  =»m-f  n 
na;     my 

ti      771         »        • 

_  +  _  =  m2  +  n* 
a?      y 

(1) 
(2) 

Multiply  (1)  by  n, 

77171         71                /               v 

1 =•-  n(m  +  7i) 

Tia;      my                    ' 

Multiply  (2)  by  m 

mn  t  m*     m(m*  +  na) 
nx      ny             n 

Subtract, 

7is  —  7M8  __  n^m  +  n)  — 

m(m2 
m* 

+  n«) 

mny                         n 

7lS- 

m27i2  +  mws  - 

1 

•••  y  — 

-m4- 

-mhi* 

Substitute  value  of  y  in  (2),     x  =-  — 


184 


ALGEBRA. 


n       a     b 

9.      -  +  -  =  m 

x     y 

b     a 

x     y 

(1) 

(2) 

Multiply  (1)  by  b, 

h—  =bm 

x      y 

ab     a* 

=  an 

x       y 

Multiply  (2)  by  a, 

Subtract, 

*=bm  —  an 

V           a*  +  6* 
*  '  *  **  6m  —  an 

Multiply  (1)  by  a, 

a*  t  ab     ntm 
—  h =  ar» 

a?      y 

*    y 

Multiply  (2)  by  b, 

Add, 

°,  +  J,  =  «ro+6» 
a*  +  6» 

,  +  6n 


Exercise  LXXTV. 


1. 


5*  +  3y-6z  =  4  (1) 
3x-y+2z  =  8  (2) 
x-2y  +  2z  =  2     (3) 

(1)  is         5x  +  3y-6z  =   4  (1) 
3  x  (2)  is   9s-3y  +  6z  =  24 

Add,       14  s  =28 

.-.  x-2. 

(l)is         5s  +  3y-6z  =   4  (1) 
3x(3)is    3s-6y  +  6z=    6 
Add,         8s-3y  =10  (4) 

Substitute  value  of  x  in  (4), 
16-3y  =  10, 
-3y  =  -6. 

.-.y  =  2. 

Substitute  values   of  x  and  y 

in  (3), 

2-4  +  22  =  2, 

2z  =  4. 

.-.«  =  2. 


2.      4x-5y  +  2z  =  6 
2a?  +  3y-z  =  20 


7a-4y  +  3z  =  35 
Multiply  (1)  by  3  and  (3)  by  2, 
12s-15y+6z=     18  (4) 
14a?-8y+6z  =     70 
Subt., -2x-  ly        =-52  (5) 
Multiply  (2)  by  3  and  (3)  by  1, 
6a;+9y-3z  =  60 
7a?-4y+3z  =  35 

Add,    13s+5y        =95      (6) 
Multiply  (5)  by  5  and  (6)  by  7, 
10a;  +  35y  =     260 
91s  +  35y=     665 

Subt.,  -81*  =-405 

.*.  x  =  5. 
Substitute  value  of  x  in  (6), 
...y  =  6. 

2  =  8. 


TEACHERS '    EDITION. 


185 


3.  x  +  y  +  2  = 

5a;  +  4y  +  32  = 

15ar  +  10y  +  62  = 

(3)  is  15aH-10y+62: 
6x(l)is  6s+  6y+62= 
Subtract, 
(2)  is 
3x(l)is 
Subtract, 

(4)  is 
4x(5)is 

Subtract, 


9a?+  4y 
5a?+4y+32  = 
3a?+3y+32  = 


2x+  y 
9a;  +  4y 
8a;  +  4y 


=  6     (1 
=  22  (2) 
=  53  (3) 
=  53 
=  36 

=  17  (4) 
=  22 
=  18 

=  4 
=  17 
=  16 

=  1 
(5), 
=  4. 
=  2. 
and 
=  6. 
=  3. 


(5) 


Substitute  value  of  x  in 

2  +  y  = 

.\y  = 

Substitute   values  of  x 

in  (1),         1  +  2  +  2  = 

.'.2  = 

L  4a?  —  3y  +  2  = 

9a;  +  y-52  = 

x  —  4y  +  32  = 

(l)i8         4  a;— 3y+     2  = 

3x(2)is  27a;+3y-152  = 

Add,       31s       -142  = 

Multiply  (2)  by  4, 

36a:+4y-202  = 
(3)  is  s-4y+  32  = 

Add,       37a?       -172  =  66  (7) 
Multiply  (5)  by  37, 

1147ar-527z  =  2046 
31 X  (7)  is  1147a;-518z  =  2107 
Subtract,  -92  =  -63 

.'.2=7. 

(7), 
=  66, 
=  185. 
=  5. 
and  y 
9, 
-18. 


9  (I 

16  (2" 

2  \Z) 

■■   9 

^8  (4) 

=  57  (5) 

64  (6) 
2 


Substitute  value  of  2  in 

37*  -119  = 

37s  = 

.*.  x  = 

Substitute   values  of  x 

in(l),    20-3y  +  7  = 

-3y  = 

.-.y  = 


5.        8a;  +  4y-32  = 

x  +  3y  —  2  = 

4x  — 5y  +  42» 

(l)is       8a:+4y-32  = 

3x(2)is  3a:+9y-32  = 

Subt.,       bx—by 

Multiply  (2)  by  4, 

4s+12y-42  = 
(3)  is      4a;—  5y+42  = 

Add,      8a?+  7y 
Multiply  (4)  by  7  and 
35  a;  —  35y  = 
40a;  +  35y  = 
Add, 


-6      m 

>8       (3) 
6 

■■     21 


>-15(i) 


:28 


=  36     (5) 
(5)  by  5, 
=  -105 
=      180 


75a; 


Substitute  value  of  x 

b-by- 

-bl 

•••y- 

Substitute  values  of 

in  (2),     1  +  12-2  = 

.*.  2  = 

6.      12a;  +  5y-42  =  29 

13a;- 2y  +  5y  =  58 

17a;  — y -2=  15 

(l)is      12x+5y-42  =  29 

4x(3)  is  68a;-4y-42  =  60     (4) 

Subt.,    56a;-9y        =31     (5) 
(2)  is      13a;-2y+52==    58 
5x(3)is85a?-5y-52=    75  (6) 
Add,      98a;-7y        =133  (7) 
Multiply  (7)  by  9  and  (5)  by  7, 

882a; -63y  =  1197(8) 
392a; -63y  =  217(9) 

Subt.,   490  a;     =  980 

.-.  x  =  2. 
Substitute  value  of  x  in  (7), 
196  -  7y  =  133. 
.-.y  =  9. 
Substitute  values  of  x  and  y 
in(l),  24+45-42  =  29. 

.'.2=10. 


18* 


ALGEBRA. 


x-y~z  =  5  (1) 
aj  +  y_2  =  25  (2) 
a.  +  y  +  2=,35   (3) 

x-y-z=*   5 

x  +  y  +  g  =  35 
2*  =40 

.%  x  =  20. 

Substitute  value  of  a;  in  (2)  and  (3), 
y-2=    5 
y  +  2  =  15 


(l)is 

(3)  is 
Add, 


Add, 


Subtract, 


2y        =20 
.-.y-10. 
-  2z  =  -  10. 
.-.2  =  5. 


8.  x  +  y  +  2  =  30  (1) 

8x  +  4y  +  22  =  50  (2) 

27*  +  9y  +  32«64  (3) 

Multiply  (1)  by  2, 

2s  +  2y  +  22  =  60  (4) 

(2)  is      8a;  +  4y  + 22  =  50 
Sub., -6a;- 2y  -10  (5) 
Multiply  (1)  by  3, 

3s  +  3y  +  32  =  90  (6) 

(3)  is    27a  +  9y  +  32  =  64 
Sub.,-24a-6y  -=26(7) 
Multiply  (5)  by  3, 

-18a;-6y  =  30  (8) 

(7)  is         -24s-6y  =  26 

Subtract,       6  a;  =4 

.•-*-!■ 

Substitute  value  of  x  in  (8), 

-12-6y  =  30. 

Substitute  values  of  x  and  y 

in(l),       1-7  +  ^  =  30, 
•\  2  =  36J. 


Multiply  (1)  by  2, 
Multiply  (2)  by  3, 

Add, 

Multiply  (2)  by  7, 
Multiply  (3)  by  16, 

Add, 

Multiply  (6)  by  154, 
Multiply  (9)  by  3, 

Subtract, 


9.  15y  =  242-10a;  +  41    (1) 

15a;  =  12y-162  +  10    (2) 
18x-(72-13)=14y 

20a;  +  30y-482=    82 
45a-36y  +  482=   30 

65a;-    6y  =112 

105a;-  84y  +  1122  =  70 
288a;  -  224y  -  112z  =  -  208 

393a; -308y      =-138 

10,010a; -924y  =  17,248 
1,179a; -924y  =  -414 


8,831a; 


- 17,662 
.  x-2. 


Substitute  value  of  x  in  (6), 


130-6y  =  112. 

-•-y-3. 

Substitute  values  of  x  and  y  in  (1), 

20  +  45-242  =  41. 


(3) 

(4) 
(5) 

(6) 

(7) 
(8) 
(9) 


teachers'  edition. 


187 


10. 


>. 

3*-y 
5a;  +  3y- 

7a;+4y- 

+  z 
>2z 
•  bz 

=  17 
=  10 
-3 

8 

(3) 

Multiply  (1)  by  2, 

(2)  is 

6a?-2y  + 
5a?  +  3y- 

2z 
2z> 

=  34 
=  10 

(4) 

Add, 

lla;  +    y 

-44 

(5) 

Multiply  (1)  by  5, 

(3)  is 

15a?  —  by  + 

7a?  +  4y  — 

bz- 
bz- 

=  85 
=    3 

(6) 

Add, 
(5)  is 

22  a;-    y 
11a;  +    y 

=  88 
=  44 

$ 

Add, 

33a; 

.  x- 

=  132 
=  4. 

Substitute  value  of  x  in  (5), 

44  +  y- 

=  44, 

From 

(1), 

z- 

=  0. 
=  5. 

(3) 


12.  a?+2y  +  3«  =  6 
2a:  +  4y+22  =  8 
3a;  -f  2y  +  82  =  101 

Multiply  (1)  by  2, 

2a;  +  4y  +  6*=12 
(2) is  2a; -f  4y  +  22  =  8 
Subtract,  42=   4 

.'.2=1. 

(2)  is     2a;+4y+  22=    8 
2x(3) is  6a;+4y+162  =    202    (7) 
Subt.,-4a;        -142  = -194    (8) 
Substitute  value  of  2  in  (8)f 

-4a; -14  =  -194, 
-4a;  =  -180. 
.\x  =  45. 

Substitute  values  of  a;  and  z  in  (1), 
45  +  2y  +  3  =  6, ' 

2y  =  -42. 
.-.y  =  -21. 


11.  x  +  y  +  2  =  5     (1) 

3s-5y  +  72=75  (2) 
9x  -112+10  =  0     (3) 

Multiply  (1)  by  5, 

5a;  +  5y  +    52=   25(4) 
(2)  is    3a;-5y  +    72=    75 
Add,   8a;  +122-100(5) 

Multiply  (5)  by  9  and  (3)  by  8, 

72a;  +  1082=  900 
72a;-   88 z  =  -80 
Subtract,  1962  -  980 

.-.2  =  5. 

Substitute  value  of  z  in  (3), 

9a?-55  =  -10, 
9a;  =  45. 
.*.  a;  =  5. 

Substitute  values  of  x  and  2 

5  +  y  f  5  =  5. 
.-.  y  =  -5. 


13.        a;-3y-22=l 

2a?-3y+52  =  -19 
5a?  +  2y-2  =  12 

Multiply  (3)  by  2, 

10a;  +  4y-22  =  24 
(1)  is  x-3y-22=  1 
Subt.,  9a;+7y  =23 

Multiply  (3)  by  5, 

25a;+10y-52=      60 
(2) is       2a;-  3y+52  =  -19 

=      41 
=      23 


Add,  27x+  7y 
(5)  is    9a;+  7y 


Sub.,  18  a; 


18 


,  x=l. 


Substitute  value  of  x  in  (5), 
9  +  7y  =  23. 
.-.y  =  2. 

Substitute  values  of  x  and  y  in  (1), 
1-6-22  =  1. 
.•.2  =  -3. 


188 


ALGEBRA. 


14.  3x-2y  =  5       (1) 

4a-3y  +  2z=ll     (2) 

ar_2y-5z  =  -7    (3) 

Multiply  (2)  by  5  and  (3)  by  2, 

20ar-15y+102  =     55(4) 

2s-  4y-102  =  -14(5) 

Add,    22x-l9y  =     41(6) 

Multiply  (1)  by  19  and  (6)  by  2, 

57z-38y  =  95     (7) 

44s-38y  =  82     (8) 


16. 


Subtract,    13  s 


=  13 
.*-l. 


Substitute  value  of  x  in  (1), 
3-2y  =  5, 
-2y  =  2. 
.-.y 1. 

Substitute   values  of  x  and  y 
in  (2), 

4  +  3  +  22  =  11, 
22  =  4. 
.\z  =  2. 


15.  x+    y  =1     (1) 

y+    2=    9     (2) 

x  +    2=   5    (3) 

Add,      2a; +  2y  + 22  =  15 

x  +  y  +  2  =  7}     (4) 
Subtract  (1)  from  (4), 

2  =  6}. 
Subtract  (2)  from  (4), 

s  =  -l£. 
Subtract  (3)  from  (4), 

y  =  2j. 


(lHs 
(2)  is 

Add, 
(3)is 


2s-3y  =  3 
3y-4z  =  7 

-5* +  4z  =  2 
2.r-3y  =   3 

3y-42=    7 


=  10 

.    2 


2) 
(3) 


2x 
-5x 


-42  = 
-t-42  = 


Add,    -3a;  =12 

.-.*  =  - 4. 
Substitute  value  of  x  in  (1), 
-8-3y  =  3, 
-3y=ll. 
.-.y  =  -3f. 
Substitute  value  of  x  in  (3), 
20  +  42  =  2, 
42  =  -18. 
.-.«  — 4J. 


17.  3s-4y  +  62=l 

2sc  +  2y  — 2  =  1 

*x  —  §y  +  72  =  2 

(l)is        3s-  4y+62  =  l 

6x(2)  isl2aj+12y-6z  =  6 

Add,      15  s+  8y        =7 

Multiply  (2)  by  7, 

14.r+14y— 72  =  7 
(3)  is         Ix-  6y+7z  =  2 

Add, 


9 


2) 
(3) 

(5) 
(6) 
(7) 


21x+  8y 

Subtract  (7)  from  (5), 

-6x  =  -2. 

Substitute  value  of  x  in  (7), 
7  +  8y  =  9, 
8y  =  2. 

Substitute  values   of  z  and  y 
in  (2), 

l  +  J-«-l, 

-*=l_i-J. 


TEACHERS     EDITION. 


189 


18. 


7»-3y 
9y--5z 

x  +  y  -f  z 
Multiply  (3)  by  7, 

7x  +  7y  +  7z 
(1)  is     7s-3y 


=  30  (1) 
=  34  (2) 
-  33         (3) 

=  231 
=   30 


Subtract,      10y+7z 
Multiply  (2)  by  10  and 
90y-   50«  = 
90y  +    63z  = 
Subtract,  -113z 


=  201       (5) 
(5)  by  9, 
=       340 
-      1809 


n 


Substitute  value  of  z  in  (5), 

lOy  +  91  =  201, 
lOy  =  110. 
.-.y  =  ll. 

Substitute  values  of  y  and  z  in  (3), 
a+ll  +  13  =  33. 
.-.  c  =  9. 


=  - 1469 
-13. 


19. 


x  +  *  +  z- 
2     3 

z      x 
y     2     3 


6, 


=  17. 


Simplify, 

6a;  +  3y  +  2z  =    36  (1) 

2x  +  6y  +  3z  =  -6  (2) 

3<c  +  2y  +  6z=102  (3) 


20. 


1     2 

-  +  - 
x     y 

3_4 

y    z 

34 

z      x 


=  5 


(1) 


_-  =  5 


=  -6       (2) 


3x(l)is  18aj+9y+6z  =  108 
(3)  is        3aH-2y+6z  =  102 
Sub.,      15x+7y        =     6       (4) 
2x(2)is4aH-12y+6z  =  -    12 
(3)  is      3s+  2y+Qz  =     102 
Sub.,        z+lOy        =-114   (5) 
Subtract  7  X  (5)  from  10  X  (4), 

143x  =  858 

.\  re  =  6. 
Substitute  value  of  x  in  (5), 
y  =  -12. 
Substitute  values  of  a?  and  y  in  ( 1 ), 
z=18. 

Multiply  (6)  by  4  and  (3)  by  3, 

2! +  2? -los 

X        z 

12     9 

+-=    15 

X       z 


(3) 


Add 


41 


=  123 


Multiply  (1)  by  3  and  (2)  by  2, 

=  15         (4) 

=  -12     (5) 


3     6 

-  +  -  = 
x      y 

_8      6 
z      y" 


Subtract, 


3     8 

-  +  _  = 
x     z 


27     (6) 


Substitute  value  of  z  in  (3), 
9-i-B. 

X 

.-.  a?  =  l. 

Substitute  value  of  x  in  (1), 


190  ALGEBRA. 


21. 


i  +  l_I=a  (1)       (4)18        i+i+1-a+ft+c 

x     y     z  w  x     y     z 

x     y  +  2  W  *     y      2 

i  +  l-l  =  c  (3)       Subtract,        -         =a  +  c 
y     z     x  y 

Add,  l  +  i  +  I  =  a+6+c(4)  "y~^+7 

x   y   z  ill 

(l)is  l  +  i-l-a  •     V     ■ 

*     y     * (3)is    -I+I+l-c 

o  %    y    z 

Subtract,  -  =  6  +  c  

2  Subt.,       -  =a  +  & 

-  *       2 


6  +  c 


a  +  6 


&s  +  cy  = 

a 

(1) 

<K  +  cx  = 

-b 

(2) 

ay  +  bx  = 

*c 

(3) 

abz  +  acy  = 

■a* 

(4) 

abz  +  6cx  = 

-V 

© 

acy  +  bcx  = 

=  c* 

(6) 

2a6z  +  2acy  +  2bcx  = 

*a*  + 

6* 

+  C3 

(7) 

2bcx  = 

*&*  +  (* 

-a2 

(8) 

2acy  = 

=  a»- 

&» 

+  C2 

(9) 

2abz  = 

-<*'  + 

6» 

-c* 

(10) 

Multiply  (1)  by  a, 
Multiply  (2)  by  6, 
Multiply  (3)  by  c 

Add  (4),  (5),  and  (6), 
Subtract  twice  (4)  from  (7), 
Subtract  twice  (5)  from  (7), 
Subtract  twice  (6)  from  (7), 

In(8),  —  *  +  *-* 

In  (9), 
In  (10), 


2  6c 
a»-6a  +  c2 


2ac 
x  a3  -f  ft2  - 
2a6 


TEACHERS     EDITION. 


191 


3.         2      4+l  =  7|  (1) 
x     by     z 

l  +  l+a.ioj  (2) 
3a;     2y     z 

l_J_+i=16A  (3) 
5a;      2y     z 

Multiply  (1)  by  60  and  (2)  by  30, 

180    48     60     AKa  ... 

+  —  =  456  (4) 

x       y      z  w 

10     15     60     OAC  /KN 

—  +  —+—  =  305  (5) 
x      y      z 


24.  ?_?  +  !=2.9      (1) 

x     y      z 


5_6_7 
x     y     z 


=  -10.4(2) 


Sub.,™L?3         =151         (6) 
x       y 


Multiply  (2)  by  60  and  (3)  by  30,       (2)  i8  £  _  5  - 1 .  -10.4(6) 


y       z      x  w 

Add,  _I  +  2  =  7.4      (4) 


Multiply  (1)  by  2, 

1-2  +  5-M     (5) 
x     y     z  w 

5     6     7 


5+S  +  ^.mo        (7) 

*J$+^-«S       .(8) 
x      y        z 

4     45 
Sub.,--+—  =127        (9) 

x     y 

Multiply  (6)  by  2  and  (9)  by  85, 

^-^  =  302       (10) 

_*»  +  ?«*_  10795  (11) 
x        y 

Add,  2®?- 11097 

y 
•••y  =  i- 

Substitute  value  of  y  in  (9), 

-1  +  135  =  127, 
x 

8a;  =  4. 

Substitute  values  of  x  and  y 
in  (5), 

RO 
20+45+— =  305, 
z 

240z  =  60. 

•\  z  =  J. 


x     y     z 


Subt.,     -1      +  —  =16.2  (7) 

a;  z  v  ' 

(4)  is  _!  +  !=    7.4  (8) 

a?     z  v  ' 


Subtract,  £=    8.8 

z 

Substitute  value  of  z  in  (4), 

-1  +  7.7  =  7.4. 
x 

Simplify,   — 1  +  7.7  a;  =  7.4  a;, 
3a;  =10. 

Substitute  values  of  a;  and  z  in  (1), 
0.6 --  +  4.4  =  2.9. 

y 

Simplify, 

0.6y-3  +  4.4y  =  2.9y. 

.-.y-lf. 


192 

ALGEBRA. 

25 

?+l-5-o 

x     y     z 

(1) 

(5)  is                   *_*  =  2 

x      z 

?_2  =  2 

2       y 

(2) 
(3) 

Mul.(3)by3,    -  +  ?  =  4 

2        Z 

X      z 
Multiply  (1)  by  2, 

Add,                   1        =6 

X 

.•.*=ii. 

Substitute  value  of  x  in  (5), 

x     y     z 
(2)is                --  +  i=2 

(*) 

24^5  =  2 

7      z 

Substitute  values  of  x  and  y  in  (1), 

y    2 

(5) 

12            1            10           yv 

Add,             *        _?  =  2 

—  +  -  —  —  =  0. 

...  y=_3J. 

ax  +  by  +  cz  *-  a  (1) 

ax  —  by  —  cz  =  b  (2) 

ax  +  cy  +  62  =  c  (3) 

Add  (1)  and  (2),  2  ox  =  a  +  b  (4) 

a  +  6 
.\  x  =  -- — • 
2a 
Multiply  (2)  by  b  and  (3)  by  c, 

afo  _&*y  -&cz  =  &2  (5) 

acx  +  chj  +  bcz=*<?  (6) 

Add,  afcx  +  acx-&2y  +  chf  =  62  +  c2  P) 

Substitute  value  of  x  in  (7), 

g26  +  a2c  +  a&2  +  abc     ^  -(*)y  =  &  +  <*. 

ab  +  ac  +  bc  —  ft2 -2c2 
"y~  2(62-c2) 

Substitute  values  of  x  and  y  in  (3), 

a  +  b     abc  +  ae*  +  be2  —  62c  —  2c8      ,   = 

2  2(&a-c2) 

,       352c-a62-a6c-52 

2  2(62-c2) 

36c—  aft  —  ac  —  h* 

'■'  *  2(&2-c2) 


teachers'  edition.  193 


81 


X—  I 


a  a  +  b  +  c 
y ~ z  =  x  —  a  —  b 

b  a  +  b  +  c 
x + z     x—a— b 


0) 


27. 

%x~ U  _  3y  -f  2z_  x  —  y  —  z^  . 
3  4  5 

Simplify,  40s  -20y-45i/  +  30  2  =  12a?  -  12y  -  12z  =  240 

40a;-20y  =  240  (1) 

45y  +  30z  =  240  (2) 

12a?-12y-12z  =  240  (3) 

Divide  (1)  by  20,  2x  -  y  =  12  (4) 

Divide  (2)  by  15,  3  y  +  2  z  =  16  (5) 

Divide  (3)  by  12,  x  -  y  -  z  -  20  (6) 

Multiply  (4)  by  3,  6a? -3y*         =36  (7) 

(5)  is  3y  +  2z  =  16 

Add,  6a?  +  2z  =  52  (8) 

(4)  is  2a; -y        =    12 


[6)  is  x-y-z=a   20 

Subtract,  x        +  z  =  -8  (9) 

(8)-*-2is  3a;  +  z  =     26 

(9)  is  x+z  =  -   8 

Subtract,  2  a;         =     34 

.-.a;  =  17. 
Substitute  value  of  x  in  (4V  y  =  22. 

Substitute  value  of  y  in  (5),  z  =  —  25. 


(1) 
(2) 
(3) 


c        a  +  6  +  c 

x(a  +  b +  c)  —  y(a  +  b +c)  =  ax  —  a2  —  ab  (4) 

-z(a  +  b  +  c)+y(a  +  b  +  c)  =  bx  —  b*  —  ab  (5) 

x(a  +  b  +  c)  +  z(a  +  b  +  c)  =  ex  —  ac  —  be (6) 

Add,  2 a; (a  +  b  +  c)  =  x(a+b+c)-at-b2-2ab-ac-bc 

x(a  +  b  +  c)  =  —  (a*  +  b2  +  2ab  +  ac  +  be). 
.-.  x=>  —  (a  +  6). 
From(4),  -(a+6)(a+6+c)-yfa+6+c}  =  -  2a2  -  2a&, 
or,  — y{a+b+c)  =  —  a2+b2  +  ac  +  be. 

_(a+6)(tt-6-c) 
a  +  6  +  c 
From(6),  -(a+6)(a+6+c)+z(a+6+c)  =  -  2ac  -  26c, 

-     z(a+6+c)  =  a2  +  2  aft  +  62  —  ac  —  6c. 

...  z=3(a  +  6-c)(a  +  6)t 
a  +  6  +  c 


194  ALGEBRA. 


Exercise  LXXV. 

1.  The  sum  of  two  numbers  divided  by  2  gives  as  a  quotient 
24,  and  the  difference  between  them  divided  by  2  gives  as  a  quo- 
dent  17.     What  are  the  numbers? 


Let                x  =  first  number, 
id                    y  =  second  number. 

Then 

*±2  =  24 
2 

id 

2 

Add  (1)  and  (2), 
Subtract  (2)  from  (1), 

3  =  41 

y-7. 

0) 

(2) 


2.  The  number  144  is  divided  into  three  numbers.  When  the 
first  is  divided  by  the  second,  the  quotient  is  3  and  the  remainder 
2 ;  and  when  the  third  is  divided  by  the  sum  of  the  other  two 
numbers,  the  quotient  is  2  and  the  remainder  6.  Find  the  num- 
bers. 

Let  x  =  first  number, 

y  =  second  number, 


and                    z  =  third  number. 

Then 

x  +  y  +  2=144 

(1) 

£n2  =  3 

(2) 

and 

x+y 

(3) 

Simplify  (2), 

x-3y  =  2 

W 

Simplify  (3), 

«-2y-2x  =     6 

(5) 

Multiply  (1)  by  2, 

2x  +  2y  +  22  =  288 

(6) 

Add  (5)  and  (6), 

32  =  294 

.-.2  =  98. 
Substitute  value  of  z  in  (1),  x  +  y  +  98  =  144, 

s  +    y  =  46  (7) 

(4)  is  a:-3y=    2 

Subtract,  4y  =  44 

.-.y-U. 
Substitute  value  of  y  in  (7),      x  +  11  —  46. 

.-.a?  =  35. 


teachers'  edition.  195 

3.  Three  times  the  greater  of  two  numbers  exceeds  twice  the 
less  by  10 ;  and  twice  the  greater  together  with  three  times  the 
less  is  24.    Find  the  numbers. 

Let  x  =  greater  number, 

and  y  =  less  number. 


Then 

3z-2y  =  10 

(1) 

and 

2s  +  3y  =  24 

(2) 

Multiply  (1)  by  2, 

6x~   4y  =     20 

Multiply  (2)  by  3, 

6x+   9y  =      72 

Subtract, 

-13y  =  -52 

•••y-4. 

Substitute  value  of  y 

in(l),  3s-8  =  10. 
.\x  -6. 

4.  If  the  smaller  of  two  numbers  is  divided  by  the  greater, 
the  quotient  is  0.21  and  the  remainder  0.0057  ;  but  if  the  greater 
be  divided  by  the  smaller,  the  quotient  is  4  and  the  remainder 
0.742.     What  are  the  numbers  ? 

Let  x  =  larger  number, 

and  y  =  smaller  number. 

Then       -  =  smaller  divided  by  larger. 

x 

-  =  larger  divided  by  smaller. 


Hence 

2^021  .0.0057 

(1) 

X                             X 

and 

x_        0.742 

y           y 

(2) 

Simplify  (1), 

y  =  0.21  x  +  0.0057 
y  -  0.21  x  =  0.0057 

(3) 

Simplify  (2), 

x  =  4y  +  0.742 
x— 4y  =  0.742 

W 

Multiply  (3)  by  4, 

4y-0.84x  =  0.0228 

(4)  is 

-4y+         x  =  0.742 

Add, 

0. 16  X-  0.7648 
.-.3  =  4.78. 

Substitute  value  of  x  in  (4), 

-4y=  -4.038 

.-.y  =  1.0095. 

1 


196  ALGEBRA. 


5.  Seven  years  ago  the  age  of  a  father  was  four  times  that  of 
his  son ;  seven  years  hence  the  age  of  the  father  will  be  double 
that  of  the  son.    What  are  their  ages? 

Let  x  =  number  of  years  in  father's  age. 

Then    x  +  7  =  number  of  years  in  father's  ago  7  years  hence 

x  —  7  =  number  of  years  in  father's  age  7  years  ago. 
Let  y  =  number  of  years  in  son's  age. 

Then   y  +  7  —  number  of  years  in  son's  age  7  years  hence, 
y  —  7  =  number  of  years  in  son's  age  7  years  ago. 

*-7  =  4(y-7)  (1) 

s  +  7  =  2(y  +  7)  (2) 

<r-4y  =  -21  (3) 

s-2y  =       7  (4) 

Subtract,  -2y 28 

.-.y  =  i4. 

Substitute  value  of  y  in  (4), 

ar-28  =  7. 
.\  x  =  35. 

6.  The  sum  of  the  ages  of  a  father  and  son  is  one-half  what 
it  will  be  in  25  years ;  the  difference  between  their  ages  is  one- 
third  of  what  the  sum  will  be  in  20  years.    What  are  their  ages? 

Let  x  =  number  of  years  in  father's  age, 

and  y  =-  number  of  years  in  son's  age. 

Then    x  +  y  —  sum  of  ages, 
x  +  y  +  50  —  sum  of  ages  in  twenty-five  years. 


t  x  4-  y  +  50 
2 
;-y-*  +  y  +  40  (2) 


*  +  y  =  -l^  (i) 


Simplify  (1),  x  -f  y  =■  50 

Simplify  (2),  2  x  -  4  y  =  40 

(3)  is  x  +    y  =  50 

(4)  -*-  2  is  a;-2y  =  20 

Subtract,  3y  =  30 

.-.y  =  10. 

Substitute  value  of  y  in  (3), 

x  +  10  -  50. 
.-.  x-40. 


teachers'  edition.  197 

7.  If  B  give  A  $25,  they  will  have  equal  sums  of  money  ;  but 
if  A  give  B  $22,  B's  money  will  be  double  that  of  A.  How 
much  has  each  ? 

Let  or  =  number  of  dollars  B  has, 

and  y  =  number  of  dollars  A  has. 

Then  x  —  25  =  number  of  dollars  B  has  after  giving  $25  to  4, 
y  +  25  =  number  of  dollars  A  has  after  receiving  $  25. 
z-25  =  y  +  25  (1) 

y  —  22  =  number  of  dollars  A  has  after  giving  $25  to  B. 
x  +  22  =  number  of  dollars  B  has  after  receiving  $22. 
z  +  22  =  2(y-22)  (2) 

Transpose  and  combine, 

x-    y  =     50  (3) 

x  _  2y  =  -  66  (4) 

Subtract,  y  =    116 

Substitute  value  of  y  in  (3), 

x-  116  =  50. 
.-.a  =166. 


8.  A  farmer  sold  to  one  person  80  bushels  of  wheat  and  40 
bushels  of  barley  for  $67.50;  to  another  person  he  sold  50 
bushels  of  wheat  and  30  bushels  of  barley  for  $85.  What  was 
the  price  of  the  wheat  and  of  the  barley  per  bushel? 

Let     x  =  number  of  dollars  received  per  bushel  of  wheat, 
and         y  =  number  of  dollars  received  per  bushel  of  barley. 

Then  30*  +  40y  =  671  (1) 

50s  +  30y  =  85  (2) 

Simplify  (1),  60s  +  80y  =  135  (3) 

Multiply  (2)  by  &    60s  +  36y  =  102  (4) 


ract, 

44y 

=   33 

.-.  y 

-i 

bitute  value  of 

yin(3), 

60* +  60  = 

135, 

60x  = 

75. 

••.  X- 

=-H. 

198  ALGEBRA. 


9.  If  A  give  B  $5,  he  will  then  have  $  6  less  than  B ;  but  if  he 
receive  $5  from  B,  three  times  his  money  will  be  #20  more  than 
four  times  B*s.     How  much  has  each? 

Let  x  =  number  of  dollars  A  has, 

and  y  =  number  of  dollars  B  has. 

Then  x  —  5  =  number  of  dollars  A  has  after  giving  B  $ 5, 

and  y  +  5  =  number  of  dollars  B  has  after  receiving  $  5. 

Hence,        x  —  5  =  y  -f  5  —  6, 
and  3(x  +  5)  =  4(y  -  5)  +  20. 

Transpose,  x  —  y  =  4  (1) 

3*-4y  =  -15  (2) 

Multiply(l)by3,  3s-3y  =     12  (3) 

(2)  is  3s-4y  =  -15 

.-.y=    27 
Substitute  value  of  y  in  (1), 

*-27  =  4. 
.\*-31. 


10.  The  cost  of  12  horses  and  14  cows  is  $1900;  the  cost  of 
5  horses  and  3  cows  is  $650.  What  is  the  cost  of  a  horse  and  a 
cow  respectively? 


Let                     x  =  number  of  dollars  a  hors 
and                       y  =  number  of  dollars  a  cow 

e  costs, 
costs. 

Then                            12a;  +  14y  -  1900 
and                                    5a;  +    3y  =  650 

a) 

(2) 

Multiply  (1)  by  3,       36*  +  42y  =  5700 
Multiply  (2)  by  14,      70*  +  42y  =  9100 

(3) 

Subtract,                       34*             =  3400 

.-.  *  =  100. 

Substitute  value  of  *  in  (2), 

500  +  3y-650, 
3y  =  150. 
.-.y-60. 

teachers'  edition.  199 


11.  A  certain  fraction  becomes  equal  to  2  when  7  is  added  to 
its  numerator,  and  equal  to  1  when  1  is  subtracted  from  its 
denominator.     Determine  the  fraction. 

Let  -  =  required  fraction. 

V 

By  conditions,       ^±_?  =  2  (1) 

and  -2—  =  1  (2) 

Simplify  (1),  x  +  7  =  2y  (3) 

Simplify  (2),  x  =  y  - 1  (4) 

Transpose  (3),  x  -  2  y  =  -  7  (5) 

Transpose  (4),  x  —    y  =  —  1  (6) 

Subtract,  y  =      6 

Substitute  value  of  y  in  (5), 

rr  -  12  =  -  7. 
.*.  x  =  5. 
.•.  fraction  =  4. 


12.  A  certain  fraction  becomes  equal  to  \  when  7  is  added  to 
its  denominator,  and  equal  to  2  when  13  is  added  to  its  numera- 
tor.   Determine  the  fraction. 

Let  -  =  required  fraction. 

y 

By  conditions,       — ^—  =  -  (1) 

y  +  7     2 

and  ^±13  =  2  (2) 

y 

Simplify  (1),        2s-y  =  7  (3) 

Simplify  (2),        x  -  2y  =  -  13  (4) 

Multiply  (3)  by  2, 

4a;-2y=      14 
(4)  is  x  -2y  =  -  13 

Subtract,  3  a;  =27 

.-.  a;  =  9. 

Substitute  value  of  x  in  (3), 

18-y  =  7. 

.-.y  =  ll. 

.•.  fraction  =  -ft. 


200  ALGEBRA. 

13.  A  certain  fraction  becomes  equal  to  J  when  the  denomi- 
nator is  increased  by  4,  and  equal  to  ff  when  the  numerator  is 
diminished  by  15.    Determine  the  fraction. 

Let  _  b  fraction. 

y 

Then  -*_  =  *  (l) 

y  +  4     9  W 

-X  — 15     20  /gv 

"«  () 

Simplify  (1),  9  x  =  ly  +  28. 

Simplify  (2),  41  a?  -  615  =  20y. 

Transpose,  9 a  -  7y  =  28  (3) 

41s-20y  =  615  (4) 

Multiply  (3)  by  20,         180a;-140y  =       560  (5) 

Multiply  (4)  by  7,  287s-140y  =     4305  (6) 

Subtract,  - 107  x  =  -  3845 

.-.a;  =  35. 

Substitute  value  of  x  in  (3),  315  -  7y  =  28. 

.\  y  =  41. 
.\  fraction  =  ff. 

14.  A  certain  fraction  becomes  equal  to  J  if  7  is  added  to  the 
numerator,  and  equal  to  §  if  7  is  subtracted  from  the  denomina- 
tor.   Determine  the  fraction. 

Let  -  =  fraction. 

y 

Then  £Jl2=2  m 

V        3 

and  -?-  -  ?  (2) 

y-7     8  K> 

Simplify  (1),  3a;  +  21  =  2y. 

Transpose,  3  x  -  2y  =  -  21                  (3) 

Simplify  (2),  8a?  =  3y-21. 

Transpose,  8  x  -  3  y  =  -  21                  (4) 

Multiply  (3)  by  3,  9a?  -  6y 63 

Multiply  (4)  by  2,  16*  -  6y  =  -  42 

Subtract,  -7x  =-21 

.\  x  =  3. 

Substitute  value  of  x  in  (3),      9  —  2y  =  —  21. 

.\  y  =  15. 
.\  fraction  = -j^. 


2     5,3 

x  +  y~2 

(1) 

?  +  5  =  2 

y    * 

(2) 

x      y 

(3) 

*  +  !?..  10 

a;       y 

(4) 

»          =7 

.•.  x  =  3. 

teachers'  editiow.  201 

15.  Find  two  fractions  with  numerators  2  and  5  respectively, 
such  that  their  sum  is  \\ ;  and  if  their  denominators  are  inter- 
changed their  sum  is  2. 

Let  x  =  denominator  of  first  fraction, 

and  y  =  denominator  of  second  fraction. 

Then 
and 
Multiply  (lfby  2, 
Multiply  (2)  by  5, 
Subtract, 


Substitute  value  of  x  in  (2),  y  =  6. 

.  \  first  fraction  =  $,     second  fraction  =  f . 

16.  A  fraction  which  is  equal  to  J  is  increased  to  -ft  when  a 
certain  number  is  added  to  both  its  numerator  and  denominator, 
and  is  diminished  to  g  when  one  more  than  the  same  number  is 
subtracted  from  each.    Determine  the  fraction. 

Let  x  equal  numerator,  y  the  denominator,  and  z  the  number 
to  be  added. 
Then  2=§  (l) 

y     3o 
y  +  z     11  li 

and  "(,  +  1)-g  (3) 

Clear  of  fractions  and  transpose, 

Sx-2y  =  0  (4) 

llz-8y  +  3z  =  0  (5) 

9s-5y-4z  =  4  (6) 

Multiply  (5)  by  4,    44a-  32y  +  12*=-   0 

Multiply  (6)  by  3,    27s- 15y-  12z  =  12 

Add,  71a-47y  =12  (7) 

Multiply  (7)  by  3,  213  x  -  141  y  =  36 

Multiply  (4)  by  71,         213s-142y~   0 

Subtract,  y  =  36 

Substitute  value  of  y  in  (1),  x  =  24. 

•\  fraction  =  ||. 


202  ALGEBRA. 


17.  The  sum  of  the  two  digits  of  a  number  is  10,  and  if  54  be 
added  to  the  number  the  digits  will  be  interchanged.  What  is 
the  number? 

Let  x  =  digit  in  tens'  place, 

and  y  =  digit  in  units'  place. 

Then  10  a;  +  y  =  number. 

By  conditions,  x  +  y  =  10  (1) 

and  10a;  +  y  -f  54  =  lOy  +  xt 

9a;-9y  =  -54. 
Divide  by  9,  x  -  y  =  -  6  (2) 

Add  (1)  and  (2),  2a;  =  4. 

r.x  =  2. 
Subtract  (2)  from  (1 ),        2y  -  16. 
.-.y  =  8. 

number  =  10  a;  +  y. 
.\  number  =  28.    • 


18.  The  sum  of  the  two  digits  of  a  number  is  6,  and  if  the 
number  be  divided  by  the  sum  of  the  digits  the  quotient  is  4. 
What  is  the  number? 


Let 

x  =  digit  in 

tens' 

place, 

and 

y  -=  digit  in 

units 

'  place. 

Then 

10  a?  +  y  =»  number, 

and 

*  +  y  =  6 

(i) 

But 

10a?  +  y^1 
6 

(2) 

Clear  of  fractions, 

10a;  +  y  =  24 
x  +  y=   6 

Subtract, 

9x        -18 
.-.  a;«2. 

Substitute  value  of 

x  in  (1), 
2  +  y  =  6. 
.-.y-4. 

.*.  number  -» 24. 

teachers'  edition.  203 

19.  A  certain  number  is  expressed  by  two  digits,  of  which 
the  first  is  the  greater.  If  the  number  is  divided  by  the  sum  of 
its  digits  the  quotient  is  7;  if  the  digits  are  interchanged,  and 
the  resulting  number  diminished  by  12  is  divided  by  the  differ- 
ence between  the  two  digits,  the  quotient  is  9.  What  is  the 
number? 

Let  x  =  digit  in  tens'  place, 

and  y  =  digit  in  units'  place. 

Then  10  jc  +  y  =  number. 

By  conditions,  *  =  7  (1) 

J  x +y  x ' 

10y  +  a.-12  =  & 

x  —  y  v  ' 

Simplify  (1),  3s-6y  =  0  (3) 

Simplify  (2),  -8a;  +  19y  =  12  (4) 

Multiply  (3)  by  f  8s-16y  =    0 

Add,  3y  =  12. 

Substitute  in  (3),  x  =  8. 

.\  number  =  84. 

20.  If  a  certain  number  is  divided  by  the  sum  of  its  two 
digits,  the  quotient  is  6  and  the  remainder  3 ;  if  the  digits  are  in- 
terchanged, and  the  resulting  number  is  divided  by  the  sum  of 
the  digits,  the  quotient  is  4  and  the  remainder  9.  What  is  the 
number? 

Let  x  =*  digit  in  tens'  place, 

and  y  =  digit  in  units'  place. 

Then  10  a?  +  y  =  number. 
By  conditions,  Wx  +  y-3  =  6  (1) 

J  s  +  y 

10y  +  a?-9  =  4 
x  +  y 
Clear  (1)  and  (2)  of  fractions,  transpose  and  combine, 

4s-5y  =  3  (3) 

-3a  +  6y  =  9 
Divide  (4)  by  3,  -  x  +  2y  =  3 

Add  4  x  (5)  and  (3),        -4s  +  8y  =  12 
4s-5y  =    3 

3y  =  15 

'.y  =  5. 
Substitute  value  of  y  in  (3),  x  =•  7. 

•\  number  =  75. 


204  ALGEBRA. 

2L  If  a  certain  number  is  divided  by  the  sum  of  its  two  digits 
diminished  by  2,  the  quotient  is  5  and  the  remainder  1 ;  if  the 
digits  are  interchanged,  and  the  resulting  number  is  divided  by 
the  sum  of  the  digits  increased  by  2,  the  quotient  is  5  and  the 
remainder  8.     Find  the  number. 

Let  x  =  digit  in  tens'  place, 

and  y  =  digit  in  units  place. 

Then  10  x  +  y  =  number. 

By  conditions,  I0x  +  y  =  5  + 1 ^ 

x+y—2  x+y—2 

and  My  +  S-5+— 3 

x  +  y+2  a?  +  y  +  2 

Clear  of  fractions,  10 a:  +  y  =  5  x  +  by  — 10  4- 1. 

lOy  +  x  =  bx  +  by  +  10  4-  8. 
Transpose  and  combine,     bx  —  4y  =  —  9  (1) 

5y-4x  =  18  (2) 

Multiply  (1)  by  5,  25  x  -  20y  -  -  45 

Multiply  (2)  by  4,       -  16  a;  +  20y  -      72 

Add,  9x  -     27 

.-.a?  =  3. 
Substitute  value  of  x  in  (1),  y  =  6. 

.\  number  =  36. 


22.  The  first  of  the  two  digits  of  a  number  is,  when  doubled, 
3  more  than  the  second,  and  the  number  itself  is  less  by  6  than 
five  times  the  sum  of  the  digits.    What  is  the  number? 

Let  x  =  digit  in  tens'  place, 

and  y  =  digit  in  units'  place. 

Then  10  a;  +  y  =  number. 

By  conditions,  2x  —  y  +  3 

and  10x  +  y  +  6  =  5x  +  5y 

Transpose  and  combine,        2  a;  —  y  =  3 

5a;-4y  =  -6 

Multiply  (3)  by  4,  8 x  -  4y  -  12 

(4)  is  5a;-4y  =  -6 

Subtract,  3  a;  =18 

.-.  x  -6. 
Substitute  value  of  x  in  (3),  y  =  9, 

•\  number  =  69. 


(2) 


TEACIIKRS*    EDITION.  205 

23.  A  number  is  expressed  by  three  digits,  of  which  the  first 
and  last  are  alike.  By  interchanging  the  digits  in  the  units' 
and  tens'  places,  the  number  is  increased  by  54 ;  but  if  the  digits 
in  the  teus'  and  hundreds'  places  are  interchanged,  9  must  be 
added  to  four  times  the  resulting  number  to  make  it  equal  to  the 
original  number.     What  is  the  number? 

Let  x  =  digit  in  hundreds'  and  units'  place, 

and  y  =  digit  in  tens'  place. 

Then  101  x  +  lOy  =  number. 

By  conditions,  110a  +  y  =  101  x  +  lOy  +  54      (1) 

4(lla  +  100y)  +  9  =  101a;+10y  (2) 

Transpose  and  combine  (1),    9r  —  9y  =  54  (3) 

Divide  (3)  by  9,  x  -  y  =  6  (4) 

Transpose  and  combine  (2), 

-57a;  +  390y  =  -9 
Multiply  (4)  by  57,  57  a;-    57y=342 


Add, 

333  y  =  333 

.-.y-i. 

Substitute  value  of  y  in  (4), 

*-7. 

.*.  number  =  717. 

24.  A  number  is  expressed  by  three  digits.  The  sum  of  the 
digits  is  21 ;  the  sum  of  the  first  and  second  exceeds  the  third 
by  3 ;  and  if  198  be  added  to  the  number,  the  digits  in  the  units' 
and  hundreds'  places  will  be  interchanged.    Find  the  number. 

Let  x  =  digit  in  hundreds'  place, 

y  =  digit  in  tens'  place, 
and  z  =  digit  in  units'  place. 

Then    lOOx  +  lOy  +  z  =  number. 

By  conditions,  x  4-  y  +  z  —  21  (1) 

and  x  +  y  -  z  =  3  (2) 

100a  +  10y  +  z  +  198  =  1003  +  lOy  +  x        (3) 
Subtract  (2)  from  (1),  2z  =  18. 

.'.2  =  9. 

Divide  (3)  by  99,  x  -  z  -  -  2  (4) 

Substitute  value  of  z  in  (3),         x  —  9  =  —  2. 

.-.  s  =  7. 
Substitute  values  of  x  and  2  in  (2),  y  =  5. 
.  \  number  =  759. 


206  ALGEBRA. 

25.  A  number  is  expressed  by  three  digits.  The  sura  of  the 
digits  is  9 ;  the  number  is  equal  to  forty-two  times  the  sum  of 
the  first  and  second  digits ;  and  the  third  digit  is  twice  the  sum 
of  the  other  two.    Find  the  number. 

Let  x  =  digit  in  hundreds'  place, 

y  =  digit  in  tens'  place, 
and  z  =  digit  in  units'  place. 

Then  x  +  y  +  z  =  9 

100a;  +  lOy  +  z  =  42(3  +  v) 

2  =  2(a;  +  y) 

From  (2),  58s  -  32y  +  z  =  0 

From  (3),  -2s-    2y  +  z  =  0 

Subtract,  60a;  -  30y        =0 

Divide  by  30,  2»-y=0  (4) 

Subtract  (3)  from  (1),  3  a;  +  3y  =  9 

Divide  by  3,  x  +  y  =  3  (5) 

(4)  is  2a;-y  =  0 

Add,  3  a;        =3 

.-.  a;=l. 
Substitute  value  of  x  in  (5),  y  =  2. 

Substitute  values  of  x  and  y  in  (1),  z  =  6. 
.*.  number  =  126. 

26.  A  certain  number,  expressed  by  three  digits,  is  equal  to 
forty-eight  times  the  sum  of  its  digits.  If  198  be  subtracted  from 
the  number,  the  digits  in  the  units'  and  hundreds'  places  will  be 
interchanged;  and  the  sum  of  the  extreme  digits  is  equal  to 
twice  the  middle  digit.    Find  the  number. 

Let  x  =  digit  in  hundreds'  place, 

y  =  digit  in  tens'  place, 
and  z  =  digit  in  units'  place. 

Then  100a;  +  lOy  +  z  =  48(a;  +  y  +  2)  (1) 

100a;  +  lOy  +  2  -  198  =  IOO2  +  lOy  +  x       (2) 

and  x  +  z  =  2y  (3) 

From(l),  52a; -38y- 472  =  0  (4) 

From  (2),  99  a;  -  992=  198. 

Divide  by  99,  x  -  2  =  2  (5) 

From  (3),  a;-2y  +  2  =  0  (6) 

Subtract  19  X  (6)from  (4), 33  x  -  66z  =  0. 

Divide  by  33,  a;-22  =  0  (7) 

Subtract  (7)  from  (5),  2  =  2. 

Substitute  value  of  2  in  (5),  x  =  4. 

Substitute  values  of  x  ana  2  in  (6),  y  =  3. 

.*.  number  =  432. 


teachers'  edition.  207 

27.  A  waterman  rows  30  miles  and  back  in  12  hours.  He 
finds  that  he  can  row  5  miles  with  the  stream  in  the  same  time 
as  3  against  it.  Find  the  time  he  was  rowing  up  and  down 
respectively. 


Let             x  =  number  of  hours  he  rowed  down, 

and                 y  =  number  of  hours  he  rowed  up. 

By  conditions, 

x  +  y  =  12 

(1) 

and 

5a;  =  3y 

(2) 

Transpose  (2), 

5a?-3y=    0 

Multiply  (1)  by  3, 

3x  +  3y  =  36 

Add, 

8*           =36 

.-.  x  =  4J. 

Substitute  value  of  x  in 

(1),  4J+y  =  12. 

28.  A  crew,  which  can  poll  at  the  rate  of  12  miles  an  hoar 
down  the  stream,  finds  that  it  takes  twice  as  long  to  come  up  the 
river  as  to  go  down.    At  what  rate  does  the  stream  flow? 

Let  x  —  rate  of  pulling, 

and  y  =  rate  of  stream. 

x  +  y  =  rate  down  stream, 
x  —  y  =  rate  up  stream. 


Then 

a;  +  y  =  12 
a;-y=    6 

(1) 
(2) 

Subtract,                 2y=   6 

.  \  y  =  3  =  rate  stream  flows. 
Substitute  value  of  y  in  (1), 
x  +  3  -  12, 

x  =  12-3. 
.-.a  =  9. 

29.  A  man  sculls  down  a  stream,  which  runs  at  the  rate  of  4 
miles  an  hour,  for  a  certain  distance  in  1  hour  and  40  minutes. 
In  returning  it  takes  him  4  hours  and  15  minutes  to  arrive  at  a 
point  3  miles  short  of  his  starting-place.  Find  the  distance  he 
palled  down  the  stream  and  the  rate  of  his  pulling. 


208  ALGEBRA. 


Let  x  =  rate  the  man  sculls, 

and  y  =  number  of  miles  he  goes. 

Then       x  +  4  =  rate  going  down  the  stream, 
and  x  —  4  =  rate  going  up  the  stream. 

(x  +  4)  |  =  number  of  miles  be  goes. 

.-.(*  +  4)t-y  (1) 

and  (a;_4)V=y-3  (2) 

4x(l)is  20a  +    80  =  12y  (3) 

3  X  (2)  is  51  x  -  204  =U2y  -  36  (4) 

Subtract,  -  31  x  +  284  =  36 

.  \  x  —  8,  rate  of  pulling. 
Substitute  value  of  a?  in  (1),      y  =  20. 

30.  A  person  rows  down  a  stream  a  distance  of  20  miles  and 
back  again  in  10  hours.  He  finds  he  can  row  2  miles  against  the 
stream  in  the  same  time  he  can  row  3  miles  with  it.  Find  the 
time  of  his  rowing  down  and  of  his  rowing  up  the  stream ;  and 
also  the  rate  of  the  stream. 


Let 

x  =  rate  of  rowing, 

and 

y  =  rate  of  stream. 

Then 

2            3 

X-y         X+y 

(1) 

x+y     x—y 

(2) 

Simplify  (1), 

i  =  5y 

(3) 

Substitute  this  value  of  x  in  (2), 

2^+20  =  10. 
6y     4y 

•••y  =  i 

From  (3),  x  =  4&. 

°0 
Therefore,  --  — ■  =  4  (time  of  running  down), 

20 
and  — — -  =  6  (time  of  rowing  up). 

4y  —  f 


teachers'  edition.  209 


31.  A  grocer  mixed  tea  that  cost  him  42  cents  a  pound  with 
tea  that  cost  him  54  cents  a  pound.  He  had  .30  pounds  of  the 
mixture,  and  by  selling  it  at  the  rate  of  60  cents  a  pound,  he 
gained  as  much  as  10  pounds  of  the  cheaper  tea  cost  him.  How 
many  pounds  of  each  did  he  put  into  the  mixture? 

Let  x  —  number  of  pounds  of  tea  at  42  cents, 

and  y  =  number  of  pounds  of  tea  at  54  cents. 

Then 

Multiply  (1)  by  42, 


x  +  y  =  30 

(1) 

42a; -t-54y  =  1800- 

-420 

(2) 

42a  +  42  y  =  1260 

42  s  +  54y  =  1380 

12y  =    120 

.-.y  =  10. 

x  +  10  -  30. 

.\  x  =  20. 

32.  A  grocer  mixes  tea  that  cost  him  90  cents  a  pound  with 
tea  that  cost  him  28  cents  a  pound.  The  cost  of  the  mixture  is 
$61.20.  He  sells  the  mixture  at  50  cents  a  pound,  and  gains 
$3.80.    How  many  pounds  of  each  did  he  put  into  the  mixture? 

Let  x  =  number  of  pounds  of  tea  at  90  cents, 

and  y  =>  number  of  pounds  of  tea  at  28  cents. 

28  y  =  number  of  cents  second  kind  cost. 
Then  90  a;  +  28  y  =  number  of  cents  whole  cost, 

x  +  y  =  number  of  pounds  in  whole  mixture, 
and  50  (a:  +  y)  =  number  of  cents  received. 


Hence,                                  50s  +  50y  =  6500 

(i) 

90  a?  +  28y  =  6120 

(2) 

Multiply  (1)  by  Jf ,              28  x  +  28  y  =  3640 

(3) 

Subtract,                               62  a;             =  2480 

.-.a;  =  40. 

Substitute  value  of  x  in  (1),               y  =  90. 

210  ALGEBRA. 


33.  A  farmer  has  28  bushels  of  barley  worth  84  cents  a  bushel. 
With  his  barley  he  wishes  to  mix  rye  worth  $  1.08  a  bushel,  and 
wheat  worth  $1.44  a  bushel,  so  that  the  mixture  may  be  100 
bushels,  and  be  worth  $1.20  a  bushel.  How  many  bushels  of 
rye  and  of  wheat  must  he  take? 

Let  x  =  number  of  bushels  of  wheat, 

and  y  =  number  of  bushels  of  rye. 

Then     2352  =  cost  in  cents  of  barley, 
144  a;  =  cost  in  cents  of  wheat, 
108  y  =  cost  in  cents  of  rye, 
and        12,000  ~  cost  in  cents  of  mixture. 

x  +  y  +  28  =  100, 

z  +  y  =  72  (1) 

144a;  +  108y  +  2352  -  12000, 

144a; +  108y  =  9648  (2) 

Divide  (2)  by  36,  4x  +  3y  =  268  (3) 

Multiply  (1)  by  3,  3a  +  3y  =  216  (4) 

Subtract,  x  =  52 

Substitute  value  of  a;  in  (1),  y  =  20. 

34.  A  and  B  together  earn  #40  in  6  days;  A  and  C  together 
earn  ft  54  in  9  days ;  B  and  C  together  earn  $80  in  15  days.  What 
does  each  earn  a  day? 


Let             x  = 

=  number  of  dollars  A  earns  in  one 

»  day, 

y- 

:  number  of  dollars  B  earns  in  one 

day, 

and                 z  = 

=  number  of  dollars  C  earns  in  one 

day. 

Then 

x  +  y-Y 
a;  +  «  =  6 

a) 

(2) 

and 

y  +  *  =  tt 

(3) 

Simplify  (1), 

3a;  +  3y          -20 

(*) 

Simplify  (3), 

3y +  3z-16 

(5) 

Subtract, 

3a;           -3z=    4 

(6) 

Multiply  (2)  by  3, 

3a;           +3«  =  18 

CO 

Add, 

6a;                   =22 
.-.  s  =  3f. 

Substitute  value  of  a;  in  (1),               y  =  3. 

Substitute  value  of  ar  in  (2),                z  =  2J. 

teachers'  edition.  21 1 


35.  A  cistern  has  three  pipes,  A,  B,  and  C.  A  and  B  will  fill 
it  in  1  hour  and  10  minutes ;  A  and  C  in  one  hour  and  24  min- 
utes ;  B  and  C  in  2  hours  and  20  minutes.  How  long  will  It  take 
each  to  fill  it? 

Let  x  =  number  of  minutes  it  takes  A  to  fill  it, 

y  =  number  of  minutes  it  takes  B  to  fill  it, 

and  z  =  number  of  minutes  it  takes  0  to  fill  it. 

-»  -»  -,  =  parts  A,  B,  C  can  fill  in  one  minute. 
x  y  z 

Hence,  l  +  I-£  (1) 


i     2      84 
1=_1 
'  z      140 


h\-±         (3) 


Add,  and  divide  by  2,  I  +  I  + 1  -  .1 

J  x     y     z      60 

Subtract  (1),  z  =  420. 

Subtract  (2),  y  -  210. 

Subtract  (3),  x  _  105. 


36.  A  warehouse  will  hold  24  boxes  and  20  bales ;  6  boxes 
and  14  bales  will  fill  half  of  it.  How  many  of  each  alone  will  it 
hold? 

Let  x  =  number  of  boxes  it  will  hold, 

and  y  -*  number  of  bales  it  will  hold. 

Then      -,  -  =»  parts  one  box,  one  bale,  can  fill. 
x  y     * 

xj  24      20     t  /tx 

Hence,  —  +  —  =1  (1) 

x       y  ' 

A  6        14        1  ,ox 

and  _  +  _  =  _  (2) 

4x(2)-(l) 

Substitute  value  of  y  in  (2), 


24 

20 

— 

+  —  = 

1 

X 

y 

6 

14 

1 

— 

+  —  = 

— 

X 

y 

2 

36 

1. 

y 

.-.  y  = 

■36. 

X- 

54. 

212  ALGEBRA. 


37.  Two  workmen  together  complete  some  work  in  20  days ; 
but  if  the  first  had  worked  twice  as  fast,  and  the  second  half  as 
fast,  they  would  have  finished  it  in  15  days.  How  long  would  it 
take  each  alone  to  do  the  work? 

Let  x  =  number  of  days  it  would  take  the  first  alone, 

and  y  =  number  of  days  it  would  take  the  second  alone. 

Then      -,  -  =  parts  they  can  do  in  one  day, 

-  +  _  =  part  both  could  do  in  one  day, 
x     y 

2      1 
and       -  +  —  =  part  they  could  do  if  first  worked  twice  as  fast, 

x     2y        an(j  Becond  worked  half  as  fast. 

.•.i  +  i-i-  (1) 

and  2  +  J_      1  (2) 


(1)-}  of  (2)  is 


x     2y     15 

4y     60' 
.-.  y  =  45. 
Substitute  value  of  y  in  (1),  x  =  36. 


38.   A  purse  holds  19  crowns  and  6  guineas ;  4  crowns  and  5 
guineas  fill  J  J  of  it.     How  many  of  each  alone  will  it  hold? 

Let  x  =  number  of  crowns  bag  holds, 

y  =  number  of  guineas  bag  holds. 

Then  -  =  part  of  bag  1  crown  occupies, 

-  =  part  of  bag  1  guinea  occupies. 


and 
5  x  (1)  -  6  X  (2)  is 

Substitute  value  of  x  in  (1), 


L9  +  ?  =  i 

x      y 

(1) 

4     5_17 
x     y     63 

(2) 

71      213 

x       63 

.\z  =  21. 

21    y 

.-.y-63. 

teachers'  edition.  213 

39.  A  piece  of  work  can  be  completed  by  A,  B,  and  C  together 
in  10  days ;  by  A  and  B  together  in  12  days ;  by  B  and  C,  if  B 
work  15  days  and  C  30  days.  How  long  will  it  take  each  alone 
to  do  the  work? 

Let  x  =  number  of  days  it  takes  A, 

y  =  number  of  days  it  takes  B, 

and  z  =  number  of  days  it  takes  C. 

Then  -,  -,  and  -,  respectively,  =  part  each  can  do  in  one  day. 

x     y     12 

a                                               15     30     -  /QX 

and  h  —  =  1  ,      (3) 


Subtract  (2)  from  (1), 


V       z 

1     60* 
\  z  -  60. 


15      1 
Substitute  value  of  z  in  (3),        —  +  -  =  1. 

y    2 

.-.y  =  30. 

Substitute  value  of  y  in  (2),      -  +  — ■  =  — -» 
x      60      12 

1  =  i- 
x     20* 

.\  x  =  20. 

40.  A  cistern  has  three  pipes,  A,  B,  and  C.  A  and  B  will  fill 
it  in  a  minutes ;  A  and  C  in  h  minutes ;  B  and  C  in  c  minutes. 
How  long  will  it  take  each  alone  to  fill  it? 

Let  x  =  number  of  minutes  it  takes  A, 

y  =  number  of  minutes  it  takes  B, 

and  z  =  number  of  minutes  it  takes  C. 

Then  -,  -,  -,  respectively,  =  part  each  fills  in  one  minute, 
x  y  z        x 

and  -  +  -  =  part  A  and  B  fill  in  one  minute. 

x     y     x 

But  -  *=  part  A  and  B  fill  in  one  minute. 


214  ALGEBRA. 


.-.1+1-1  (i) 

x     y     a 

\+\'\  (2) 

X       Z        0 

1+1-1  (3) 

y    z    c 

Add,  and  divide  by  2,  1  + 1  + 1  -  hc  +a M  +  ab     (4) 

1 


Subtract  (1)  from  (4), 
Subtract  (2)  from  (4), 
Subtract  (3)  from  (4), 


z 
.  z-- 

1 

y~ 

•y= 

i 

a; 


be 

+  OC  + 

a6 

2abc 
ac  +  ab  — 

6c 

2abc 
2abc 

ac  +  ab- 
aft—ac+ 

6c 
6c 

2abc 
2abc 

ab 
ac 

—  ac  + 

—  ab  + 

6c 
6c 

2abc 
2abc 

ac  —  a6  +  6c 


41.  A  man  has  $10,000  invested.  For  a  part  of  this  sum  he 
receives  5  per  cent  interest,  and  for  the  rest  4  per  cent;  the  in- 
come from  his  5  per  cent  investment  is  $50  more  than  from  his 
4  per  cent.    How  much  has  he  in  each  investment? 

Let  x  =  number  of  dollars  invested  at  £%, 

and  y  =  number  of  dollars  invested  at  4%. 

Then    x  +  y  =  total  number  of  dollars  invested. 

.-.  x  +  y  =  10,000  (1) 

5a; 
As  he  receives  5%  on  x  dollars,  —  =  interest  at  5%. 
10  100 

As  he  receives  4%  on  y  dollars,  —^  =  interest  at  4%. 

But  interest  at  5%  is  $50  more  than  that  at  4%. 

...  l£_i£  =  50  (2) 

100     100  W 

Simplify  (2),  5s  -  4y  *       5000 

Multiply  (1)  by  5,  5s  +  5y=*     50000 

Subtract,  -  9  y  =  -  45000 

.-.  y  =  5000. 

Substitute  value  of  y  in  (1),  x  =  5000. 


teachers'  edition.  215 

42.  A  sum  of  money  at  simple  interest  amounted  in  6  years 
to  $26,000,  and  in  10  years  to  #30,000.  Find  the  sum  and  the 
rate  of  interest. 

Jet  x  =  number  of  dollars  at  interest, 

and  y  =  rate  of  interest. 

Then       -^-  =  interest  on  x  dollars  for  one  year, 
100 

-— 2£  =  interest  on  x  dollars  for  six  years, 
100  * 

and  — ?&.  =  interest  on  x  dollars  for  ten  years. 

100 

...|^  +  x  =  26,000  (1) 

and  ^+  x  =  30,000  (2) 

Multiply  (1)  by  5,  y^  +  5  x  =  130,000 

Multiply  (2^  by  3,  ^-  +  3  x  =  90,000 


Subtract,  2x  =  40,000 

.-.  x  =  20,000. 
Substitute  value  of  x  in  (1),  y  =  5. 


43.  A  sum  of  money  at  simple  interest  amounted  in  10 
months  to  $26,250,  and  in  18  months  to  $27,250.  Find  the  sum 
and  the  rate  of  interest. 

Let  x  =  sum, 

and  y  =  rate  of  interest. 

Then  -^-  =  interest  for  one  year. 

100  J 

Since  10  months  equals  J  of  a  year, 

-  of  -^-  =  interest  for  10  months, 
6      100 

and  -  of  ^-  =  interest  for  18  months. 

2      100 

But  26,250  -  x  =  interest  for  10  months, 

and  27,250  —  x  =  interest  for  eighteen  months. 


216  ALGEBRA. 


.-.  1^=26,250  -x  (1) 

600  v  ' 

and  ^  =  27,250-*  (2) 

200 

Multiply  (1)  by*,  |g,  2**0-9, 


Subtract,  0^100,000-4* 

4a;  =100,000. 
.-.a;  =  25,000. 
Substitute  value  of  x  in  (2),  y  =  6. 

44.  A  sum  of  money  at  simple  interest  amounted  in  m  years 
to  a  dollars,  and  in  n  years  to  b  dollars.  Find  the  sum  and  the 
rate  of  interest. 

Let  x  =  sum, 

and  y  —  rate  of  interest. 

Then      —^  =  interest  on  sum  for  m  years, 

and  ^2£  =  interest  on  sum  for  n  years. 

.-.^+a;  =  a  (1) 

100  W 

and  ™%  +  x  =  b  (2) 

100  W 
Multiply  (1)  and  (2)  by  100, 

mxy  +  100.t  =  100a  (3) 

nxy  +100*  =1006  (4) 

Multiply  (3)  by  n,       mnxy  + 100  nx  =  100  an 

Multiply  (4)  by  m,      mnxy  +  lOOmx  =  1006m 

Subtract,  lOOnx  -  lOOma;  =  100an  -  1006m 

Divide  by  100,  nx  —  mx  =  an  —  6m, 

an  —  bm 

.*.  x  = 

n  —  m 

Substitute  value  of  x  in  (3), 

many  —  m26y      100an— 1006m 

t =  lUUa. 

7i  —  m  7i  —  m 

Multiply  by  n  —  m, 

many  -  m%by  +  100  cm  —  100  6m  =  100  an  —  100  am, 

many  —  m26y  =  1006m  —  100  am, 

any  -  m6y  =  1006  -  100a. 

100(6 -a) 

y        an  —  bm 


teachers'  edition.  217 

45.  A  sum  of  money  at  simple  interest  amounted  in  a  months 
to  c  dollars,  and  in  &  months  to  d  dollars.  Find  the  sum  and  the 
rate  of  interest. 


Let                 x  = 

=  sum, 

and                     y  = 

=  rate  of  interest. 

Then          -3U 
100 

=  interest  of  $x  for  one  year, 

1200 

=  amount  of  $#  for  a  months, 

1200 

=  amount  of  $rc  for  6  months. 

™!L  +  x  =  c 
1200 

(i) 

*%L  +  x  =  d 

(2) 

1200 

Simplify  (1), 

axy  +  1200  x  =  1200c 

(3) 

Simplify  (2), 

%  + 1200  a  =  1200  d 

(4) 

Multiply  (3)  by 

6  and  (4)  by  a, 

abxy  +  12006a?  «=  12006c 

(5) 

abxy  +  1200oa;  =  1200ac? 

(6) 

(6) -(5)  is 

1200s(a-&)  =  1200  (ad- 
ad  —  be 

.*.  x  = — • 

-be) 

a  —  b 
Substitute  value  of  x  in  (3), 

ay/«^|£\  +  12(X)(^^U  1200c. 

Simplify,  ay  (ad-  be)  + 1200  (ad  -  be)  =  1200(ac  -  be). 
Transpose  and  unite,         ay  (ad  —  be)  =  1200 a(c  —  d). 

,  1200(e-rf) 

ad—  6c 

46.  A  person  has  a  certain  capital  invested  at  a  certain  rate 
per  cent.  Another  person  has  $  1000  more  capital,  and  his  capi- 
tal invested  at  one  per  cent  better  than  the  first,  and  receives  an 
income  $  80  greater.  A  third  person  has  $  1500  more  capital,  and 
his  capital  invested  at  two  per  cent  better  than  the  first,  and  re- 
ceives an  income  $  150  greater.  Find  the  capital  of  each,  and 
the  rate  at  which  it  is  invested. 


218  ALGEBRA. 

Let  x  =  capital, 

and  y  —  rat§. 

Then (*  +  1000)(y  +  fi  -  interest  on  capital   $1000  greater,  at  1% 
100  greater, 

and       (x  + 1500)  (y  +  2)  =  interest  on   capital  $150o   greater,  at  2% 
100  greater. 

12L  +  80  =  interest  on  first  capital,  increased  by  $80, 

^-  +  150  =  interest  on  first  capital,  increased  by  $150. 

(s-HOOO)(y  +  l)  =  i^     8Q  (1) 

100  100  W 

(x-M500)(y  +  2)  =  ^_     15Q  (2) 

100  100  W 

Simplify,  xy  +  lOOOy  +    x  +  1000  =  xy  +  8000, 

ay  +  1500y  +  2x  +  3000  =  xy  +  15000 

Combining,  1000 y  +    a;  -  7000  (3) 

1500y  +  2x=  12000  (4) 

Multiply  (3)  by  2,  2000y  +  2x  =  14000  (5) 

(4)  is  1500y  +  2s  =12000 

Subtract,  500y  =    2000 

.-.  y  =  4. 
Substitute  value  of  y  in  (5),  x  =  3000. 

.%  the  capitals  are  $3000,  $4000,  $4500;  and  the  rates  4%,  5%,  6%. 

47.  A  person  has  $12,750  to  invest.  He  can  buy  three  per 
cent  bonds  at  81,  and  five  per  cents  at  120.  Find  the  amount 
of  money  he  must  invest  in  each  in  order  to  have  the  same 
income  from  each  investment. 

Let  x  =  number  of  dollara  in  three  per  cent  bonds, 

and  y  =  number  of  dollars  in  five  per  cent  bonds. 

Then  =  interest  of  money  invested  in  three  per  cents. 

81 

But  J**i#  =,  interest  of  money  invested  in  five  per  cents. 

120 

300a  =  500y  ,^ 

'""    81         120 

x  +  y  =  12750  (2) 

Reduce  (1),  8a;-9y  =  0 

Multiply  (2)  by  8,  8a;  +  8y=*  102000 

Subtract,  17y-  102000 

.-.  y  =  6000. 
Substitute  value  of  y  in  (2),  x  =  6750. 


teachers'  edition.  219 

48.  A  and  B  each  invested  $1500  in  bonds;  A  in  three  per 
cents  and  B  in  four  per  cents.  The  bonds  were  bought  at  such 
prices  that  B  received  $5  interest  more  than  A.  Both  classes  of 
bonds  rose  ten  points,  and  they  sold  out,  A  receiving  $50  more 
than  B.    What  price  was  paid  for  each  class  of  bonds? 

Let  x  =  amount  paid  for  $1  three  per  cents, 

and  y  =  amount  paid  for  $  1  four  per  cents. 

=  face  value  of  three  per  cents. 

x  r 

1500 

=  face  value  of  four  per  cents. 

y 

X  —  =  income  from  three  per  cents. 

x         100  r  • 

- X —  income  from  four  per  cents. 

y        100 


Then    »XA\_«X    3  N  =  5 
\   y        lOOJ      \    x        lOOJ 

V  *     loo)    V  y     iwy 


Simplify, 

60     45&5 

y       z" 

(1) 

5-2-1 

x     y 

(2) 

Multiply  (2)  by  20, 

_«2  +  60_20 

y      x 

(3) 

Add  (1)  and  (3), 

X 

.-.  3  =  0.60. 

Substitute  value  of  x  in 

(2),      6-2-L 

.-.y-0.75. 

That  is,  the  three  per  cents  were  bought  at  60  and  the  four 
per  cents  at  75. 


220  ALGEBRA. 


49.  A  fferson  invests  $10,000  in  three  per  cent  bonds,  $16,500 
in  three  and  one-half  per  cents,  and  has  an  income  from  both  in- 
vestments of  $  1056.25.  If  his  investments  had  been  §  2750  more 
in  the  three  per  cents,  and  less  in  the  three  and  one-half  per 
cents,  his  income  would  have  been  G2\  cents  greater.  What 
price  was  paid  for  each  class  of  bonds? 

Let  x  =  amount  paid  on  $  1  three  per  cent  bonds, 

and  y  =  amount  paid  on  $  1  three  and  one-half  j><  r 

cent  bonds. 

10000        3 
Then     X =  number  of  dollars  income  from  first  invest- 

*  10°         ment, 

and  x  -^-  =  number  of  dollars  income  from  second  in- 

y         100         vestment. 

.30000     115500  _  105625  (1) 

lOOx        200y 

12*750        3 

— —  X  —  =  number  of  dollars  income  of  3  per  cents  if 

x         100         the  stated  addition  in  the  amount  invest- 
1  ohca      oi  ed  had  been  made, 

— - —  x  — *-  =  number  of  dollars  income  of  3J  per  cents  if 

V         100         the  stated  deduction  in  the  amount  invest- 

<iA9*n    QA9*n        e<*  na(*  Deen  made. 
Then    ^^  +  ^^  -  number  of  dollars  income  on  both. 
lOOx      200y 

...  S  +  W"?1056'87}  ^ 

Multiply  (1)  by  5,  ±^-  +  ^2  =  5281.25  f  (3) 

Multiply  (2)  by  6,  Sgg+fgf-  6341.25  (4) 

Subtract,  1^2=1060, 

100a; 

106000  a;  =  79500. 

.-.  x=0.75. 

That  is,  the  3  per  cent  bonds  were  bought  at  75. 

Substitute  value  of  x  in  (1), 

30000.  +  115500  =  105625> 


75  200y 

115500 


656.25. 


200y 

.-.  y  =  0.88. 

That  is,  the  3£  per  cent  bonds  were  bought  at  88. 


teachers'  edition.  221 

. '  - c _. 

50.  The  sum  of  $2500  was  divided  into  two  unequal  parts 
and  invested,  the  smaller  part  at  two  per  cent  more  than  the 
larger.  The  rate  of  interest  on  the  larger  sum  was  afterwards 
increased  by  1,  and  that  of  the  smaller  sum  diminished  by  1 ;  and 
thus  'the  interest  of  the  whole  was  increased  by  one-fourth  of  its 
value.  If  the  interest  of  the  larger  sum  had  been  so  increased, 
and  no  change  been  made  in  the  interest  of  the  smaller  sum,  the 
interest  of  the  whole  would  have  been  increased  one-third  of  its 
value.    Find  the  sums  invested,  and  the  rate  per  cent  of  each. 

Let  x  =  number  of  dollars  in  larger  part, 

and  y  =  number  of  dollars  in  smaller  part. 

Then  x  +  y  =  2500  (1) 

Let  z  =  rate  per  cent  on  larger  part, 

and  z  4-  2  =  rate  per  cent  on  smaller  part. 

Then  xz  +  y(z  +  2)  —  interest  on  whole  amount. 

Changing  rate  per  cent, 

2  +  1  =  rate  per  cent  on  larger  part, 
and  2  +  1  =  rate  per  cent  on  smaller  part. 

Then         x  (z  + 1)  -f  y  (z  + 1)  =  interest  on  whole  after  change. 
Then        z(2  +  l)+y(2  +  l)  =  $  [xz+y(z  +  2)]  (2) 

Changing  rate  per  cent  again, 

2  +  1  =  rate  of  larger  part, 
2  +  2  =  rate  of  smaller  part. 
Then      x(z+  1)  +  y  (2  +  2)=  %[xz  +  y  (2  +  2)]  (3) 

Simplify  (2),  4  x  —  6  y  —  xz  —  yz  =  0. 

Simplify  (3),  3  x  -  2y  -  xz  -  yz  =  0.  (4) 

Subtract,  x  —  4y  =  0  (5) 

Subtract  (5)  from  (1),  by  =  2500. 

.-.  y  =  500. 
Substitute  value  of  y  in  (4),  x  =  2000. 

Substitute  values  of  x  and  y  in  (3), 

6000  -  1000  -  20002  -  5002  =  0, 

~2500z  =  -5000. 
.-.  2  =  2. 


222  ALGEBRA. 


51.  If  the  sides  of  a  rectangular  field  were  each  increased  by 
2  yards,  the  area  would  be  increased  by  220  square  yards ;  if  the 
length  were  increased  and  the  breadth  were  diminished  each  by 
5  yards,  the  area  would  be  diminished  by  185  square  yards. 
What  is  its  area? 

Let  x  =  number  of  yards  in  length, 

and  y  =  number  of  yards  in  width. 

Then        xy  —  number  of  yards  in  area. 

(*  +  2)(y-f2)  =  sy  +  220  (1) 

(x  +  5)(.y-5)  =  :ry~185  (2) 

Simplify  (1),  xy  +  2x  +  2y  +  4  =  xy  +  220,  . 

2x  +  2y  =  216, 

x  +  y  =  108  (3) 

Simplify  (2),  xy  -  bx  +  by  -  25  =  xy  -  185, 

5a-5y  =  160 

x-y  =  32  (4) 

Add  (4)  and  (3),  2  x  =140 

.-.3  =  70. 
Subtract  (4)  from  (3),  2  y  =  76. 

.-.y  =  38. 
.*.  xy  =  2660  square  yards. 


52.  If  a  given  rectangular  floor  had  been  3  feet  longer  and  2 
feet  broader  it  would  have  contained  64  square  feet  more ;  but  if 
it  had  been  2  feet  longer  and  3  feet  broader  it  would  have  con- 
tained 68  square  feet  more.  Find  the  length  and  breadth  of  the 
floor. 

Let  x  =  number  of  feet  in  length, 

and  y  =  number  of  feet  in  breadth. 

Then        xy  =  number  of  feet  in  surface. 

(s  +  3)(y  +  2)  =  zy  +  64  (1) 

(z  +  2)(y  +  3)  =  zy  +  68  (2) 

Simplify  (1),  xy  +  Zy  +  2x  +  6  =  xy  +  64. 

3y  +  2a  =  58  (3) 

Simplify  (2),  xy  +  2y  +  3a;  +  6  =  xy  +  68. 

2y  +  3x  =  62  (4) 

Multiply  (3)  by  2, 
Multiply  (4)  by  3, 
Subtract, 

Substitute  value  of  x  in  (3), 


6V 

+  4z  = 

116 

Oy 

+  9z  = 

186 

~bx  = 

-70 

.-.  x  = 

14. 

32/ 

+  28  = 

3y  = 

.-.  y  = 

58, 
30. 
10. 

TEACHERS     EDITION. 


223 


53.  In  a  certain  rectangular  garden  there  is  a  strawberry-bed 
whose  sides  are  one-third  of  the  lengths  of  the  corresponding 
sides  of  the  garden.  The  perimeter  of  the  garden  exceeds  that 
of  the  bed  by  200  yards ;  and  if  the  greater  side  of  the  garden 
be  increased  by  3,  and  the  other  by  5  yards,  the  garden  will  be 
enlarged  by  645  square  yards.  Find  the  length  and  breadth  of 
the  garden. 


I 

* 

Let  x  =  number  of  yds.  in  length  of  garden, 

and  y  =  number  of  yds.  in  width  of  garden. 

Then  2x  +  2y  =  perimeter  of  garden, 
and  xy  =  area  of  garden. 

Also,  |  =  number  of  yds.  in  length  of  bed, 


and 

Then 

Add  3  to  one  side  of  garden,  x  +  3, 


|  =  number  of  yds.  in  width  of  bed. 

o 


—  +  -^  =  perimeter  of  bed. 


Add  5  to  other  side  of  garden,  y  +  5. 
Then  (s  +  3)(2/  +  5)  =  area. 


Simplify, 

Simplify, 

Multiply  (1)  by  5, 
Subtract, 


+  =^  =  200. 
3        3  ) 


x  +  y  =  150 
(a>  +  3)(y  +  5)  =  zy +  645. 
5z  +  3y  =  630 
5a;  +  5y  =  750 


Substitute  value  of  y  in  (1), 


2y  =  120 
.-.  y  =  60. 
x  4-  60  =  150. 
.-.a;  =  90. 


(1) 
(2) 


224  ALGEBRA. 


64.  In  a  mile  race  A  gives  B  a  start  of  100  yards,  and  beats 
him  by  15  seconds.  In  the  second  trial  A  gives  B  a  start  of  45 
seconds,  and  is  beaten  by  22  yards.  Find  the  rate  of  each  in 
miles  per  hour. 

Let  x  =  number  of  yards  A  runs  in  one  second, 

and  y  =  number  of  yards  B  runs  in  one  second. 

Since  there  are  1760  yards  in  one  mile, 

and   =  number  of  seconds  A  ran  in  first  and  second 

x  x  trials  respectively, 

and  =  number  of  seconds  B  ran  in  first  and  second 

y  y  trials  respectively. 

Then  1^_1^=15  0) 

and  I?k_i™  =  _45         (2) 

x  y 

Multiply  (1)  by  88,  146080  _  154880  =     ^ 

y  x 

™  u-  i    /ox  k    Qo           146080     144254         0„0K 
Multiply  (2)  by  83, + =  —  3735 


Add,  -^^  =  -2415 

x 

Therefore,  A  runB  4^*^  yards,  or  ^fa  of  a  mile,  in  one  second, 
and  in  one  hour  (=  3600  seconds),  9  miles. 

Substitute  value  of  a?  in  (1),  y  =  4. 

Therefore,  B  runs  4  yards  in  one  second,  or  8-j^-  miles  in  one 
hour. 

55.  In  a  mile  race  A  gives  B  a  start  of  44  yards,  and  beats  him 
by  51  seconds.  In  the  second  trial  A  gives  B  a  start  of  1  minute 
and  15  seconds,  and  is  beaten  by  88  yards.  Find  the  rate  of  each 
in  miles  per  hour. 

Let  x  =  number  of  yards  A  ran  in  one  second, 

and  y  =  number  of  yards  B  ran  in  one  second. 

,    =  number  of  seconds  A  ran  in  first  and  second 

x  x            trials,  respectively. 

1 71 6  1 7fi0 

-- — ,  -— —  =«  number  of  seconds  B  ran  in  first  and  second 

V         y  trials,  respectively. 


teachers'  edition.  225 


Then 

1716     1760  _ 

y        x 

(1) 

and 

1672      1760  _ 
x           y 

(2) 

Multiply  (1)  by  19, 

33440 +32&04_       ^ 
x             y 

Multiply  (2)  by  20, 

33440      35200         1rAA 

==  — 1500 

•    x             y 

Add, 

_?5?6  =  _    531 

y 

Therefore,  B  runs  4f  yards  per  second,  or  10  miles  per  hour. 

Substitute  value  of  y  in  (1),  x  =  5|$. 

Therefore,  A  runs  5||  yards  per  second,  or  12  miles  per  hour. 

56.  The  time  which  an  express-train  takes  to  go  120  miles  is 
T9T  of  the  time  taken  by  an  accommodation-train.  The  slower 
train  loses  as  much  time  in  stopping  at  different  stations  as  it 
would  take  to  travel  20  miles  without  stopping;  the  express- 
train  loses  only  half  as  much  time  by  stopping  as  the  accommo- 
dation-train, and  travels  15  miles  an  hour  faster.  Find  the  rate 
of  each  train  in  miles  per  hour. 

Let  x  =  the  rate  of  the  accommodation-train, 

and  y  =  the  rate  of  the  express-train. 

120 

=  number  of  hours  accommodation-train  goes  120 

x  miles  without  stopping, 

120 

=  number  of  hours  it  takes  express-train  to  go  120 

V  miles  without  stopping, 

20 

—  =  number  of  hours  accommodation-train  loses  in 

x  stopping, 

—  =  number  of  hours  express- train  loses  in  stopping, 

120     20 

1 =  number  of  hours  accommodation -train  goes  120 

x        x  miles  including  stops, 

1 =  number  of  hours  express-train  goes  120  miles 

V        x  including  stops. 


226  ALGEBRA. 


120      10      9/120      20  \  m 

—  +  — =  tt    —  +  —  )'  0) 
y        x       14  \  x        x  J 

y-x=15  (2) 


Simplify  (1),  120a;  -  80y  -  0 

Multiply  (2)  by  80,  ♦       -80a  +  80  y  =  1200 

Add,  40s  =  1200 

.\x  =  30. 
Substitute  value  of  x  in  (2),  y  —  45. 


57.  A  train  moves  from  P  towards  Q,  and  an  hour  later  a 
second  train  starts  from  Q  and  moves  towards  P  at  a  rate  of  10 
miles  an  hour  more  than  the  first  train ;  the  trains  meet  half-way 
between  P  and  Q.  If  the  train  from  P  had  started  an  hour  after 
the  train  from  Q,  its  rate  must  have  been  increased  by  28  miles  in 
order  that  the  trains  should  meet  at  the  half-way  point.  Find 
the  distance  from  P  to  Q. 

Let  x  =  number  of  hours  first  goes  half  the  distance. 

Then    x  —  1  =  number  of  hours  second  goes  half  the  distance. 

Let  y  =  rate  of  first. 

Then  y  +  10  =  rate  of  second. 

Hence,  xy  =  one-half  of  the  whole  distance. 

But  (x—  l)(y  -l- 10)  =  one-half  the  whole  distance. 
.•.(*-l)(y  +  10)-ay. 

Simplify,    10a;-y  =  10  (1) 

In  Becond  statement, 
if  x  —  2  =  number  of  hours  "first  goes  half  distance, 

and  y  +  28  =  rate  of  first, 

then    (x  —  2)(y  +  28)  =  one-half  of  the  whole  distance. 

But,  from  (1),     xy  =  one-half  of  the  whole  distance. 
.-.(a-2)(y  +  28)-*y  (-) 

Simplify  (2),  28z-2y  =  56 

2x(l)is  20z-~2y  =  20 

Subtract,  8x  =36 

.\*-4J. 
Substitute  value  of  x  in  (1),  y  =  35. 

Therefore,  one-half  the  distance,  xy,  is  157J  miles,  and  the 
whole  distance  is  315  miles. 


teachers'  edition.  227 


58.  A  passenger-train,  after  travelling  an  hour,  meets  with  an 
accident  which  detains  it  one-half  an  hour ;  after  which  it  pro- 
ceeds at  four-fifths  of  its  usual  rate,  and  arrives  an  hour  and  a 
quarter  late.  If  the  accident  had  happened  30  miles  farther  on, 
the  train  would  have  been  only  an  hour  late.  Determine  the 
usual  rate  of  the  train. 

Let         x  =  number  of  miles  train  usually  goes  per  hour, 
and  y  =  number  of  miles  train  travels. 

^ =  number  of  hours  usually  required, 

x 

and     2L —  =  number  of  hours  actually  required  after  accident. 

y 

Since  the  detention  is  J  hour,  and  the  train  is  1 J  hours  late,  the 
loss  in  running-time  is  }  of  an  hour. 

.  y-x     y-x     3  (1, 

4z  x     =  4  K  ' 

5 
If  the  accident  had  occurred  30  miles  farther  on,  the  loss  in 
running-time  would  have  been  J  an  hour. 

y-g-30     y  -  a:  -  30  _  1  ,™ 

\x  -      -o  v; 

5 
Simplify  (1), 

Simplify  (2), 

Subtract,  x  =  30 

59.  A  passenger-train,  after  travelling  an  hour,  is  detained  15 
minutes ;  after  which  it  proceeds  at  three-fourths  of  its  former 
rate,  and  arrives  24  minutes  late.  If  the  detention  had  taken 
place  5  miles  farther  on,  the  train  would  have  been  only  21  min- 
utes late.  Determine  the  usual  rate  of  the  train. 
Let  x  =  usual  rate  of  train  per  hour, 

and  y  =  number  of  miles  train  has  to  run. 

y  —  x  =  number  of  miles  train  has  to  run  after  detention, 

=  number  of  hours  usually  required  to  run  y  —  x 

x  miles, 

and        r  =  number  of  hours  actually  required  to  run  the 

~*  y  —  x  miles. 

4  ^ 


x 

2 

y- 

-4ar 

=    0 

y- 

-3* 

=  30 

228  ALGEBRA. 


Since  the  detention  was  15  minutes,  and  the  train  is  24  minutes 
late,  the  loss  in  running- time  is  9  minutes  =  fa  of  an  hour. 

.   y~x     y~x=S  (I) 

"     'Sx  a;         20  w 

4 

If  the  detention  had  occurred  5  miles  farther  on,  the  loss  in 
running-time  would  have  been  6  minutes  =  ^  of  an  hour. 

•  .V~-T~5  y—x—h^  1                        ,2) 

3j  x             10                       K  ' 
4 

Simplify  (1),  20y-29s  =      0 

Simplify  (2),  20y  -  26 x  =  100 

Subtract,  3  x  =100 

.-.  s  =  33J. 

60.  A  man  bought  10  oxen,  120  sheep,  and  46  lambs.  The 
cost  of  3  sheep  was  equal  to  that  of  5  lambs ;  an  ox,  a  sheep,  and 
a  lamb  together  cost  a  number  of  dollars  less  by  57  than  the 
whole  number  of  animals  bought;  and  the  whole  sum  spent  was 
$2341.50.  Find  the  price  of  an  ox,  a  sheep,  and  a  lamb,  respec- 
tively. 

Let  x  -=  number  of  dollars  paid  for  an  ox, 

y  =  number  of  dollars  paid  for  a  sheep, 
and  z  =  number  of  dollars  paid  for  a  lamb. 

lOx  +  120y  +  46z  =  number  of  dollars  paid  for  all. 

.-.  lOar  +  120y  +  48z  =  2341.50  (1) 

x  +  y  +  z  =  119  (2) 

3y  =  5z  (3) 

(1)  is  lOx  +  120y  +  46z  =  2341.50 

Multiply  (2)  by  10, 10 x  +    lOy  +  lOz  =  1190 

Subtract,  110y  + 36z  =  1151.50  (4) 

Multiply  (4)  by  3,  330y  +  108  2  =  3454.50. 

Multiply  (3)  by  110,         330y  -  550z  =  0. 
Subtract,  658  z  =  3454.50. 

.-.  z  =  5.25. 
Substitute  value  of  z  in  (3),  y  =  8.75. 

Substitute  values  01  y  and  2  in  (2),   x  =  105. 


teachers'  edition.  229 


61.  A  farmer  sold  100  head  of  stock,  consisting  of  horses, 
oxen,  and  sheep,  so  that  the  whole  realized  $  11.75  a  head ;  while 
a  horse,  an  ox,  and  a  sheep  were  sold  for  $110,  $62.50,  and 
$7.50,  respectively.  Had  he  sold  one-fourth  of  the  number  of 
oxen  that  he  did,  and  25  more  sheep,  he  would  have  received  the 
same  sura.  Find  the  number  of  horses,  oxen,  and  sheep,  respec- 
tively, which  were  sold. 

Let  x  =  number  of  horses, 

y  =  number  of  oxen, 
and  z  =  number  of  Bheep. 

Then  a;  +  y  +  2  =  100  (1) 

110*  +  62Jy  +  7i2  =  1175  (2) 

and      no*  +  (lx^)+lf+^=1175  (3) 

Multiply  (3)  by  8,  880a;  +  125y  +  602=  7900 
Multiply  (2)  by  8,  880  x  +  500  y  +  60  z  =  9400 


Subtract, 

-375y 

=  -1500 

-y- 

=  4. 

Substitute  value  of  y  in  (1),         x  +  z  =  96 
Substitute  value  of  y  in  (2), 

110*  + 250 +  7*2  =  1175. 
110a:  +  7*2  =  925. 

Multiply  by  2, 
Multiply  (4)  by  15, 

220  a; +  152  = 
15a;  +  152  = 

=  1850 
=  1440 

Subtract, 

205  a; 

-    410 

,:x  =  2. 
*  Substitute  values  of  x  and  y  in  (1),       2  —  94. 


(4) 


230  ALGEBRA. 


62.  A,  B,  and  C  together  subscribed  $100.  If  A's  subscrip- 
tion had  been  one-tenth  less,  and  B's  one-tenth  more,  C's  must 
have  been  increased  by  $  2  to  make  up  the  sum ;  but  if  A's  had 
been  one-eighth  more,  and  B's  one-eighth  less,  C's  subscription 
would  have  been  $  17.50.     What  did  each  subscribe? 

Let  x  —  number  of  dollars  A  subscribed, 

y  =  number  of  dollars  B  subscribed, 
and  z  =  number  of  dollars  C  subscribed. 

-^  =  -j^  of  A's  subscription, 

-— ^  =  ii  of  B's  subscription, 

z  +  2  =  $  2  more  than  C's  subscription, 

9x 

—  =  {  of  A's  subscription, 
8 

-%  —  J  of  B's  subscription, 

8 

100  =  number  of  dollars  all  subscribed, 

..*  -|-  — V-  +  z  +  2  =  number  of  dollars  all  subscribed, 
10        10 

—  +  — %-  +  17.5  =  number  of  dollars  all  subscribed. 
8        8 


.-.  x  +  y  +  z  =  100 

(1) 

?*  +  litf  +  1  +  2-100 
10       10 

(2) 

^  +  Iy  +  17.5  =  100 

(3) 

Multiply  (1)  by  10, 
Simplify  (2), 

10a; +  10y  +  10z=  1000 
9a;  +  lly +  10a=    980 

Subtract, 

x  -      y             =20 

(4) 

Simplify  (3), 
Multiply  (4)  by  7, 

9a;  +  7y  =  660 
.7z-7y  =  140 

Add, 

lGa;  =  800 

.-.  a;  =  50. 
Substitute  value  of  x  in  (4),  y  —  30. 

Substitute  values  of  x  and  y  in  (1),  z  =  20. 


TEACHERS*    EDITION.  231 

63.  A  gives  to  B  and  C  as  much  as  each  of  them  has ;  B  gives 
to  A  and  C  as  much  as  each  of  them  then  has ;  and  C  gives  to  A 
and  B  as  much  as  each  of  them  then  has.  In  the  end  each  of 
them  has  $6.     How  much  had  each  at  first? 

Let  -       x  =  number  of  dollars  A  had  at  first, 

y  =  number  of  dollars  B  had  at  first, 

and  z  —  number  of  dollars  C  had  at  first. 

x  —  y  —  z  =  number  of  dollars  A  had  at  1st  distribution, 

2y  —  number  of  dollars  B  had  at  1st  distribution, 

2z  =  number  of  dollars  C  had  at  1st  distribution, 

2y— {(%— y— z)+2z}  =  number  of  dollars  B  had  at  2d  distribution, 

or  3y  —  x  —  z  =  number  of  dollars  B  had  at  2d  distribution, 

2x  —  2y  —  2z  =  number  of  dollars  A  had  at  2d  distribution, 

4z  =  number  of  dollars  C  had  at  2d  distribution, 

4z-{(2x— 2y-2z)+(3  y-x-z)}}  or  7z  -  x  -  y 

=  number  of  dollars  C  had  at  3d  distribution, 
4  a;  —  4y  —  4z  =  number  of  dollars  A  had  at  3d  distribution, 
Qy  —  2x  —  2z  =  number  of  dollars  B  had  at  3d  distribution. 
.\  7z-x-y  =  6  (1) 

4a;-4y-4z  =  6  (2) 

Gy-2x-2z  =  6  (3) 

Multiply  (1)  by  4,        28z  -  4s  -  4y  =  24 

(2)  is  -4z+4a;-43/=    6 

Add,  24z  -8y  =  30  (4) 

Multiply  (1)  by  2, 

(3)  is 
Subtract,  16z  -83/=    6  (5) 

(4)  is 
Subtract,  8z  =24 

.-.  z  =  3. 
Substitute  value  of  z  in  (5),  y  —  5J. 

Substitute  values  of  y  and  z  in  (1),   x  =  9}. 


24z 

-8y  =  30 

14z- 

-2a;-2y=12 

-2z- 

-2x  +  6y=    6 

16z 

-83/=    6 

24z 

-8y  =  30 

232  ALGEBRA. 


64.  A  pays  to  B  aud  C  as  much  as  each  of  them  has ;  B  pays 
to  A  and  C  one-half  as  much  as  each  of  them  then  has ;  and  C 
pays  to  A  and  B  one-third  of  what  each  of  them  then  has.  In 
the  end  A  finds  that  he  has  $1.50,  B  $4.16f  C  $0.58*.  How 
much  had  each  at  first? 

Let  x  =  number  of  dollars  A  had  at  first, 

y  =  number  of  dollars  B  had  at  first, 

and  z  =  number  of  dollars  C  had  at  first. 

x  —  y  —  z  =  number  of  dollars  A  has  left  after  giving  to  B 
andC, 
2y  —  number  of  dollars  B  has  after  A  pays  him, 
2z  =  number  of  dollars  C  has  after  A  pays  him, 
x~    y  ~ — -  =  number  of  dollars  A  has  after  B  pays  him, 

y  ~g~J  „  number  of  dollars  B  has  left  after  paying  A  and  C, 
3z  =  number  of  dollars  C  has  after  receiving  B's  money, 
x~ — y^ — -  =  number  of  dollars  A  has  after  C  pays  him, 

y  ~  z  =  number  of  dollars  B  has  after  C  pays  him, 

— Z~~X~~V  =  number  of  dollars  C  has  after  paying  A  and  B. 

4<r-4y-4«_1J0  {1) 

2  v  ' 

10y-2x-2z  =  416?  (2) 

O 

ll*-*-y==0.58t  (3) 

Simplify  (1),  x-y-z=  0.75  (4) 

Simplify  (2),  10y  -  2x  -  2z  =  12.50  (5) 

Simplify  (3),  11  z  -  x  -  y  =  1.75  (6) 

(4)  is  x-y-z=  0.75 

Add  (4)  and  (6),  10z  -  2y  =  2.50  (7) 

(5)  is  10y-2x-2z  =  12.50 
Multiply  (4)  by  2,             -2.y  +  2x-2g-    1.50 

Add,  8y           -4z  =  14.00                    (8) 

Multiply  (7)  by  4,  -  Sy  +  40  2  =  10.00 

(8)  is  8y-    4z  =  14.00 

Add,  36  z  =  24.00 


.\  z  =  0.66f. 
y  =  2.08} 
Substitute  values  of  y  and  z  in  (4),         x  «  3.50. 


Substitute  value  of  2  in  (8),  y  =  2.08J. 

ad  z  ' 


teachers'  edition.  233 


Exercise  LXXVI. 


5. 


V2as&V 


1.   (a*)2  2.   (z5)3  3.   (afy3)8 

=*a8xS  =  i5x3  =a;2xay3xa 

=  a6.  -a;15.  =  z*y«. 

a3x*&ax4  ^   35a;axV 

"       24  ""^a8"5^5 

^ala68  _  243  x10  y5 
16  '  32  a15  610' 

6.  (s  +  2)8 

=  (^  +  3(^(2)  +  3(s)(2)'  +  (2)» 
=  a*  +  6a?+12a;  +  8. 

7.  (a:  -2)* 

=  a*-  4(a;)8(2)  +  6(af(2)a-  4(a;)(2)8  +  2* 
=  z*-8a*+24a*-32ar  +  16. 

8.  (x  +  Sf 

=  (af  +  5(a;)4(3)  +  10(^(3)'  +  10(a;)*(3)8  +  5(a;)(3)4  +  (3)5 
=  a*  +  15a*  +  90a*  +  270ar»  +  405a;  +  243. 

q      /I     i    9<r\5 

'  =  l5  +  5(2a;)  +  10(2a;)a  +  I0(2xf  +  5(2a;)4  +  (2a;)5 
=  1  +  10a;  +  40  a?  +  80  x9  +  80  a*  +  32a£ 

10.  (2m -l)8  13.  (-7m8naV)3 

=  (2m)8-3(2m)2+3(2m)-(l)8  =  -  7am3x*naa*xVx2 

=  8m8  -  12ma  +  Sm  -1.  =  49m«n2  »V- 

11.  (2aa&c»)4  1A    /     2a?yV 

=  2*a2x46icSx4  14, ' 

=  16a86icls. 


5 


(- 

_      25a?x*y5 
12.  (-bax*y*?  =     S^^c5 

=  -58a8a^xSy2x3  32a;15v5 

=  -125a8a*y«.  .  =  ~2l3^W 

15.  (3  a; +  1)* 

=  (3a?)4  +  4(3a;)8  +  6(3a;)a  +  4(3a;)  +  1 

-  81a*  +  108ar*  +  54a?  +  12a;  +  1. 

16.  (2a; -a)* 

=  (2a;)4  -  4(2a;)3(a)  +  6(2a;)a(a)a  -  4(2a?)(a)8  +  (a)4 
=  16a*  -  32aa*  +  24a2a?  -  8  a3a;  +  a4. 

17.  (3x  +  2af 

-  (3a;)5  +  5(3a?)4(2a)  +  10(3a;)8(2a)2 

-f  10(3a;)a(2a)s  +  5(3a;)(2a)4  +  (2a)5 

-  243a*  +  810aa*  +  1080a2*3  +  720a8a:2  +  240a4a;  +  32a5 


234  ALGEBRA. 


18.  (2a; -y)* 
=  (2*)*- 
=  16a?- 

19.  (x*y-2ry*f 


(2a; -y)* 

=  (2xf-A(2xf(y)  +  e(2xyW-^(2x)(yf^(yY 

=  16a?-  32a?y  +  24a?y*  -  8xy*  +  y*. 

(xty-2xy1f 

=  (^yy-G(^yf(2xy^)+l5(^y)\2xy^-20(x*vf(2xy^ 

+  I5(x*yn2xy')*-6(x*y)(2xyfy+(2xy*j 
-  a?V  -  12a?»y7  +  60a?°y«  -  160a?y»  +  240a?y*° 


if 

-  192a?  y11^-  64a?y» 

20.   (ab-Sy 

=  (oft)7  -  7(o5)«(3)  +  21(oft)5(3)»  -  35(oft)*(3)s 

+  35(a&)*(3)*  -  21  (aftf (3)*  +  7 (a6)(3)«  -  (3)' 
=  aT67  -  21  a6  66  +  189a5  6*  -  945  a*  6* 

+  2835  a«  6s  -  5103  a*  6s  +  5103  ab  -  2187. 


21. 


(-3a'6'C)*  05    /     a?yVY 

=  -35a2*5&2*5c5  *°"   I 2~"J 

243a»ft»A  V      ^xTyixy^ixr 


22.  (-3  ay3)*  27 
=  -3*a?yS*«                                               a;1**'1** 
=  729a?y»                                         ^3— 

23.  (-5  a2  6a?)5 

=  -55a2x*ft5a^x* 

-  S12WW.  26.  (l-a-a«)* 

0>(     /     3a6»V  =  {l-(a+a*)}* 

24.  -— )  -l»-2(a  +  tf)  +  (<i  +  «fr 
V  **"/  =l-2a-2as+a2  +  2as  +  a4 
=  81  a4 y  =  1  -  2a-a2  +  2os  +  0*. 

256  <?* 

27.  (2-3*  +  4x»f 

-[2-(3x-4^)]8 

=  (2)»  -  3(2)3(3jc  -  4  a?)  +  3(2)(3x  -4a?)»  -  (3a;  -  4  a?)8 

=  S  -  36a;  +  48a?  +  54a?2  -  144a? 

+  96  a?  -  27a?  +  108a?  -  144a?  +  64a* 

-  8  -  36ar  +  102a?  - 171  a?  +  204a?-  144a?  +  64a?. 

28.  (l-2x  +  a?)» 

=  {(l-2z)  +  a?}» 

=  (1  -  2xf  +  3  (1  -  2x? a?  +  3(1  -  2a;)(a?)a  +  (a?)3 

=  1  -  6x  +  12a?  -  8a?  +  3 a?  -  12a?  +  12a?  +  3a?  -  6a?  +  a? 

=  1  -  6a;  +  15a?  -  20a?  +  15a?  -  6a?  +  a?. 

29.  (l-a;+a?)s 

=  U-(z-a?)}* 

«  I  _  3  (x  -  a?)  +  3  (x  -  a;*)*  -  (x  -  a?)* 

=  1  -  3a;  +  3a?  +  3a?  -  6a?  +  3a?  -  a?  +  3a?  -  3a?  +  a? 

-  1  -  3 x  +  6a?  -  7a?  +  6a?  -  3a?  +  a?. 


TEACHERS     EDITION. 


235 


(1  +  X  +  3*)* 

=  {l+(*  +  3*)}* 

=  1*  +  4(l)3(a;  +  a?)  +  6(l)2(ar  +  a*)'  +  4(1)  (a?  +a?)3  +  (a;  +  a?)* 
=  1  +  4a;  +  4a;2  +  6a;2  +  12a?  +  6a?  +  4a?  +  12a*  +  12a? 

+  4a;6  +  x*  +  4a?  +  6a?  +  4a;7  +  a? 
=  1  +  4a;  +  10a;2  +  16a?  +  19a?  +  16a?  +  10a?  +  4a?7  +  a?. 


Exercise  LXXVII. 


1.  Va*  =  ±a*. 
v^?  =  ±a?, 

^64  =  4, 

v^l6a12W  =  ±2a36c2, 
v'-32a*  =  -2a3. 

2.  -fr-1728  <?d"ay=  -12cWgy»t 
^33756"?*=  15  6V, 
Mfcv^X  3*  x  74, 
v^2*x3*x7W=  ±  42  c4*. 


3.    V53361 6*  (*y»«» 


32 

7 

7 

ll2 


53361 


5929 


847 


121 


V53361=V32x72xll2. 
•  ^63361 6*C8yi2zi«=  ±  231  Pfety«cP. 

▼      343  22* 

2a? 


66<? 


i 


64  a^8 
729  z30* 


3s5 


4.  V25a2Mc2+v/8a36V-^81a*y<?-v^32as610(? 
=  V52a26V+^23a36«c3-v/3*a*68c*-v/26a56l0c5 
-  5a£2c  +  2a62c  -  3a62c  -  2a62c 

=  2a62c. 

5.  v^7^6x^243^xVl6^V 
=  3ay2x3yzx4a;2z 

=  36a?y322. 

6.  4V2x -y/abxy  +  5y/a?b*xy  ! 
=  4  V2  x  2  -  Vl  x3_x2x6+5Vl2x33x2x6 

=  4\/4-V36  +  5V324 
=  4x2-6  +  5x18 
=  92. 

7.  2ay/Sax  +  by/\2by  +  4a5a;Vkcy 

=  2xlV8xlx2+3v^l2x3x6  +  4xlx3x2V3x2x6 
=  2Vl6  +  3^216  4-  24V36 
=  2x4  +  3x6  + 24  XG 
=  170, 


236 


ALGEBRA. 


8.    Vas  +  2ab  +  » X v'a8  +  3a2  6  +  3a&2  +  6s 
=  (a  +  6)  x  (a  +  6) 
=  (l  +  3)x(l  +  3) 
=  16. 


9.    vfo  -  362a  +  36a*  -  a8  -s-  V62  +  a2-  2ab 
=  ^(5_a)8_5_V(6-a)a 
=  tf-a)-*-(6-a) 


Exercise  LXXVIII. 

1. 

a*  +  4a8  +  2a*  -  4a  +  lla2  +  2a-l 
a* 


2a*  +  2a 

4a8 
4a8 

+  2a2 
+  4a2 

2a8  +  4a 

-1 

-2a2-4a  +  l 
-2a2-4a  +  l 

x*-2x* 

2. 

y +  33y-2ay +  y4|32- 

-  ay  -f  y3 

2^-^-2^  +  3^^ 
y|-2r»y  +    s2y2 


2a2  —  2xy  +  ya 


2a?y2-  2ay +y* 

2s2y2-2a;y3  +  y* 


3. 


4a*-12q5a;  +  5a*j2  +  6a33r»  +  ayi2q8-3a2j-q32 
4a6 
4a8-  30*71  -12a5a?  +  5aV 
h9aV 


to* 
30^]-12a5a:  + 

l-12a5a?4- 


4a8  —  6a2jc  —  oar* 


-4aV  +  6aV  +  aV 
-4aV  +  6a8ar»4-aV 


9a*-24a;V-12^v3+16a;V+16xv5+4v613^-4a^-2y3 
9a* 


6a*-4:cya 


-24a;y-12^y8+16xy 
24a74y2+16ar>y*+16g2y4 


6arl-8a;y2-2y3|-12ic3y8 
I -12  ay 


+16.Ty5+4y6 
+16  ay5 +4  y* 


teachers'  edition. 


237 


5. 

4a8  4-  16a6  c2  -  32a2  c6  4-  16c8!  2a4  4-  4a2  c2  -  4c* 
4a8 
4a4  4-  4a2^21     16a'c2  ~  32a2 c6 


|     16  a6 


c^iea4^ 


4a44-8a2c2-4c4 


-16a4c4~32a2d64-16c8 
lGa4c4-32a2c64-l(>c8 


4ar4  -  20ar>  4-  37a;2  -  30a;  4-  9  |2s2-5.r4-3 
4a* 


±x*-bx 


-20^  +  37^ 
-203s  4- 25  s2 


4s2 -10a;  4- 3 


12a* -30a;  4- 9 
12a;2 -30a;  4- 9 


16a*-  16a6a*  4-  1662a*  4-  4a262  -  8a63  4-  464|4a;2-  2a6  4-  262 
16a* 
8a*-2^-16a&a*4-16&2a*  +  4a2&2 
|-16a6a;2  4-4a262 

1662a*  -8a63H-464 

1662a*  -8a634-464 


8a*-4a&4-262 


8. 


16  -  24a;  +  25a* -  20a*  4- 10a* -4a* 4- a*[4- 3s  4- 2a* -a* 
16 


8-3a; 

-24  a;  4- 25  a;3 
-24a;  4-    9xa 

-20  a*  4- 10a* 
-12a*  4-    4a* 

8-( 

\x  +  2xl 

16a^ 
16  a* 

8-6a?4-4a* 

-a* 

-  8a*  4-    6a*-4a*4-a* 

-  8a*  4-    6a*-4a*  4- a* 

9. 

a*-4a*y4-8aV-10a^y34-8a;2y4-4^v54-/lar>-2a^.v4-2a;.v2-y3 
a* 


2x*-2x*y 


2a*-4x2y  +  2xy2 


-4a*y+8a*y2 
-  4  a*y  4- 4a*  ya 


2a*-4a*y4-4a^2  — if 


4a*y2-10a*ys4-8a*y* 
4a*y2-  8ar>y34-4ar'y4 


—  2a*y34-4a*y4— 4a:y54-y6 
2  a*  y8  4-  4  x2 1/4—  4  x?f  4-  y6 


238 


ALGEBRA. 


10. 

4a« _  4a5_  iia*  +  i4as  +  5aa_  12a  +  4l2a8-c^-3a  +  2 
4a« 


4a8 -"a8" 

—  4a{ 
-4a! 

>-lla4 

>  +      a* 

14a8  +  5a* 
6a8  +  9a8 

4  a8- 2a' 

-3a 

-12a4  + 
-12a4  + 

4a8 -2a1 

-6a  +  2 

8a8-4a8-12a  +  4 
8a8-4a2-12a+4 

11. 

1 3a-b+ 5c+d 
9  a8  -6  a&+68+30  ac-10  6c+25c8+6aa^-26d+10  cd+d8 
9a8 


6a-6 


-6a&+68 
-6a&+68 


6a-26  +  5c 


30OC-106C+25C8 
3000-1060+25^ 


6a-2b  +  10c  +  d 


12. 


Gad^&d+lOcd+d8 
6ad-26aT+10cd+d2 


|5a?8-3a8y-4gyl  +  y8 

25  a*  -  30a*y  -  31a*y8  +  34*V  +  10a"y*  -  8xy5  +  y6 
25  a* 


10a* -3*^ 


10a*-6a*y-4zy8 


-30a^y-31a*y" 
-300*7+    9g*.Va 


10a*- 6a*y- 80^  +  3* 


40a?*y8  +  343*^  +  lOa^y4 
-  40a*y8  +  243*3*  +  16  a*  y4 


103*3*-    ea^^-Sa^  +  y6 
10aV-    6a?V-  Sgy5  +  y» 


13. 

|  m4— 2  m8+3  ro8— 4  m+5 
m8-4m7+10m6-20m5+35m4-44m8+46m2-40wH-25 


2m4-2ma 


-4m7+10m6 
-4mT+  4m6 


2m4-4m8+3m8 


6m«-20m5+35m4 
6m6-12m5+  9m4 


2  m4— 4  m3+6  m8— 4  m 


-  8ra5+26m4-44m8+46m8 

-  8m5+16m4-24m8+16m8 


2m4-4m8+6m8-8m+5 


10m4-20m8-f30m8-40m+25 
10m4-20m8+30m8-40m4-25 


TEACHERS     EDITION. 


239 


14. 

x*  —  x*y  —  Jafy2  +  xy*  +  y4  \x*  —  jxy  —  y2 
x* 


2x*-ixy 


2x*-xy-y2 


•  a?y  +  j&y* 


-  2a?y  +  xy*  +  y* 
2s2y2  +  sy3+y4 


15. 


a4-4aV+6aV-6a^,  +  By4-—  +  ^l 
a4  xx 


x2  —  2xy  +  y*  —  * 


2xi-2xy 


—  4a?y  +  6afy2 

—  4afy  +  4a2y2 


2a?  — 4a;y  +  ya 


2afy2-6;Fy8  +  5y4 
2sy-4gy8+    y* 


2a?-4a;y  +  2y2 - 


y8 


-2^.+  4^-3£  +  g 
-W  +  4/-tf  +  S 


16. 


^_^j?4-  43 g2a;2     3 as3     a^la2     3aa?  .  a? 
9       2  48  4         4    3,42 


_9 
2a2     Sax 
3         4 


asa?  43a2a;2 
2  48 

a8a?  27  oV 
2  48 


.)/^_3aa2\     of 
-U        4    y      2 


aV_3ar»  ^ 

3  4  4 

aV_3aa?  a£ 

3  4  4 


240 


ALGEBRA. 


17. 

10     20,25      24. 

X2         X*        X4,         X6 


1+-  +  --5  +  -3  +  -Z  -f  —  + 


f      2      3       4 


9 

2  +  - 

X 

4 

-  + 

4 

-  + 

X 

10 
4 

X* 

20      25 
12      9 

X*  +  X* 

"+1 

X 

♦s 

4      6 

X       X2 

4 

8      16     24     16 
x3      x*      b6      je6 
8      16     24     16 

X3        X*         if        Xs 

18. 

is      2o  ,  o      26^6*|a     ,  ^b 
>*       o  a       ar\b  a 


2a 

-1 

-^  +  3 

b 

b 

6 

?,b 

2a 

«     6 

^V 

z  +  - 

2- 

+  — 

o             a 

a 

a1 

o 

2b 

+  *' 

a 

a« 

19. 

^  +  ^      -      3  +  9       +2      3 


12 


2*2  +  f 


2xJ  +  a;-i  -  — 


X8 

_5r* 

12 

^-T" 

4 

1 

2z* 

3     9 

3 

3 

2s2 

-2  +  1 
3      9 

3 

teachers'  edition. 


241 


Exercise  LXXIX. 


(1) 


(2) 


(3) 


(4) 


(5) 


120409(347 
9 

(1) 

16803.9369(129.63 
1 

64)304 

22J68 

256 

44 

687)4809 

249)2403 

4809 

2241 

2586)16293 
15516 

4816.36(69.4 

25923)77769 
77769 

36 

129)1216 

(2) 

4.5449976i  (2.1319 

1161 

4 

1384)  5536 

4lj54 

5536 

41 
423)1349 
1269 

1867.104i  (43.21 

4261)8097 

16 

4261 

83)267 

42629)383661 

249 

383661 

862)1810 
1724 

(3) 

0.24373969(0.4937 

8641)  8641 

16 
89)837 

8641 

'     801 
983)3639 

1435.652i  (37.89 
9 

2949  » 
9867)69069 

69069 

67)535 

469 

(4) 

0.5687573056(0.75416 

748)6665 

49 

5984 

145J787 

7569)68121 

725 

68121 

1504)6257 
6016 

64.128064(8.008 

15081)24130 

64 

15081 

16008)128064 

150826)904956 

128064 

904956 

242 


AU1KBRA. 


3. 


(1)  0.9000000000(0.94868 

81 
184)900 
736 
1888)16400 
15104 


(5) 


18966) 129600 
113796 


189728)1580400 
1517824 


17.00(4.1231 
16 
8ljl00 
81 
822)1900 
1644 
8243)25600 
24729 


82461)87100 
82461 


(2) 


(3) 


W 


6.21(2.4919 

4 
44J221 

176 
489)4500 
4401 


4981)9900 
4981 
49829)491900 
448461 


0.43(0.6557 
36 
125)700 
625 
1305)7500 
6525 
13107)97500 
91749 


0.008620(0.0923 

81 
182)420 
364 
1843)5600 
5529 


(6) 


(7) 


i29.O0060606(lL3578 

2l}29 
21 
223)800 


2265)13100 
11325 


22707)177500 
158949 
227148)1855100 
1817184 


347.2590(18.6348 
1 

28 J247 
224 

366)2325 
2196 
3723)12990 
11169 


37264)182100 
149056 


372688)3304400 
2981504 


TEACHERS*    EDITION. 


Ui 


(1) 


(2) 


(3) 


14295.3870(119.5633 
1  ' 

21)42 
21 
229)2195 
2061 
2385) 13438 
11925 


(4) 


23906)  151370 
143436 


239123)793400 
717369 


2391263)  7603100 
7173789 

2.50000(1.5811 
1 


25)150 
125 
308)2500 
2464 
3161)3600 
3161 


31621)43900 
31621 

2000(44.7213 
16 
84)400 
336 

887)640Q 
6209 


8942)19100 
17884 


89441)121600 
89441 

0.3006o6o6  (0.5477 
25 

104J500 
416 
1087)8400 
7609 


10947)79100 
76629 


(5) 


(6) 


(1) 


(2) 


(3) 


0.03000000(0.1732 
1 

27)200 
189 
343)1100 
1029 
3462)7100 
6924 


111(10.5356 


205)1100 
1025 


2103)7500 
6309 
21065)119100 
105325 


210706)1377500 
1264236 


O.OOlil  (0.0333 
9 
63)210 
189 
663)2100 
1989 


O.06460606  (0.0632 
36 
123)400 
369 
1262)3100 
2524 


0.0050(0.07071 

49 

1407)10000 
9849 


14141)15100 
14141 


244 


ALGEBRA. 


(5) 


(6) 


(<) 


iOOOOOOOO  (1.4142 
1 
24jl00 
96 
281)400 
281 
2824)11900 
11296 
28282)60400 
56564 


5.00(2.2360 
4 
42)100 
84 


443)1600 
1329 


4466)27100 
26796 


3.25(1.8027 
1 
28)225 
224 
3602)  10000 
7204 
36047)279600 
252329 


8.600000(2.9325 
4 


G. 

(1) 

vT-J. 

(2)- 

Vii  =  f 

(3) 

v7ji-h=j 

(4) 

vib-h. 

(5) 

V|!|  =  «. 

(6) 

V«{  =  H=! 

7. 

(1) 

J  =  0.5. 

0.50)0.7071 
49 

(2) 


(3) 


49)460 
441 
583)1900 
1749 
5862)15100 
11724 


1407) 10000 
9849 
14141)15100 
14141 

}= o,6mm. 

0.666666(0.8164 
64 
16lj266 
161 
1626)10566 
9756 

f  =  0.75. 

0. 750000  (0.86(50 
64 
166)1100 
996 
1726)10400 
10356 
1732)4400 

(4)        ^  =  0.03125. 

0.031250(0.1767 

1 


27)212 
189 


346)2350 
2076 


3527)27400 


teachers'  edition. 


245 


(5)   ^  =  0.0546876. 

m 

f  =  0.857142. 

0.05468760(0.2338 

0.857142(0.9258 

4 

81 

43)146 

182)471 

129 

364 

463)1787 

1845)10742 

1389 

9225 

4668)39850 

18508)151700 

37344 

148064 

(8) 

^  =  0.08333333. 

0.08333333(0.2886 

(6)      xt  3  =0.048. 

4             V 

0.0480(0.2190 

48)433 

4 

384 

41)80 

568)4933 

41 

4544 

429)3900 

5766)38933 

3861 

34596 

Exercise 

LXXX. 

1. 

3a* 


(3a;  +  2y)2,y  =  6a;y  +  4ya 


3a;2  +  6a;y  +  4ya 


a*  +  6x*y  +  12a-y2  +  8y*\x  +  2y 

^6a?y  + 12a^2  +  8y8 
6a;2y +  12a?y2  +  8y8 


2. 

g8-9a2  +  27a-27|a-3 


3a2 
-3(3a-3)=        -9a  +  9 


3a2-9a  +  9 


-  9a2  +  27a -27 

-  9a2  +  27a -27 


(3<c  +  4)4  =  _ 


3a;2 


12*  +  16 


3. 

a"  +  12a?  +  48a;  +  64[a;  +  4 

~*12a;a  +  48a;  +  64 
12a;2  +  48  a;  +  64 


3a;2  +  12a;  +  16 1 


246 


ALGEBRA. 


3** 


(3x*-ax)(-ax)  =      -SaaP+atx2 


x*  —3a3?+ba*j!*-3a?x—a*  \x*—ax-a2 

3*  

-3aa-5+5a8z3-3a5a; 


-Zax*+Za2x*-a*x* 


3ar*-3aa^+aV 
3(z*-aar)2  =  3^-60^+3^^ 
(3xJ-3ax-a*X-<»,)  =  -3aV+3a8z +a* 


$x*Sax* 


+3a8a?+a* 


-3a2ar*+6aV  -3a5a?-a« 
-3aV+6a8a*-3a5x-a6 


i6  +335+6rf*+7a*+6a?*+3a:+l  la^+ar+l 
a*  


(3a*+a:)c= 


3ar* 


+3a8+3f 


3ar*+3a*-Kr* 
3(ar*  +  a?)*=3ar*+6a*+3a* 
(3a"+3ap+l)(l)- 3s*+3a:-H 


3a*+6a?*+7a* 
3a*+3a>*+  Xs 


3ar*+6a*+6ar»+3a!+l 
Sa^+ea^+ea^+Sar+l 


3o?4+6ar,+6xa+3a;+l 

6. 

[l-3g  +  4s» 
1  -  9x  +  39  x1  -  99a*  +  156a?*  -  144a*  +  64a* 

1 


3 
-9a?  +  9x2 

-9x+39a?*- 
-9a?  +  27a?»- 

-99ar» 

3-9a;  +  9a^ 

-27 x* 

3~18x  +  27aa 

12x2-36a!, 

+  16ar* 

12s2- 
12a*- 

-  72ar»  +  156a*  -  144  a*  4-  64  a* 

3  -  18s  +  39a?- 36 a^ 

*  +  16ar* 

-  723?  +  156a*  -  144s6  +  64a* 

7. 

|of-2o-l 
a*-6a*+  9a*  +  4a8-9a2-6a-l 


3  a* 


(3o*-2aX-2a)=  -6a84-4a2 

3a*-6as  +  4a2 

3(a2-2a)2  =  3a*-12a8+12a2 

(3a2-6a-l)(-l) 


-6a6+  9a*+4a8 
-6a6  +  12a*-8a8 


3a2+6a+l 


3  a* -12  a8  +  9a2+6a+l 


-3a*  +  12a8-9a2-6a-l 
-3a*  +  12a8-9a2-6a-l 


TEACHERS     EDITION. 


247 


48** 
(12*3+4*)  4s  48a8+16*2 


8.  |4*2+4*-l 

64*«+ 192**+144**-  32*8-36*2+ 12*- 1 
64a* 

192*6+144**-32orl 


48**+48*8+16*2 
3(4x*+4aj)2  =48**+96*8+48*2 
(12x«4-12*-lX-l)= 


192.U5 +192**  f64*» 


12*2-12*+1 


48**+96s8+36*a-12*+l 


-48**-96*«-36*a+12*- 1 
-48**-96*8-36*J+12*-l 


9.  li-g+s*-*8 

1  -3*+6*»-10*8+12ar*-12*6+10a*-6*7+3*8-*» 
1 


3 

-3*+** 

Sx+Gx'-lOar1 
-3*+3*2-    x9 

3-3*+** 

3-6*+3** 

*»+** 

3*2-  9*3+12**- 
3**-  6*»+  6s*- 

-12s8+10s8 

-  3**+    a* 

3-6*+6*2-3*3+** 
3-6*+9*J-6*s+3** 

-3*s+3**-3*5+*6 

-  3*»+  6**- 

-  3*8+  6s*- 

-  9**+  g^-e^+Sa8-*8 

3-6*+9*a-9s8+6a!* 

-3s*H 

-s6 

-  9**+  9s8-6s'+3s8-s9 

10.  |a2+3a6-962 

a6  +9a56-135as6»+729a66-7296« 


3a* 


9a86+9a262 


3a*+9a86+9a2&2 
3a*+18a86+27a»6a 
-27a262-81oy+81M 


3a*+18a86 


-&la&+m4' 


9a*6-135a8&8+729a&8 
9a66+27a*62+27a8&3 


-27a*62-162a363+729a68-729&* 
-27a*62-162a868+729a68-72966 


11.  |c2-46c+4&2 

<J8-126c8+6062c*-160&8c8+240Mc2-19268c+6468 


3c*  I  -126c5+6062c*-160J8c8 

-126c8+166Vl-126cs+48yc*-  64W 


S^-^W+ieto2 

3c*-246c8+4862c2 

12i2c2-4868c+166* 


3c*-246c8+6062c2-4868c+166* 


1262c*-  9668c3+2406*c2-19265c+646* 
126V-  96&8c8+2406*c2-19266c+646* 


248 


ALGEBRA. 


12. 

|2a*+4o&-3y 
8a«+48a56+60aW-  80a368-90a264+108a^-276« 
8a« 

12a4  48a5&+60a*&*-  SOaW 


24a86+16a*&2 


12a4+24a8&+16a26* 
12a4+48a3&+48a*&» 


48a56+96a4&*+  64a868 


-18aW-36aP+96* 


12a4+48as&+30a2&*-36a68+9&4 


36a462-144a8i8-90a264+108a55-27is 
-36a4y-144a86a-90a»64+108a65-27y 


Exercise  LXXXI. 


68  = 

274625165 
216 

63  = 

262144164 
216 

3(60)*  =  10800 

3(60x5)=  900 

5s  =   25 

58625 

3(60)*  =  10800 

3(60x4)=   720 

4*=   16 

46144 

11725 

58625 

11536 

46144 

2. 

4. 

110592148 
48  =      64 

884.73619,6 

9s  =      729 

3(40)*=  4800 

3(40x8)=   960 

8*=   64 

46592 
46592 

3(90)»  =  24300 

3(90x6)=  1620 

6*=   36 

155736 

6824 

25956 

155736 

4s  - 


5. 


1092153521478 
64 


3(40)*=   4800 
3(40x7)=   840 

7*  =    49 
5689 

45215 
39823 

3(470)*  =  662700 

3(470x8)=  11280 

82=    64 

5392352 

674044 

5392352 

TEACHERS     EDITION. 


249 


Is  = 

3(10)2  = 

3(10x1)  = 

12  = 

6. 

300 
30 

1 

i48i544| 
1 
481 

331 

114 

9. 

Is  = 

3(10)2=  300 

3(10x4)=   120 

42=   16 

2.80322J  |1.41 
1      

1803 

331 

436 

1744 

3(110)2  = 

3(110x4)  = 

42  = 

36300 

1320 

16 

150544 
150544 

3(140)2  =  58800 

3(140x1)=  420 

P=    1 

59221 

37636 

59221 

59221 

10. 


18= 


1601.613J11.7 
1 


3(10)2=  300 

3(10x1)=   30 

12=    1 

601 

331 

331 

3(110)2  =  36300 

3(110x7)=  2310 

(7)2  =   49 

270613 

38659 

270613 

Is- 

3(10)2  = 


300 


3(100)2  =  30000 

3(100x8)=  2400 

&2= 64 

32464 


12597121108 
1 

"259712 


259712 


1»  = 

3(10)2  = 

3(10x9)  = 

92; 


7077888|192 


3(190)2  = 
3(190x2)  = 


300 

270 

81 

6077 

651 

108300 

1140 

4 

5859 
218888 

109444 

218888 

11. 


2»  = 

3(20)2  = 

3(20x3)  = 

32  = 


3(20x3)  = 

2(3)2. 

3(230)2  = 

3(230x4)  = 

42  = 


12.81290412.34 


1200  - 
180 
9 

8 
4812 

1389 

4167 

180 
18 

645904 

158700 

2760 

16 

161476 

645904 

250 


ALGEBRA. 


12. 

38  = 

3(30)a  =  2700 

3(30x8)=  720 

8*=  64 


3484 


3  (380)»  =  433200 

3(380x4)=     4560 

4*=         16 


437776 


56.623104|3.84 

_27  

29623 


27872 


1751104 


1751104 


13. 


38  = 


33076.16J  |32.1 
27 


3(30)*  =  2700 

3(30x2)=  180 

(2)*=   4 

6076 

2884 

3(30x2)=  180 
2(2)a  =   8 

5768 
308161 

3(320)*  =  307200 

3(320x1)=   960 

1*=     1 

308161 

308161 

14. 


48  = 


102503.232146.8 
64 


3(40)*=  4800 

3(40x6)=   720 

6*=    36 

38503 

5556 

33336 

3  (460)*  =  634800 

3(460x8)=  11040 

8*  =    64 

5167232 

045904 

5167232 

TEACHERS     EDITION. 


251 


15. 

9»  = 
3  (90)*  =  24300 
3(90x3)=  810 
3*=    9 

820.02585619.36 
729 
91025 

25119 

75357 

810 
18 

15668856 

3(930)2  =  2594700 

3(930x6)=   16740 

62  =     -36 

2611476 

15668856 

16. 

2»  = 
3(200)*=  120000 
5(200x5)=    3000 
(of  =     25 

8653.002877120.53 
8 

653002 

123025 

615125 

3(2050)2  =  12607500 

3(2050X3)=   18450 

(3)2  =      9 

37877877 

12625959 

37877877 

17. 

Is  - 
3(10)2  =    300 
3(10X1)=     30 
1»-      1 

i.37i33063i  11.111 
~ 371 

331 

3(110)*=   36300 

3(110  x  1)  =    330 

(1)2=      1 

331 
40330 

36631 

36631 

3(1110)2  =  3696300 

3(1110x1)=   3330 

12=      1 

3699631 

3699631 

3699631 

252 


ALGEBRA. 


18. 


3(20)*  = 
3(20x7)  = 

7*  = 


3(270)*  = 

3(270x5)  = 

5*  = 


3(2750)a  = 

3(2750  x  5)  = 

5*  = 


20910.518875127.55 
8 


1200 

420 

49 

12910 

1669 

11683 

420 

98 

1227518 

218700 

4050 

25- 

222775 

1113875 

4050 
50 

113643875 

22687500 

41250 

25 

22728775 

113643875 

19. 


4s  = 

3(40)*  =  4800 

3(40x5)=  600 

(5f  =  25 

9J.398648466125 14.5045 

64 

27398 

5425 

27125 

600 
50 

273648466 

60750000 

3(4500x4)=   54000 

42=     16 

243216064 

60804016 

54000 

32 

30432402125 

6085804800 

3(45040x5)=    675600 

52  =       25 

6086480425 

30432402125 

teachers'  edition. 


253 


20. 


18  = 

3(10)*  = 

3(10  X  7)  = 

(7J»» 


3(170)*  = 
3(170x4) 


5.340104393239|1.7479 
1 


X4)  = 


3(1740)*  = 

3(1740x7)  = 

7*  = 


3(17470)*  = 
3(17470  x9)- 


=  300 
=  210 

-  49 

4340 

559 

210 

98 

3913 
427104 

=86700 
=  2040 

■   16 

88756 

355024 

2040 
32 

72080393 

-  9082800 

.  36540 

49 

9119389 

63835723 

36540 
98 

82446702 

=  915602700 

471690 

81 

916074471 

82446702, 

21. 


(1) 


3(10)*  = 

3(10x3)  = 

3*  = 


3(130)*  = 

3(130x5)  = 

5*  = 


3(1350)*  = 

3(1350x7)  = 

7*  = 


2.500|1.3572 

1 

300 

1500 

90 

9 

1197 

399 

303000 

90 

18 

50700 

1950 

25 

263375 

52675 

39625000 

1950 

50 

5467500 

28350 

49 

38471293 

5495899 

1153707000 

254 


ALGEBRA. 


00 


5*  = 


>.206006006006|0.5848 
125 


3(50)*=  7500 

3(60x8)=  1200 

8*=   64 

75000 

8764 

70112 

1200 
128 

4888000 

8(580)*  =  1009200 

3(680x4)=   6960 

4*  =     16 

1016176 

4064704 

6960 
32 

823296000 

3(5840)*  =  102316800 

3(5840x8)=   140160 

8*  =     ,  64 

102457024 

819656192 

(3) 


0.( 
2"  = 
3(20)*  =  1200 
3(20x1)-  60 
1*=   1 

)10000000000 10.2164 
8 
2000 

1261 

1261 

60 
2 

739000 

3(210)'=  132300 

3(210  X  5)  =   3150 

5*  =    25 

135475 

677375 

3150 
50 

61625000 

3(2150)*  =13867500 

3(2150x4)=   25800 

4*=      16 

13893316 

55573264 

TEACHERS     EDITION. 


255 


w 


Is  = 

3(10)*  = 
3(10x5^  = 


3(150)*  = 
.     3(150x8)  = 

8*  = 


3(1580)*  = 

3(1580x7)  = 

V- 


3(15870)*  = 

3(15870x4)  = 

4*  = 


4.00000000000011.5874 
1  


300 

150 

25 

3000 

475 

2375 

150 
50 

.  625000 

67500 

3600 

64 

71164 

569312 

3600 
128 

55688000 

7489200 

33180 

49 

7522429 

52657003 

33180 
98 

3030997000 

755570700 

190440 

16 

755761156 

3023044624 

(5) 


7s- 


0.40000000000010.7368 
343 


3(70)*  =  14700 

3(70x3)-  630 

3*=    9 

57000 

15339 

46017 

630 
18 

10983000 

3(730)*  =  1598700 

3(730  X  6)  =   13140 

6*=     36 

1611876 

9671256 

13140 
72 

1611744000 

3(7360)"=.  162508800 

3(7360x8)=   196640 

8*=      64 

162705504 

1301644032 

256 


ALGEBRA. 


Exercise  LXXXII. 


1. 

81a4  -540a86+1350aa5a-1500a6a*+625&4l  9  a*-30ab +25b> 
81a4 


18aa-30a& 


-540a8&+1350oa&* 
-540a8&+  900aa62 


18tt*-60ai+2562 


450aa6a-1500a&,+625&4 
450a2y-1500qy+6256* 


9«'-30a6+256a13a-56 
9a» 
6a-^56]-30a6+25&1 
-30a&+25&» 


2x* 


a;8  -4o;7+10a!«-16a*+19«4-16a!»+10«2-4a;+l  1  s4-2s8+3a;a-2g+l 

x8  " 

2a7|-4a;7+10a* 
|-4a;7+  4a* 
2a74-4a*+3ar' 


6a*-16ar>+19z* 
6a*-12a*+  9s* 


2a?4-4a*+6z*-2a> 


-4s»+10a*-16z8+103s 
-42*+  8^-12^+  4x* 


2a!*-4a8+6<r1-4z+l 


2a?4-  4a*+  6a*-4aM  1 
2s4-  4a»+  6s»-4s+l 


a*  -2ss+3za-2aH-l  |g«— a?+l 
a?4  

2aj»-a7Ufcr»+332 

2aa-2*+l 


2a*-2a;+l 
2xa-2a;+l 


3. 

64- 192s  +  240  a?- 160  a*  +  60  a;4- 12a? +  a*18-12s  +  6a?-s' 
64 


16-12* 


-  192a  +  240  a* 
-192*  +  144  a3 


16  -  24a:  +  6s2 


16-24a?+12xJ-x» 


96xa-160aj»+p0aJ* 
96*a-144a8  +  36&4 


-  16a*  +  24a4- 12a*  +  a* 

-  16a*  +  24  s4 -12a*  +  a* 


teachers'  edition. 


257 


2*  = 
3(2)2=12 
(6 -*)(-*) 


8  -  12x  +$x2  - a?\2-x 
8 

12a; +  6^-3^ 


-6x4-  x2 


12-6»+V 


•  12a;  4- 6a;2 -s8 


4. 

729a*  -1458g5-hl215g*-540a!a+135g>-l8g4-l|9a*-6x-fl 
(9a?)8**  729a* 
243ar*  -1458a*+1215a*-540a* 


-162a8 


4-36a? 


243a^l62*8+36<ra 


-14583*4-  972a^-216a* 


243a^324ar»4-108a* 

27^-18  x 


4-1 


243  a*-324  ar*4-135  a*-18  s+1 


243a*-324ar»4-135a*-18a;4-l 


243^-324^4-1 35a*-18a;4-l . 


9a?-6a;  +  l|3a?  — 1 
9a? 


6»-l 


-6»4-l 
-6x  4-1 


1  -8t/4-28t/>-56vg4-70v4-56v54-28v6-8y74-v8 1  l-4y4-6y2-4y84-y4 


2-4y 


-8y4-28y2 
-8y4-16y* 


12a;g-48y84-36y4 


2-8y+6y2 


2-8y+12y2-4y8  |_  8y84-34y*-W4-28y* 
|-  8y84-32y*-48y»4-16y« 


2-8y+12y»-8y»+y* 


2y*_  8i/54-12y6-8v74-y8 
2y*~  8y54-12yg-8yT4-y8 


1  -4y4-6y*-4ys4-y4ll--2y4-y* 


2-2y 


-^y+Sy1 
— 4y4-4ya 


l-2y4-y2|l-y 


2-y 


2-4y4<^|2y2-4y84-y* 
1 2y2— 4y*4-y* 


-2y4-y2 
-2y4-y* 


258 


ALGEBRA. 


Exercise  LXXXIII. 


1. 


s3  -  3  =  46, 

x>  =  49. 
.-.  x  =  ±7. 


2.  2(x*-l)-3(x»+l)+14  -  0. 
Simplify, 
2x*-2-3x*-3+14  =  0, 
x*  =  9. 

.-.  x  =  ±3. 


3. 


s*-5      2x»  +  l 
6 


3 

Simplify, 
2xa-10  +  2x*  +  l  =  3, 
4x»«12, 

a?  =  3. 
.-.  x  =  ±V3. 


-*-  +  -*-  -8. 

1  +  x      1  — x 

Simplify, 

3-3x  +  3  +  3x  =  8-8x», 
8x*  =  2, 

.-.x  =  ±}. 


5.  J— -L- 1 
4x»     6x*     3 

Simplify,  9-2  =  28x», 
-28x*  =  -7, 

x»  =  J. 
.\  x  =  ±  J. 

6.  5x*-9  =  2x*+24, 

3x»  =  33, 
a*  =  11. 
.-.  x  =  ±Vll. 

7.  (x  +  2)a  =  4x+5. 
Simplify, 

3*  + 4a: +  4  =  4s +  5. 
Transpose  and  combine, 
x*  =  l. 
.\  x  =  ±l. 


8     ^_?1-10 


■  7- 


50+^ 
25   ' 


5        15 
Simplify, 

15a*-W+50  =  525-150-32*. 
Transpose  and  combine, 
13  a2  =  325, 
x»  =  25. 
.*.  a;  =  ±5. 


9    3x^-27     90  +  4xV 


x*  +  3 


x*  +  9 


■7. 


Simplify,   (3x*-27)(x2  +  9)  +  (904-4x»)(x*  +  3)  =  7(xa  +  3)(*«  +  9). 
3x4  -  243  +  4x*  +  402x»  +  270  =  7x*  +  84x*  +189, 
18  a?  =  162, 

x*  =  9. 
.-.  x=±3. 


10. 


8x+^  = 


65x 

x  7  ' 
Simplify,  56  x2  +  49  =  65  x*. 
Transpose  and  combine, 

-9x*  =  -49, 
x2  =  ^. 
.-.x  =  ±J 


11. 


4x*+5     2x»-5       7x*-25 


15 


20 


10 
Simplify, 

24  x*+30-8  x»+20  -  21  x*-75, 
24x*-8x*-21  x*  -  -75-30-2 
-5x»  =  -125, 
x»  =  25. 
.\  x«=±5. 


teachers'  edition. 


259 


12. 

10**  + 17      12s2 +  2     5a;2 -4 


18  11^-8  9 

Simplify,         110s*  +  107s2  -  136  -  216s2-  36  -  110s*-168s2  +  64. 
Transpose  and  combine,  59  s2  =  236, 

s2  =  4. 

.-.  s  =  ±2. 


13. 

14s2 +  16      2s2 +  8 


2tf 
'  3  ' 


21  8s2 -11 

Simplify,  112s4  -  26s2  -  176  -  42s2  -  168  -  112s*  -  154s2. 

Transpose  and  combine,  86  s2  =  344, 

s2  =  4. 

.-.  x~±2. 


14. 


s2  +  bx  +  a  =  6s(l  —  bx), 
Xs  +  bx  +  a  =  bx  —  62s2, 
s2+62s2  =  — a, 
a 


s2  =  -- 


1  +  62 


>     1  +  ft2 


15.  wis2  -t-  n  =  g, 

mx2  =  q—n. 


16.     s2  —  ax  +  b  =  as(s  —  1), 
s2  —  as  +  b  =  as2  —  axt 
x*-ax*  =  -bt 
x*(l  -a)  =  -6, 


*2  =  - 


a-1 


Exercise  LXXXIV. 


1.  s2  +  4s  =  12, 

Complete  tbe  square, 

s2  +  4s  +  4=16. 
Extract  the  root, 

x  +  2  =  ±  4. 
.-.  s  =  2or-6. 


2.  x*-6x  =  16. 

Complete  the  square, 

s2-6s  +  9  =  25. 
Extract  the  root, 

x  -  3  =  ±  5. 


,  s  «=  8  or  -  2. 


260 


ALGEBRA. 


3.  a*- 12* +  6 -J. 

x»-12x=-V. 

Complete  the  square, 

x*  -  12x  +  36  =  ifi. 

Extract  the  root, 

x-6  =  ±V- 
.\  s  =  llJor}. 

4.  s3-7s  =  8. 
Multiply  by  4, 

4**- 28a?  =  32. 
Complete  the  square, 

4x2-()  +  49  =  81. 
Extract  the  root, 

2x-7  =  ±9, 
2x  =  7  ±  9. 
.\x  =  8or-l. 


5.  3s»-4s=7. 

Multiply  by  3, 

9x*-12s  =  21. 
Complete  the  square, 
9s*- 12s +  4  =  25. 

Extract  the  root, 

3x-2  =  ±5, 
3s  =  7  or -3. 
.\  x  =  2§or-l. 


7.  x2  -  x  =  6. 

Complete  the  square, 

4      4 
Extract  the  root, 

x-l  =  ±5, 

2         2 
1±5 


,  x  =  3or-2 


8.  5s»-3x  =  2. 

Multiply  by  5, 

25x*-15x  =  10, 
Complete  the  square, 

25*-()  +  |-i?. 

Extract  the  root, 


5—|-±7 


5x 


2        2 
3±7 


,  a?  =  lor-§. 


6.  12x8+s-l  =  0. 

12s2  +  x=l. 

Multiply  by  3, 

36x*  +  3x  =  3. 
Complete  the  square, 

36x*  +  ()  +  JL-^. 
W      16      16 
Extract  the  root, 

6x  +  £  =  ±  J, 
6x  =  f  or -2. 
.*.  x  =  J  or  —  J. 


9.  2x2-27x  =  14. 

Multiply  by  8, 

16x2-216x  =  112. 

Complete  the  square, 

16^-0  +  729  =  841. 

Extract  the  root, 

4x-27  =  ±29, 

4x  =  56or-2. 
.  •.  x  =  14  or  -  J. 


teachers'  edition. 


261 


10. 


12 


3 

a2- 


2£  =  _Jl 
'  3  12* 

Complete  the  square, 

rf-^  +  i-l. 

3       9     36 

Extract  the  root, 

3       6 

-14 

3    6 

2      6 


12. 


3a  ,  _£  . 
4       3a  = 


13 
6* 


Simplify, 

9a2 -26  a  =  -16. 
Complete  the  square, 
169  =  25 
9        9* 
Extract  the  root, 
13 


9r>_26*  +  ^  =  ^. 


-a-4 

3s-6orf. 
3 

.-.  a  =  2or- 
9 


11. 


12*  +  24, 
24. 


2      3 
Simplify,  3a2-2a 
3a2 -14a 
Multiply  by  3, 

9a* -42a  =72. 
Complete  the  square, 

9^-0  +  49=121. 
Extract  the  root, 

3a-7  =  ±ll, 
3a  « 18  or 


=2(a  +  2).        13, 


-4. 


,  a?  ~  6  or  —  1  \. 


a  +  1 
a; +  4 
Simplify, 
(a  +  l)(a  +  6)  = 


2a-l 
a  +  6  ' 


=  (2a-l)(a  +  4> 
a2  +  7a  +  6  =  2a2  +  7a-  4, 
a2-2a2+7a-7a  =  -  4  -  6, 
-a2  =  -10, 

a2  =     10._ 
.\a  =  ±Vl0. 


14. 


Simplify,  18  a2*  72  a 

Transpose  and  combine, 
Multiply  by  10, 

Complete  the  square, 
Extract  the  root, 


x  x  +  S    ^      1 

a  +  1      2(a  +  4)  =      18* 
-  9a?  -  36a  -  27  =  -  a2  -  5a 
10^  + 41a  =  23. 
100a2  +  410a  *-  230. 


lOa  +  L1^, 
2  2 


10a  = 


41±51 
"  2      2' 


,  a  =■  J  or  —  4|. 


262  ALGEBRA. 


15.               2     -     3     +     2    • 

x— 1      x— 2      x  —  \ 

Simplify,                2  a?  - 

\2x  +  16  =  3a*  -  15x  +  12  +  2a?  -  6x-  +  4. 

Transpose  and  combine, 

-3a?  +  9x  =  0. 

Divide  by  —3, 

a?-3a;  =  0, 

orx(x-3)  =  0. 

,\  a;  =  3or0. 

16. 

5<c(x  -  3)  -  2  (a?  -  6)  =  (x  +  3)(x  +  4). 

Simplify, 

5  a?  -  15s  -  2a?  +  12  =  a?  +  7x  +  12. 

Transpose  and  combine, 

2a?-22*«0, 

a?-llx  =  0, 

or  x(x-ll)  =  0. 

.-.a;  =11  orO. 

17. 

3a;          5       3a?           23 

2(x  +  l)     8      a?-l      4(x-l) 

Simplify, 

12a2  -  12a;  -  5a?  +  5  =  24x2  - 46a?  -  46. 

Transpose  and  combine, 

17a?  -34  a;  =  51, 

Divide  by  17, 

a? -2a;  =  3. 

Complete  the  square, 

a?-()  +  l  =  4. 

Extract  the  root, 

a-l  =  ±2. 

.-.  x  =  3  or  -  1. 

18. 

(*-2)( 

x-4)-2(x-l)(x-3)  =  0. 

Simplify,                      a?  - 

-6x  +  8-2a?  +  8x-6  =  0. 

Transpose  and  combine, 

a?-2x  =  2. 

Complete  the  square, 

a?-2x+l  =  3. 

Extract  the  root, 

s-l-i  V3. 

.%  x  =  l±V3. 

19. 

i(a-4)-|(x-2)  =  i(2x  +  3). 

Simplify, 

5a?  -  20a;  -  14a;2  +  28a;  =  70a;  +  105. 

Transpose  and  combine, 

9a?  +  62a;  =  -105. 

Complete  the  square, 

324x2  +  ()  +  (62)2  =  64. 

Extract  the  root, 

18x  +  62  =  ±8, 

18*  =  -54  or  -70. 

...  x  =  -3or-3f 

Simplify, 


Multiply  by  17, 
Complete  the  square, 
Extract  the  root, 


TEACHERS     EDITION. 


263 


20. 

5)-I(x"-l)-2(*-2>>. 

18a;2  -  6s  -  30  -  5s2  +  5  =  30s2  -  120s  +  120, 
17s2 -114s  =  -145. 
289s2  -  1938s  =  -  2465.  (  (.  7 

289  s2 -0  +  3249  =  784. 
17s -57  =  ±28. 
,\  s  =  5  or  l|f. 


21. 


-50 


Simplify, 


2x      3s- 

15     '3(10  +  s)  =        ...    ^ 
760s  +  71s2  +  5^0s  -  9500  -  Sfs^TBlOs  +  2100, 
40s2*  760s  =  11600.     \ 
Divide  by  10,  4  s2  +  76  s  =  1160. 

Complete  the  square,  4  s2  +  ( )  +  361  =  1521. 

Extract  the  root,  2  s  +  19  =  ±  39. 

.\s=10or-29, 


(7) 


Simplify, 

Multiply  by  7, 
Complete  the  square, 
Extract  the  root, 


_15-7s 

-1     8(l-s)' 

7s -15 


(s  +  l)(s-l)     8(s-l) 

8s  =  7s2  -8s  -15, 
7s2 -16s  =  15. 
49s2 -112s  =105. 
49^-0  +64  =  169. 
7s- 8  =±13. 
.*.  s  =  3  or  —  ^. 

23. 

2s-l      1  =  2s -3 
s-1       6       s-2* 

Simplify,  12s2  -30s  +  12  +  s2-3s  +  2  =  12s2-  30s  +  18. 

Transpose  and  combine,  s2  —  3  s  =  4. 

Complete  the  square,  4^  —  0  +  9  =  25. 

Extract  the  root,  2  s  —  3  =  ±  5. 

.*.  s  =  4or  —  1. 


264  ALGEBRA. 


24. 

x  +  2  _  4-x  ^  7 
x-l        2x   **  3* 

Simplify,                 6x*  + 
Transpose  and  combine, 
Multiply  by  5, 

12a?  +  3x»  -  15*  +  12  -  Hi1  -  14x. 
5*»-llx=12. 
25V -55*  =  60. 

Complete  the  square, 

25^-()  +  l|i  =  ?61. 
4         4 

Extract  the  root, 

K        11      ^19 
2          2 

5x  =  15  or  —4. 

.-.  x  =  3or-£ 
25. 

14x-9     *»-3 

X  —     ■  '■'          "»  -          • 

8x-3       x  +  1 

Simplify,  8x*  +  5x*  -  3x  -  Ux3 - 5a;  +  9  =  8*«- 24x- 3s2  +  9. 

Transpose  and  combine,  —  6x*  +  16x  =  0. 

Divide  by  -  2,  3*1  -  8x  =  0, 

x(3*-8)  =  0. 

.\  x  =  0or  2}. 
26. 

1      J?  -1-5  =  x  —  6 
2x  +  l~x-2* 
Simplify,  2x*  -  3x  -  2  -  x»  -  3*  +  10  =  2**  -  11*  - 6. 


Transpose  and  combine, 

*»-5x  =  14. 

Complete  the  square, 

*-o+f-fi. 

Extract  the  root, 

*          5          4-9 

x =  ±  — 

2        2 

.-.  x  =  7or~2. 

27. 

7  —  x        x 

Simplify,  10**  +  490  -  140*  +  lOx2  =  203*  -  29c2. 

Transpose  and  combine,  49  x*  —  343  x  =  —  490. 

Divide  by  49,  x*  -  7x  =  -  10. 

49     9 
Complete  the  square,  ^-(1  +  7*7 

4      4 

7        3 
Extract  the  root,  *  —  -  =  ±  — 

2        2 
.  •.  x  =  5  or  2. 


teachers'  edition.  265 

28. 

2a  +  3  7-a;        7-3a; 


2(2a?-l)      2(as  +  l)      4-3a; 
Simplify,  -14a;2-12ar»+22a;+24-12a?+106a;2-162a;+56 

-  44a;2-10a;-24a?-28. 
Transpose  and  combine,  48  x2  —  180  a;  =  —  108. 

Divide  by  12,  4  x*  - 15  x  =  -  9. 

Multiply  by  16  and  complete  the  square, 

64^-0  +  225  =  81. 
Extract  th«  root,  8a;  — 15=<±9, 

8a;  =24  or  6. 
.  \  x  =  3  or  #. 
29. 

12a?-lla;2  +  10a;-78     -.     _1 
8a?-7a;  +  6  ""   *  X      2* 

Simplify,  24ar»-22a;2+20a;-156  =  24a?-21a?+18a;-8a;2+  7a;-& 

Transpose  and  combine,  7  a?  —  5  a?  =■  150. 

Multiply  by  28,  196a?  -  140a;  -  4200. 

Complete  the  square,  196  a?  -  (  )  +  25  -  4225. 

Extract  the  root,  14  x  —  5  =  ±  65. 

.-.  a?  =  5or-4f 
30. 

6  18    =     7  8 

a;  — 1      x  +  5     x  +  1     a?  — 5 

6  7  18  8 


Combine, 


or  '        -.  n  c  e 

a;  — 1      a?  +  l      a;  +5      a;  — 5 

13 -a      10a; -130 


a?-l  a?-25 

13 -x  _  10(13 -a;) 

OT  a?-l  a?- 25 
.-.  a;  =  13.% 
Hence,  if  13  —  a?  «=  0,  the  equation  is  satisfied. 

Otherwise  we  may  divide  by  13  — x. 

1  -10 


a?-!      a?-25 
Simplify,  a?  -  25  =  10  -  10a;2, 

11a?  =  35, 


*     35 


155 


266 


ALGEBRA. 


Exercise  LXXXV. 

1.  a*  +  2ax  =  a*.  ^    ^   bnx    3n2^Q 

Complete  the  square,  2         2 

x*  +  2ox  +  a*  =  2a2.  2a?- 5  tut- 3  w2  =  0, 

Extract  the  root,  4  a*  —  10  nx  =  6  w2, 

x  +  a  =  ±a\/2.  ,.     ,,,2571*     49n2 


.  a;  =  —  a±a 


V2.  «-()  +  ^£-ffi! 


a?  =  4aa;  +  7a2.  2.-5? -±1?, 


2  2 

2a;  =  6n  or  -n. 


Transpose, 

a^-4aa?  =  7a2. 

Complete  the  square,  . '.  x  =  3  n  or 

3*-4aa;  +  4a2  =  lla2. 

Extract  the  root, 

a;-2a  =  ±aVTl.  5.    —*— *L_. 

.-.*-2a±aVlL  (*  +  a£     £-«>" 

a2(a;-a)2  =  62(a;  +  a)2, 

3.                         a»-^5?-3ii«.  *(*-a)-±b{x  +  a), 

4  ax  —  a2  =»  6a:  +  a5, 

4a?2  =  7m2  -  I2mx,  x(a  -  b)  =  a2  +  a&. 

4x2  +  12mx=7m2,  a(a  +  6) 

4a?  +  ()  +  (3m)2=16m2,  ■'•*-    a_h    > 

2  a;  +  3  m  -  ±  4  m,  0r  oa;  -  a2  =  -  bx  -  ab, 

2a;  ~±  4m -3m.  a?(a  +  &)  =  a2-o&. 

2       2  ••*-- jrnr 

6*        ca;  =  az2  +  6*2--^-. 
a  +  6 

aca:  +  Jca;  =  a2x2  +  2a&c2  +  i2*2- 
a2a;2  ^  2a6a;2  +  62a;2  -  aca;  -  Jca;  =  ac, 
x*(a?  +  2a&  +  62)_a;(ac  +  6c)  -  ac, 

a^(a2+2a6  +  62)~() +  £2=Bi«l±4 
4  4 

*(a  +  &)-g-±V4ac  +  * 
2  2 

2 

2(a  +  b) 


teachers'  edition.  267 

7  a2a^  +  —  =  2ax  8  (a2  +  1)  .t  =  aa?  +  a. 

b2       c2        c  a2a;  +  a;  =  ax2  +  a, 

a«Asi4.ft*-9flW«B  as2-(a2+  l)a;  =  -a, 

dW-2aW«--^  4aV-()  +  (a2+l)2  =  a*-2a2+l, 
aW  -  H  +  *  =  0  2aa;-(a2  + 1)  =  ±  (a2  - 1), 

aca;  =  62,  .\  a?  =  a  or  - 

52  a 

.•.  »  =  — 

9. 

a  b     =    2c 

x  —  a     x—b     x—c 
a(x -  b)(x  -  c)'+  5  fa  —  a)fa  -  c)  -  2c(a;  —  a)(x-  b), 
ax*-  abx-acx  +  abc  +  bx*  -  abx  -  bcx  +  abc  =  2  cx2-2  aca-2  &ca:+  2  ata, 
rf(a  +  &-2c)  +  «(ae  +  &c-2a&)-0.       2ab-ac-bc 

,\  x=*Q  or • 

a  +  b  —  2c 

10. 

1        -M  +  -1 


a  +  6  +  a;     a     6     a; 
afoc  =  afoc  +  62a:  +  6a?2  +  a?x  +  abx  +  ax2  4-  o26  +  o52  +  afa, 
a;2(a  +  b)  +  x(a  4-'  6)2  =  -  a6(a  +  b). 
Divide  by  (a  +  b),       x*  +  (a  +  b)x  =  —  ab, 

4  a;2  +  (  )  +  (a  +  bf  =  a2  -  2  ai  +  &2, 
2a;  +  (a  +  6)  =  ±  (a  —  6), 

2x  =  -2&or-2a. 
.  •.  x  =  —  &  or  —  a. 

11. 

_J. 1  3  +  ar2 

a— a:      a  +  a;     a2  — ar2 

a  +  a;— a  +  a;  =  3  +  0?2, 
a:2-  2  a:  =-3, 

a?-2*  +  l«-2f 

.-.  .r=l±V^2. 


268 


ALGEBRA. 


13 


(2»-«r  aft, 


x-a  +  2b 

4x*  -  4ox  +  a*  =  26x-a&  +  26', 

4x»-  2x  (2a  +  6)  =  -a*-a6  +  26», 

16*»-()+(2a+&)2  =  9&*, 

4x-(2a  +  6)  =  ±36, 

4a?  =  2a  -26, 

or  2a +  46. 

.  ~     a-6_a  +  26 

.*.  x  = or ■ 

2  2 

14.  x*  +  ax  =  a  +  x. 

a?  +  ox  —  x  —  a, 
x*  +  (a-l)x  =  a, 
4x2+()+(a-l)2  =  a2+2a  +  l, 
2x  +  (a  -  1)  =  ±  (a  +  1), 

2x  =  -(a-l)±(a+l). 
.  \  x  =  1  or  —  a. 


15.  x2  +  ax  =  bx  +  a6. 

x*  +  (a  —  6)  x  =  ab, 
4x2  +  ()  +  (a-6)2  =  (a  +  6)2> 
2x  +  (a  -  b)  =  ±  (a  +  6), 
2x  =  -2aor26. 
.•.  x=«  —  a  or  6. 


16. 


x     a_x     6 
a     x     b     x 


x*b  +  a26  =  ax*  +  aft1, 
att-a^a-a^-a2*, 
x»(6-a)  =  a6(6-a). 
Divide  by  (b  —  a), 

x*  =  a&.  _ 
.*.  x  =  ±  Va6. 


17. 


1111 

-  + r  -  -  + r 

x     x+b     a     a+ 6 

a2x  +  a26  +  a6x  +  ati*  +  a2x  +  a&x  =  ax1  +  a&x  +  bx2  +  6*x  +  ax*  +  ate, 

20X3  +  Jx2-  2a*x  +  6»x  =  a*b  +  a&2, 

(2a  +  tyx2  - (2a*  -  6»)  x  =  a26  +  aft2, 

4x*(2a+  6)2-  ( )  +  (2a*  -  &*)'  -  4a*  +  8as6  +  8a262  +  4a6»  +  b\ 

2x(2a  +  b)  -  (2a'  -  62)  -  ±  (2a8  +  2a6  +  b*), 

2x(2a  +  6)  =  (4  a2  +  2a6)  or  -(2a6  +  262). 

2a  +  6 


18. 


a     5x_  x2  ^Q 


3       4       3a 

4a2  +  15ax-4x2  =  0, 

4x2  —  15ax  =  4a2, 
64x2-()  + 225  a2  =  289a2, 
8x  —  15a  =  ±  17a, 

8x  =  32aor-2a. 

.*.  x  —  4a  or  —  -• 


19 


=  a  + 


=  a, 


x  +  3 


x  +  3 

x-3 
x+3  x-3 
x-3     x+3 

x2  +  6x  +  9-x2 

+  6x-9  =  ax2-9a, 
ax»-12x  =  9a, 
aV-12ax  =  9aJ, 
a2x2-()  +  36  =  9a2  + 36, 

ax-6 


ax^ 


,  x«=  - 


±V9(a'+4), 
GiSVtf  +  l 
6±3V?+4 


TEACHERS     EDITION. 


269 


20. 

mx2  —  1  ■■ 


x(m?  —  n*) 


m2nx2  —  mn  =  x(m?  —  n2), 
m2nx2  —  (m8  —  n2)x  =  mn, 
4m*n2a?  —  ( )  +  (ra8  —  n2)2  =  m6  +  2m*n2  +  n4, 
2  m2  nx  —  (ra8  —  ?i2)  =  ±  (m8  +  n2), 

2m2nx  =  2m8  or  —  2n2. 

771  71 

.\  x  =  —  or »• 

n  m? 


21. 

(oa;  —  b)  (bx  —  a)  =  c2, 
a&r2  —  b2x  —  a?x  +  ab  =  <?, 

abx2  —  (a2  +  b2)x  =  c2  —  a6, 
4a262ar»  -  4a&  (a2  +  &2)z  =  4a&c2  -  4a2  62, 
4a2 b*x*  -  ( )  +  (a2  +  ft2)2  =  (a2  -  i2)2  +  4aic2, 

2ata-(a2+&2)  =  ±  Vfa2 


62)2  +  4a6c2. 

a2  +  62±V(a2-62)2  +  4a6c2 
2a6 


23. 


ra  +  a; 

m  —  a; 

m2 

-mx  +  rri 

*-J-77ia;  = 

cm?  — 

ca;2, 

ca;2  = 
xVc  = 

cm?  — 

■2m2, 
'c-2. 

.*.  a?  — 

±m-i 

/? 

22  oa;  +  6  _^  ttkp  +  n 

bx  +  a     nx  +  m 
aTia?  4-  6na;  +  aTna;  -f  bm 

=  fois2  +  6na;  +  amx  +  an, 
aTix2  —  bmx2  -=an  —  bm, 
x*(an  —  bm)  =  an  —  bm, 
z2=l. 
.\  x  =  ±  1. 


24. 

(a-l)2x2  +  2(3a-l)a;     -. 

4a-l 
(a-l)2a?-f2(3a-l)a;  =  4a-ll 
4 (a  -  l)4 x*  +  ( )  +  (6  a  -  2)2  =  16  a3, 

2(a  -  l)2x  +  (6a  -  2)  =  ±  4aVa\ 

2(a-l)2a:  =  2-6a±4aVS. 
.   „,      1  —  3  a  ±  2  a  Va 


(a -If 


270  ALGEBRA. 


26. 

(aJ -&»)(*» +  1) 

■  2x. 

a2+62 

flW-P^  +  tf-P- 

■  2a2x  + 

262x, 

a2  x2 

-62x2-2a2x-262x  = 

=  62-a2, 

*»( 

a2-62)-2x(a2  +  62)  = 
2-62)*_2x(a*-6*)  = 
-PP-O  +  ^  +  Pr- 

:62-a2, 

Xs  (a 

*-a*  +  2 

a*P-ft*, 

^(a2- 

4^^, 

x(a2  -  6s)  = 

.±2a6, 

=  a2  +  62±2a6 

-(a±6y. 

a  +  &,™ 

a  — 6 

.%  x  = 

1 01 

a-b 

• 

a  +  6 

26. 

x2  — 4mnx 

■  (m  —  nV 

m 

(m  +  n)2 
x1  —  4  mnx  =  m*  —  2m2  n2  +  n*, 
«•  —  ()  +  4m,n1  =  m4  +  2m,nl  +  n*, 
aj-2mn  =  ±  (m2  +  n2), 

x  =  2mn±(m24-n2) 
=  (m  +  n)2  or  —  (m — n)2. 

27. 

.     a-6r_14a2-5a6-1062     (2a-3b)* 
a62  18a262  2a6 

18a262x2-(18a26-27a62)x  =  -4a2+13a6-1062, 
144a2  62x2  -  (144a2  6  -  216a62)x  =  -  32a2  +  104 a*  -  80 J2, 
144a262x2  -  ( )  +  (6a  -  96J2  -  4a2  -  4aA  +  J2, 
12a6x  -  (6a  -  96)  =  ±  (2a  -  6), 

12a6x  =  8a  - 106  or  4a  -  86. 
4a-56      a-26 
""'*"     6a6  3a6  " 

28. 

abx*  +  h*x  =<  6a*  +  ^  ~  2ft>     3o*x 
c  c2  c 

06c2 x2  +  62cx  =»  6a2  +  a6  -  262  -  3a2 ex, 
a6c2x2  +  (3a2c  +  62c)x  -  6a2  +  a6  -  262, 
4a262c2x2  +  4a6cx  (3  a2  +  62)  -  24  a8  6  +  4a2 62  -  806s, 
4a262c2x2  +  ( )  +  (3a2  4-  62)2  =  9a*  +  24a86+10a262-8a62+64, 
2a6cx  +  3a2  +  62  -  ±  (3a2  +  4a6  -  6s). 

2a-6  3a  +  26 

.-.  x  = or = 

ac  be 


teachers'  edition.  271 

29. 

x2  m2  —  4  a2  _  x 

3m-2a     4a-6m      2* 

2a;2  4a2-m2 

=  xt 

3m  — 2a      3m— 2a 

2s2  —  4a2  +  m2  =  3ma:  — 2aar, 
2s2  +  (2a  -  3m) a;  =  4a2  -  m2, 
16a?  +  ( )  +  (2a-  3m)2  =  (36a2  -  12am  +  m2), 
ix  +  (2a  -  3m)  =  ±  (6a  -  m), 

4a;  =  —  8a  +  4m  or  4a  +  2?/t. 

.*.  x  =  m  —  2a  or  a  +  — • 
2 

30. 

6«  +  feL+#_5(«-6)  +  ?*2* 
£  6z 

36a?  +  6(a  +  6)2  =  30a?(a-  6)  +  25a6, 
36a;2  -  30a;  (a  -  6)  =  25a6  -  6  (a  +  6)2, 

4  4 

6*-|(a-6)  =  ±^±*, 

6g==6a-46      4a-^66. 

2  2 

.   m     3a-26   rt„    2a-36 

.*.  x= or    • 

6  6 

31. 

i  (x2  +  a2  +  ab)  =  |a;(20a  4-  46). 

8a?  +  8a2  +  8a6  =  20aa;  +  46a;, 

fa?  -  (20a  +  46)a;  =  -  8a2  -  8a6, 

16a?  -  2(20a  +  46)a;  =  -  16a2  -  16a6, 

16a?-()  +  (5a+6)2  =  9a2-6a6  +  62, 

4z-(5a+6)  =  ±(3a-6), 

4a;  =  (5a+6)±(3a-6). 

...  a.=s2aor^±-6. 
2 

32. 

a?  —  (6  —  a)  c  =  ax  —  bx  +  ex. 
x2  +  bx—ax  —  cx  =  (b  —  a)  c, 
x2  +  (b  —  a  —  c)x  =  (6  —  a)c, 
4a?  +  (  )  +  (b  -  a  -  c)2  =  a2+  62  +  <?-2a&-2ac  +  26c, 
2a;  +  (6  —  a  —  c)  =  ±  (a  —  6  —  c), 
2a;  =  2a -26  or  2c. 
.  \  x  =  a  —  6  or  c. 


272  ALGEBRA. 


33. 

x2  —  2  mx  =  (n  —  p  +  m)  (n  — p  —  m) . 
a2  —  2mx  =  n2— 2?ip  +p?—m*t 
s2  -  (  )  +  m2  =  n2  —  2np  +|)2, 
a?  — m  =  ±(n  — />). 
.•.  x  =  m±(n  —  p). 

34. 

z1      (m  +  n)x  =  J(l>  +  S'  +  m  +  n)(P+5r-m-"n)- 
4  a2  —  4(m  +  n)x  =  (p  +  a+m  +  n)(p  +  a— m— n), 
4x2-(  )  +  (m  +  n)2  = jo2  +  2pq  +  y2, 
2a?-(m+n)  =  ±(p  +  a), 

2x  =  m  +  n  ±  (j?  +  j). 
.  wi  +  n±(p  +  y) 

2 

35. 

mnx*—  (m+n)(mn+l)x+  (m+rif  =  0. 

mnx2 —{m*n  -f  mn2  +  m  +  n)x  =  —  ?ra2  —  2mn  —  n2, 
4  mWx*—  4  win  (m2n  -|-  mn1+  m  +  n)  x  =  —  4  rasn  —  8  m2n2  —  4  mns, 
4mW-( )  4-  (wi2n  +  mn2  +  m-f  n)2  =  m¥+2mV-  2m3n— 4  mV 

+ra2n4— 2  mns+m,+2mn+n8, 
2mnx  —  (nfn  +  ??in2  +  ra  +  n)  =  ±  (m2n  +  ran2  —  w  —  n), 

2mna;  =  2m2n  +  2mn2  or  2m  +  2n. 

.-.  x  =  m+n  or ^±7i. 
mn 

36. 

26-a:-2a     46-7a  =  a?-4g 
6.r  oa:  —  6a:     o6  —  62 

4a6-oa;-2a2-262  +  6a;  +  462-7a6  =  32-4aa;, 

a*-3aa>-6a;  =  262-3a6-2a2, 
x2  -  (3a  +  b)x  =  2 62  -  3  a6  -  2a2, 
4z2  -  4(3a  +  b)x  -  862  -  12a6  -  8a2, 
4s2  -  ( )  +  (3a  +  bf  =  a2  -  6a6  +  962, 
2a:-(3a  +  6)-±(a-36), 

2a; -4a -26  or  2a +  46. 
.\  »—  2a  -  6  or  a  +  26. 


teachers'  edition.  273 

37. 

2x*(a*  -  62)  -  (3a2  +  62)(a?  -  1)  =  (362  +  a2)(.r  +  1). 

2a*(a*-  62)-  3a2x  -  to;  +  3a2  +  62  =-  362x  +  a2z  +  362  +  a', 

2z2(a2 -  62)  -  4a2z  -  462a;  -  262  -  2a2. 

Divide  by  2,  z2  (a2  -  62)  -  2  (a2  +  62)  a:  =  62  -  a2, 

x2 (a2  -  62)2  -  2(a*  -  A*) a  -  (62  -  a*)(a*  -  J2), 

ar^a2  -  ft2)2  -  ( )  +  (a2  +  62)2  =  4a262, 

s(a2-62)-(a2  +  62)  =  ±2a6, 

a;(a2  -  62)  -  (a  +  ft)2  or  (a  -  6)2. 

a  +  6      a  —  b 
.«.  a;  =  ^-£-f  or- — £. 

a— o      a  +  6 


38. 

a-26-a;       56-a;       2a-a;-196     Q 
a2  — 462       as  +  26a:         2bx  —  ax 


a-2b-x 56-a?       2a-a?-196_  Q 

(a-26)(a  +  26)     (a  +  26)x        (a-26)« 
ax  -  2bx-x*-  bob  +  ax  +  1062  -  26a?  -  2a2 

+  as+  15a6  +  26a?  +  3862  =  0, 
3aa?  -  2bx  -  a?2  +  lOab  +  48  62  -  2a2  _  0. 

«•  -  (3a  -  26)a?  -  -  2a2  +  10a6  +  48 62, 
4a?2  -  (  )  +  (3a  -  26)2  =  a2  +  28a6  +  19662, 
2a?-(3a-26)  =  ±(a+146), 

2x  =  4a  +  126  or  2a  -  166. 
.%  a?  =  2a +  66  or  a  —  86. 


s  +  13a  +  36     1  =  a  — 26 
5a  —  36  —  x  x  +  26 

s?  +  13az  +  56a;  +  26a6  +  662 

—  box  +  5bx  +  x2  +  662  -  10a6  =  5a2  -  13a6 -  ax  +  66*  +  26a?, 
2a2  +  (9a  +  86)a?  =  5a2  -  29  a6  -  66*. 
16  t2  +  8(9a  +  8  b)x  =  40a2  -  232a6  -  48  62, 
16s2  +( )  +  (9a  +  86)2  =  121a2  -  88a6  +  1662, 
4a?  +  (9a  +  86)  =  ±(lla-46), 

4a;  =  2a  -  126  or  -  (20a  +  46). 

...  iC-a.-36or-(5a  +  6). 
A 


274  ALGEBRA.. 


40. 

x  +  3b  36  a+36 


=  0. 


8a2-12a6"96s-4a2     (2a  +  3b)(x-3b)  = 

s  +  36 3j> a  +  36  Q 

4a(2a-36)     (2a-3&)(2a  +  36)     (2a+36)(x-36)       ' 
(a*-9&*)(2a  +  3&)+12ata-36a&*-4a(2aa  +  3a&-962)  =  0. 
(2a  +  3  ty's2  +  I2abx  =  8as  +  12a26  +  18a^  +  276s, 
4(2a  +  3J)2 x*  +  (  )  +  (12a&)2  -  64a*  +  192as5  +  432a262 

+  432a&3  +  3246*, 
2(2a  +  36)*  +  12a&  =  ±  (8a2  +  12a6  +  18&2), 

(2a  +  Sb)x  =  4a*  +  96*  or  -(4a2  +  12aft+9&2). 


2a+36 


41. 

nx*  +px  —px*  —  mx+  m  —  n  =  0. 

tub2  — jbc2  +  px  —  mx  =  n  —  w, 
^(n  — j>)  +  &  (j>  -  m)  =  n  -  m, 
4z*(n  -  j»y  +  (  )  +  (p  -  m)2  =  4na-4mn-4pn+4pm+p«-2pm-Hn,1 
2s(n  -j>)  +  (p  -  m)  =  ±  (2n  -p  -  m), 

2s(n  -jj)  =  m -p  +  2w  -p  -  m, 

or  m-j3  —  2n+^>  +  TO« 
...«-l  or^LZl^. 


42. 

(a  +  6  +  c)x*  -  (2a  +  6  +  c)x  +  a  -  0. 

(a  +  b  +  c)x*  -(2a  +  b  +  c)x  =  -a, 
4a?(a+&  +  c)2-()  +  (2a  +  &  +  c)2  =  &2  +  2&c  +  c2, 
2x(a  +  6  +  c)  -  (2a  +  b  +  c)  =  ±  (6  +  c), 

2z(a  +  6  +  c)  -  (2a  +  6  +  c)  ±  (6  +  c), 
2s  (a  +  6  +  c) «  2a  +  2 6  +  2c  or  2a. 

.-.  x=l  or 2 

a  +  b  +  c 


teachers'  edition. 


275 


43. 

(ax  —  b)(c  —  d)  =  (a  —  b)(cx  —  d)  x. 
acx  —  be  —  adx  +  bd=>  acx2  —  adx  —  bcx*  +  bdx, 
bcx*  —  acx*  +  acx  —  bdx  —  be  —  bd, 
(be  —  ac)x*  +  (ac  —  bd)x  =  be  —  bdt 
4  (6c  -  acfx*  +  ( )  +  (ac  -  bd )2  =  4  b V  -  4 abc*  -  4  62crf 

+  2a6co*  +  a2c2  +  62d2, 
2(6c  —  ac)a?  +  (ac  —  bd)^±  (26c  —  ac  -  W), 

2(6c  —  ac)x  =  —  (ac  —  bd)  ±(2bc  —  ac  —  bd), 

—  ac  +  bd  +  2bc  —  ac  —  bd 

2(bc-ac) 

—  ac+bd  —  2bc  +  ac  +  bd 


2(bc-ac) 


c(a  —  b) 


44. 


2x 


6  x\b     a) 


3s+l 


2x  +  \     a- 


26_3s+l 
a 


b  abx 

2ax*  +  ax  —  a  +  26  =  36s2  +  bx, 
2axi  —  3bxi  +  ax  — 6a;  =  a  — 26, 
x*(2a-3b)  +  x(a-b)=a-2bt 
4a*(2a-36)2  +  4a;(a-6)(2a+36)  =  (4a-86)(2a--36)l 
ix*(2a  -  36)2  +  ( )  +  (a  -  6)2  -  9a2  -  30a6  4-  25ft2, 
2a;(2a-36)+(a-6)  =  ±(3a-56), 

2a; (2a  -  36)  «  -  a  +  6  ±  (3a  -  56), 
2a;(2a-36)  =  2a-46or-4a  +  66. 
a-26 


2a-36 


or-1. 


276  ALGEBRA. 

45. 

1  1  =       a       _  26x  +  6 

2x2  +  x-l      2x2-3x+l      2bx-b      ax*-a 

1  1 a 2  6s  +  6 

(2x-l)(x  +  l)     (2x-l)(x-l)  =  6(2x-l)     a(x-l)(x  +  1)' 

L.C.D.  =  a6(x-l)(x  +  l)(2x-l). 

Simplify,      a6x  —  ab  +  a6x  +  a6  =  a2x2  —  a*  —  4  6V  +  J2, 

2a6x  =  oV  -  a2  -  46V  +  62, 

46V  -  aV  +  2a6x  =  62  -  a2, 

x2(462  -  a2)  +  x(2a6)  =  62  -  a2, 

4x2(462-  a2)2  +  4x(2a6)(462  -  a2)  =  166*-  2062a2  +  4a*, 

4x2(462-a2)2  +  ( )  +  (2a6)2  -  166*  -  1662a2  +  4a*t 

2x(462-  a2)  +  2a6  -  ±  (46*  -  2a2), 

2x(462-a2)  =  462-2a6-2a2 

or2a2-2a6-462, 

262-a6-a2      a2-a6-262 

x  *= or • 

462-a2  462-a2 

.   «      h  —  a  ^      6  +  a 

.*.  x  = or • 

26-a  26  +  a 


Exercise  LXXXVI. 

1.    (x  +  l)(x-2)(x2  +  a;-2)  =  0. 
(x  +  l)(x  -  2)(x  -  l)(x  +  2)  -  0. 
.-.  x  =  -1,2,  1,-2. 

2.  (x2-3x+2)(x2-x-12H0.      4  2x»  +  4x2-70x-0 
(x-2)(x-l)(x-4)(x+3)  =  0.      *'          J'  +  Jf^J 

"X  =    '    '    '        '  2x(x+7)(x-5)  =  0; 

3.  (a:  +  1)  (x  -  2)  (x  +  3)  =  -  6.         which  is  satisfied  if     a?  =  0, 

x8  +  2x2  -  5 X  -  6  =  -  6,  3  +  7  =  o 

*(a»  +  2»-5)-0;  or  lf                      *-5-C. 

which  is  satisfied  if      x  =  0,  .\  x  —  0,  —  7,  5. 
or  if              x2  +  2x-5  =  0. 

By  solving      x2+2x-5  =  0,  5.    (x2-x-6)(x2-x-20)  =  0. 

x  -  - 1  ±  V6.  _  (x-3)  (x+2)  (x-5)  (x+4)  =  0. 

.-.  x  =  0,  -1  ±V6.  .-.  x=3,  -2,  5,  -4. 


teachers'  edition.  277 

6. 

x(x  +  l)(s  +  2)  =  a(a  +  l)(a  +  2). 

ar* +  33*+  2x  =  a*  +  3a?  +  2a, 

x*  +  3a;*  +  2x  -  a8  -  3a2  -  2a  =  0, 

(a8  -  a8)  +  (3s2  -  3a2)  +  (2*  -  2a)  =  0, 

(xi  +  ax+a*)(x-a)  +  (3x+3a)(x-a)  +  2(x-a)  =  0, 

(x*  +  ax  +  a2  +  3a?  +  3a  +  2)  (a;  -  a)  =  0. 

•  \  a;  — a  =  0, 

and  a;  =  a. 

Or,  a?2  +  as  +  a2  +  3a;  +  3a  +  2  =  0. 

x2  +  aa?  +  3x  =  -  a2  -  3a  -  2, 
a;2  +  a;(a  +  3)  =  —  a2  —  3a  —  2, 
4s2  +  ( )  +  (a  +  3)2  =  1  -6a -3a2, 

2x  +  (a  +  3)  =  ±  VI- 6a -3a2. 

.-.  s=--^±£vl-6a-3a2. 


7.  ars-a^-a:  +  l  =  0.  10.  a*-l=0 

fr+ig;=l!i::ii:S         f  ^J^-}!-* 

.-.3  =  1   1  -1  (*  +  l)(*»-a  +  l) 

(a?-l)(a^  +  a;  +  l)  =  0, 

8.  8ar»-l  =  0.  and  a;  =  -1,1. 
(2a;-l)  (4a*+2a;+l)  =  0.                       From     x*  -  x  +  1  -  0, 
From  the  first  factor,                                             ar2  -  a:  =  -  1, 

s  =  J,  4s2-()  +  l«=-3, 

or        4z2  +  2a:  =  -l.  2a?-l=±VT3; 

16a^  +  ()  +  l 3,  1±VZ3 

4^+1  =  ^x^3.  and                  *  =  ~ J"2' 

•••  *  =  J(-l±  V^3).  From    ar2  +  a:  +  1  =  0, 

&*  +  a;-=-l, 

9.  8s8 +  1=0.  4a^  +  ()  +  l  =  -3, 
(2*+l)(4a:2-2.T+l)  =  0.  2x  +  1  =  ±  V^ 
From  the  first  factor,  _1  ±  y^3 


a:  =  -  J,  and 

or        4ar2-2a?  =  -l! 

4a*-2a:  =  -l,  .-.*-!, 


l±V-3 


lGx*-()  +  l  =  -3t  '         '        2 

1  =  ±V^3. 
ar=J(l±VZ3), 


4x-l  =  ±VT3.  and  rl*V^ 


278  ALGEBRA. 


11.  a:(x-a)(3*-&s)  =  0. 
x(x-a){x  +  b)(x-b)  =  0. 

.-.  x  =  0,  a,  ±b. 

12.  n(x*  +  1)  +  (x  +  1)  =  0. 
(x  +  l)(nx*  -  nz  +  n  +  1)  =  0. 
(n  +  l)(x  +  l)(x*  -x  +1)  =  0. 
If  a;  + 1  =  0, 

s  =  -l; 

or  if  nz*  —  na:  +  n  +  1  =  0.     ,       

By  solving,  a;  =  J  ±  —  V-3n*-4n. 


"2w 


Exercise  LXXXVII. 


1.   (s-2)(s-l)  =  0.  c   /       2\/    +3\_0 

a*-3a;  +  2  =  0.  V       3y  V       2> 

^  +  ^-1=0, 


2.  (*-7)(s  +  3)  =  0.  Ma« 

^-4s-21-0. 


6a*  +  5:r-6  =  0. 


3.  (._IW._J)_a  6  (*  +  5)KH 

V       2^       3/  or    ^  +  11^  +  5  =  0 

(2*-l)(3*-l)-4  °r    ^+    2   +2      ' 

6js*-5;r  +  l  =  0.  or2s^  +  ll*  +  5  =  0. 

I  (.  +  2)(-|)-o. 

(9a+7)(7a;-9)  =  0t 
63  a*-  32a;  -63  =  0. 

t.(-3)(.  +  8)(.-f)(.  +  f)-a 

,     153a*     81  _0 

*~ i6"  +  16-0' 

16s* -153z»  + 81=0. 

8.  (s-0)(:r-l)(a;-2)(a;-3)  =  0. 
a*-6a*  +  lla;2-6a;  =  0. 

9.  (-o)(.  +  |)(.-|)(.  +  i)-a 

4a*-7&*-3a>  =  0. 


TEACHERS     EDITION. 


279 


10.  {x-(a-2b)}{x-(3a  +  2b)}=0. 


(x-a  +  2b)(x-3a- 
x2  —  4a#  +  3a2  —  4a&  —  4&2 


=  0. 


11.  {a -  (2a  - b)}{x  -  {b  -  3a)}  =  0. 
(x  -  2  a  +  6)  (*  -  6  +  3  a)  =  0, 
a2  +  as  -6a2  +  5a6  -  b2  =  0. 

12.  {s-(a2+a)}{a;-(l-a)  =  0.      17.  32  +  4s+l  =  0. 


(a  —  a2  —  a)(x—  1  +  a)  =  0, 
a2  —  a2x  —  x  —  a8  +  a  =  0. 


13.  rf-7*  +  12-0. 

In  this  equation  p  is  7  and  q 

is  12.    

.-.  Vp2-4?=V49-48 

-VI. 
.••  roots  are  rational,  and  both 
positive. 

14.  a?*-7a:-30  =  0. 

In  this  equation  p  is  —  7  and 
q  is  -  30. 

v.  Vp2-4y=V49  +  120 

=  13. 
.*.  roots  are  rational,  and  of 
opposite  signs. 

15.  s2+4a-5  =  0. 

In  this  equation  p  is  4  and  q 
is  —  5. 

.-.  V^2~4y=V16  +  20  =  6.. 

.*.  roots  are  rational,  and  of 
opposite  signs. 

16.  5s2 +  8  =  0. 

In  this  equation  p  is  0  and  q 

is  8.  

.-.  Vp2-4y=V0^32 

.*.  roots  are  imaginary. 


In  this  equation  p  is  4  and 
q  is  1. 

.-.  Vp2-4y=\/l6^4 

=  Vl2. 
.  \  roots  are  surds,  and  neg- 
ative. 

18.  x2~  2x  +  9  =  0. 

In  this  equation  p  is  —  2 
and  j  is  9. 

...Vp2-4y=V4^36 

=  v^32. 
.\  roots  are  imaginary. 

19.  3x*-4*-4-0. 

*  a;2-4a;--  =  0. 
*        3 

Herep--!     ,-- * 

>9      3 
.•.  roots  are  rational,  and  of 
opposite  signs. 

20.  s*  +  4a;  +  4  =  0. 

In  this  equation  p  is  4  and 
q  is  4.  

.-.  Vp2-4y=Vl6-16 
=V0  =  0. 

.•.  roots  are  rational,  equal 
in  value,  and  both  negative. 


280  ALGEBRA. 

Exercise  LXXXVIII. 

1. 

Let  4+6x  —  x*  =  wi. 

x2—  6x  — 4  =  — m, 

Xs  —  6x  =  4  —  in, 
x*-()  +  9  =  13-m, 


x-3  =  ±Vl3-ro. 


,  x  =  3±Vl3-m. 


Since  V13  — m  cannot  be  negative,  w  cannot  be  greater  than  13 ; 
that  is,  the  maximum  value  is  13. 


Let  (x  +  a? 


m. 
x 


Then  x*  +  2ax  +  a*  =  mx, 

and  4ac*  +  4x(2a  —  m)  =  —  4a*. 

4a*  +  ( )  +  (2a  - mf  =  m»  - 4am, 

2x  +  (2a  — m)  =  ±Vm(m— 4  a), 


2a: =  —  (2a— m)±Vwi(m  —  4a). 
a;  =  _  2«Lzi»  ±  i  y/m{m _  4a)t 


x  «=  —  J(2a  —  m)  t  Vm(m  —  4a). 

Since  <Jm  —  4a  cannot  be  negative,  m  cannot  be  less  than  4  a. 
Hence  4  a  is  the  minimum  value. 


Let 


3. 

x*  +  l 

=m. 

x 

x2  -f  1  =  mx, 

x*  — mx  =  —  1, 

4x*-()  +  (ro)2  =  m»-4, 


2  x  —  m  =  ±  Vm*  —  4. 
Since  ^m*  —  4  cannot  be  negative,  m  cannot  be  between  +  2 
and  —2,  but  may  have  any  other  values.     Hence   +<»   is  the 
maximum  and  —  »  is  the  minimum  value. 


TEACHERS '    EDITION.  281 


4. 

Let  (a  —  x)(x  —  b)  =  m. 

Then  ax  —  ab  —  x1  +  bx  =  m, 

or  x2  —  x  (a  +  b)  =  —  ab  —  m, 

4  a?  -  (  )  +  (a  +  J)2  =  (a  -  by  -  4  m, 


2a?-(a+6)  =  ±V(a-&)*-4 


m. 


.-.  a;  =  i  (a  +  6) ± V(a - bf  -4m. 
Now,  for  all  possible  values  of  xt  (a  —  6)a  —  4m  cannot  lie  nega- 
tive ;  that  is,  m  cannot  be  greater  than  ^ — — *-  ;  hence  this  is  the 
maximum  value. 

5. 


Let 


1  +  z2      "" 

x=  m-h  ra2, 

mx2  — » x  =  —  m, 
4wW-()  +  l  =  l-4m2, 


2ma>-l  =  ±  Vl-4m2. 

...s=^-(l±  Vl-4m2). 

For  all  possible  values  of  a:,  1  —  4m2  cannot  be  negative ;  that  is, 
m  cannot  be  greater  than  £,  and  for  this  value  x  =  1. 
.-.  i  is  the  maximum  value. 

6. 

Let  a?*  +  8a;  +  20  =  m. 

x*  +  8a;=m  —  20, 
a;*  +  (  )  +  16  =  m  -  4, 

x  +  4  =  ±  Vra  —  4. 
,\  a?  =  —  4  ±Vm  —  4. 

For  all  possible  values  of  a;,  m  cannot  be  negative;  that  Is,  m 
cannot  be  less  than  4. 

*\  4  is  the  minimum  value. 


ALGEBRA. 


7. 

Let  x,-2a:  +  9  =  m. 

x*-2x  =  m-9t 
rf-O  +  l-iii-8, 


x  —  1  =  iVm-  8. 
.-.  x  =  1  ±Vm-  8. 

For  all  possible  values  of  x,  m  —  8  cannot  be  negative ;  that  is,  m 
cannot  be  less  than  8,  and  for  this  value  x  «=  1. 
.*.  8  is  the  minimum  value. 

8. 

Let 


{x  +  a)(x-b) 

x9  =  mx*  +  amx  —  bmx  —  abm, 
x2  —  mx*  —  amx  +  bmx  =  —  abm, 
x*(l  —  m)  —  x(am  —  6m)  =  —  abm, 
4(1  -  m)»3*  -  (  )  +  (am  -  6m)*  =  ahn?  +  2abm*  +  6*m*-  4o6m, 
4(1  -  m)8^2  —  (  )  +  (am  —  6m)*  —  (am  +  6m)1  —  4a6m, 

2a?(l  —  m)  —  (am  -  6m)  =  ±  V(am  +  6m)*  —  4 abm, 
2x(l  —  m)  —  (am  —  6m)  =  ±  Vm* (a  +  6)*  —  4a6m. 

.^  =  i(?!lz^±_J_Vm*(a  +  6)*-4a6m}. 
I     1  — m       1  — m  J 

For  all  possible  values  of  x,  m*(a  +  6)*  —  4o6m  cannot  be  nega- 
tive; that  is,  m*(a  +  6)*  cannot  be  less  than  —  4o6m,  and  for  thia 

value  s=am-6m. 
2(1  -m) 

is  the  minimum  value. 


(a  +  by 
Let 


9. 


a  +  a* 

x  =  am  +  mjj*f 
mx*  —  x  =  —  am, 
4m»at  -0  +  1-1  -4  am*, 

2mx—  1  =  ±\/l  —  4am*. 

...  x  =  — (l±Vi-4am*). 
2m 


For  all  possible  values  of  a;,  1—4  am*  cannot  be  negative ;  that  is, 

m  cannot  be  greater  than  +  Ja/-»  an(*  for  this  value  x  = 

VI.  a  2m 

-  is  the  maximum  value. 
a 


teachers'  edition.  283 

10.  Divide  a  line  20  in.  long  into  two  parts  so  that  the  sum  of 
the  squares  on  these  two  parts  may  be  the  least  possible. 

Let  x  =  number  of  inches  in  first  part, 

and        20  —  x  —  number  of  inches  in  second  part. 

x2  +  (20  -  x)2  =  m, 
x*  +  400  -  40  x  +  x2  =  m, 

2a* -40  a:  =  m- 400, 
4z*  -  (  )  +  400  =  2m -400, 

2  x  -  20  =  ±  V2m-400. 


Then,  as  v2m  — 400  cannot  be  a  negative  expression,  2m  cannot 
be  less  than  400. 

.•.  200  is  the  minimum  value. 
For  this  value,  2  x2  -  40  x  =  -  200, 

x2-  20  s  =  -100, 
s*-()  +  100  =  0, 
x  -  10  =  0. 
.-.  x  =  10, 
20-s  =  10. 


11.  Divide  a  line  20  in.  long  into  two  parts  so  that  the  rect- 
angle contained  by  the  parts  may  be  the  greatest  possible. 

Let  x  =  one  part, 

then  20  —  x  =  the  other  part. 

20x  —  x2  =  m, 
x2  —  20x  =  —  m, 
g«-()+  100  =  100 -mt 

x  -  10  =  ±V100-m. 
.-.  a=10±Vl00-m. 

For  all  possible  values  of  x,  100  —  m  cannot  be  negative ;  that  is, 
m  cannot  be  greater  than  100,  and  for  this  value  x  =  10. 
.•.  100  is  the  maximum  value. 


Substitute  value  of  m,  x  =  10  ±V100—  100, 

x  =  10, 
20-3  =  10. 


284  ALGEBRA. 

12.  Find  the  fraction  which  has  the  greatest  excess  oyer  its 
square. 

Let  x  =  the  fraction, 

then  2s  =  the  square  of  the  fraction. 

x  —  x*  =  m, 
x*  _  x  =  —  m, 
4a*-()  +  l  =  l-4m,_ 
2x - 1  =  ± Vl  -4m. 
.\x  =  J(l±Vl-4m). 
For  all  possible  values  of  x%  1  —  4m  cannot  be  negative ;  that  is. 
m  cannot  be  greater  tlian  J. 
.'.  \  is  the  maximum  value. 
For  this  value,  x  =  J. 

Exercise  LXXXIX. 

1.  a*  +  7x8  =  8.  2.         z4-5a?a  +  4  =  0. 

4a*  +  ( )  +  49  =  81,  a4 -5a?  =  -4, 

2^+7-* 9  4*4-()  +  25  =  9, 

2ar»=-7±9,  2a*  =  5±3, 

x»  =  -8orl.  a*  =  4orl. 

Since  a?  =  —  8,  .\a?=±2,  ±1. 

or  (*+2)(z'-2*+4)  =  0.  *           4af-  37a*  =  -9, 

Whence      x  +  2  =  0,  64a*  -  (  )  +  (37)a  =  1225, 

and                       x  =  -2%  8x*-37  =  ±35, 

or          ar2-2x  +  4  =  0.  8a;2  =72  or  2, 

x*-2x=-4,  a*  =  9  or  f 

aB»  —  ()  +  l  —  —  3,  .\a?~±3or±J. 

x-l  =  ±V^3.  4.                     16^=17jc*-1. 

.-.a:  =  l±v^3".  16a*- 17a4  =  -1, 

Since  *»=!,  1024^i>i(1^  =  *** 

.     1      a  32a?4  - 17  =  ±  15, 

^_1==u'  32a-4  =  32  or  2, 

or  (*-l)(*»+*  +1)  =  0.  xa  -  1  or  A- 

Whence      a;  —  1  =  0,  Since                a;4  =  1, 

and                       a?=l,  **-l  =  0f 


or 


a*  +  x  +  1  =  0,'  or  («I+1X*+1X*-1)-  0. 
a*  +  a;  =  -  1  .\  *  =  ^v^,  - 1.  or  1. 

4a»  +  ()  +  l--3,_  Since        ^      fl^' 

2z+l  =  ±V-3  Or(a*HX*+iX*rlH0' 

.•.a:=$(-W-3).  ...*=±iV-lf-},ori. 

.  a?  =-  —  2,  1,  1  ±  V-  3,  .-.  the  roots  are 

andJC-liV11^).       ±1.  ±  V-lf±J,  ±iV^l. 


teachers'  edition. 


285 


5.  32^-33^  +  1  =  0. 

32a:10-  33s5  =  -1, 
4096z10-()  +  (33)2  =  961, 
64  a* -33  =  ±31, 
64a^  =  64or  2, 

.*.  x=  1  or  {. 
Other  roots  may  be  found  by 
methods  given  later. 

6.  (s»-2)'  =  K*2+12). 


x*  —  4a:2  +  4  = 


a* +  12 


4a?*-16x2+16  =  s2+12, 

4a*-17a*  =  -4, 
643*-()  +  (17)2  =  225, 
8a? -17  =  ±15, 
8  a?  =  2  or  32, 
x9  =  £  or  4. 
.*.  3  =  ±£  or  ±2. 

7.  s4*_5^_25  =  0 
3        12 
12z*»-20s2»-25  =  0, 
12a*»-20a2»  =  25, 
36a?*»-()  +  25  =  100, 
6x2»-5  =  ±10, 
6x2*  =  15  or -5, 
3?w  =  f  or-f 


a;. 
0, 
0, 
0, 

0. 


-1, 
-i 

-J±jV-3. 
i(-l±V=3> 


•.x=±*/\ot±*/—  j. 


(aJ»-9)2  =  3+ll(a^-2). 
a?*  -18a* +81  =  lis2  -19, 

3^-293?  =  -100, 
4^_()  +  (29)2  =  441, 
2a;2  -  29  =  ±  21, 

x2  =  25  or  4. 
.\  a;  =  ±5  or  ±2. 


10.     19**  + 216  a;7  = 

216z7+19s*-a;  = 

*(2163*+19a*-l)  = 

^(273^-1X8^+1)  = 

a;(3a:-lX9iCa+3x+l) 

(2x+l)(4r»-2x+])  = 

.*.  x  = 

From 

9a;a  +  3x  +  l  = 

9a^  +  3a?  = 

9^  +  () +  t  = 

3aJ  +  J  = 

3a;  = 

From 

4a*-2a;  +  l  = 

4a;2-2a;  = 

4a?-2ar+  J  = 

2*-*  = 


roots  are  0,  J,  - 
i(-l±V^3), 


11.  ^  +  223* +  21  =  0. 

a*  +  22a^  =  -  21, 
Xs  +  ( )  +  121  =  100, 
x*  +  11  =  ±  10, 

a:* --lor -21. 
.*.  a?  =  ±V— 1 

or  ivQn. 
That  is,  the  roots  are  imaginary. 


a*  +  14ar,  +  24  =  0. 
3*  +  143*  =  -24, 
a*  +  ()  +  49  =  25, 
3*  +  7  =  ±5, 
a?  =  -2  or 
•.*=V^2or  #-12. 


-12. 


12.    a^"»  +  3a«l-4«0. 
4a^»»  +  12a^-16  =  0, 
4z2"*  +  ()  +  (3)2  =  25, 
2a^  +  3  =  ±5, 
.     2a^=2or-8. 
.-.  x  =  1  or  V  -  4. 


286 


ALGEBRA. 


13. 

4a?-20a?  +  23a?  +  5a;  =  6. 
4a?  -  20a?  +  23a?  +  5x  -  6  =  0. 
4  a?  -  20  x8  +  23  a2  +  5a;  -  6  [2^^5^-J 
4a? 


4a?  — 5s 


-  20  a?  +  23  a? 

-  20  a?  +  25  a? 


4a? -10a; -J 


-2a?+5a;-6 
-2a;2  +  5a;  +  } 


-¥ 


If  ^  were  added  to  both  members  the  square  would  be  complete, 
and  the  equation  would  read 

4a?  -  20a?  +  23  x*  +  5a;  +  i  =  *£. 
Extract  the  root,        2a?-5a;-j  =  ±f, 

2a?-5x  =  3or-2, 
16a?-()  +  25  =  49or9, 

4a;  —  5  =  ±  7  or  ±  3, 

4a;  =  12,  or -2,  or  8,  or  2. 
.-.  s  =  3,  -h  2,}. 

14. 

_L+ 1-20  =  0. 
£**      x11 
1  +  3a^-20a?*  =  0, 
20a?»-3a?»  =  1, 
1600a?»-()  +  9  =  89i/_ 
40a^-3  =  ±V89. 


•  :x=V&±&^. 


■0. 


15. 

x*_4a?-10a?  +  28a-15  = 
Extract  root  of  left  side, 

x*  -  4a*  -  10a?  +  28a;  -  15|a?-2a?-7 
x* 


2a?-2a; 


-4a? -10a? 
-4a?  +  10a? 


2a?-4a;  -7 


-14a? +  28  a; -15 
-  14a?  +  28a;  4-49 


-64 
Add  64  to  both  sides  to  complete  the  square, 

x*  -  4a?  -  10a?  +  28  a;  +  49  =  64. 
Extract  the  root,  a?  —  2a;  —  7  =  ±8, 

a?-2a?=15or-l, 
43*_()  +  4  =  64or0, 
2a;-2  =  ±8or±0, 
2a;  =  10, -6,  2,  2. 
.-.  a;  =  5,  -3,  1,  1. 


teachers'  edition. 


287 


16. 


a?  -  2s?  -  13a;2  +  14a;  =-  24. 
Extract  root  of  left  side, 

a?-  2a? -  13a;2  +  14a;  +  24|a?-a;-7 


2a?-4 


-2a?-13a? 

-2a?  +      a? 


-  14a?  +  14a; +  24 
-14a?  +  14a;  +  49 


-25 
Add  25  to  both  sides  to  complete  the  square,     - 

a?  -  2a?  -  13a?  +  14a;  +  49  =  25. 
Extract  the  root,  a?-a;-7  =  ±5, 

a?  -  x  =  12  or  2, 
4a?-()  +  l  =  49or9, 
2a;-l=±7or±3. 
.-.  a;  =  4,-3,2,  -1. 


17. 

108  a?  =  20  a;  (9  a?- 
108a?  -  108a?  +  51  a?  +  20a;  -  7. 
Multiply  by  12,  and  add  16  to  both  sides, 
1296a?-  2160a?  +  612a?  +  240a;  +  16  =  100. 


■l)-51a?  +  7. 


1296a?-  2160a?  +  612a?  +  240a;  +  16136a;2  -  30a;  - 
1296  a?  ' 


72a? -30a; 


-  2160a?  +  612a? 

-  2160a?  +  900a? 


72a? -60  a; -4 


-  288a?  +  240  a;  +  16 

-  288  a?  +  240  a;  +  16 


36a?-30a;-4  =  ±10, 

36a? -30a;  =14  or- 6, 
144a?-()  +  25  =  81orl, 

12a;-5  =  ±9or  ±  1, 
12a;  =14,  -4,  6,  4. 

18. 

(a?  -  l)(a?  -  2)  +  (a?  -  3)(a?  -  4)  =  a?  +  5. 
Simplify,  a?- 3a?  +  2+  a?- 7a?  + 12  =  a?  +  5. 
Transpose  and  combine,  a?— 10  a?  =  —  9. 
Complete  the  square,    a?- (  )  +  25  =  16. ' 
Extract  the  root,  a?  —  5  =  ±  4, 

a?  =  9  or  1. 
.\ai  =  ±3or±l. 


288  ALGEBRA. 


Exercise  XC. 

1.  The  sum  of  the  squares  of  three  consecutive  numbers  is 
865.    Find  the  numbers. 

Let  x  =  first  number, 

x  +  1  =  second  number, 
and  x  4-  2  =  third  number. 

.-.  x*  +  x*  +  2x  +  l+x*  +  4x  +  ±  =  365, 
3a?  +  6a;  =  360, 
a*  +  2x  =  120, 
a*  +  ()  +  l  =  121, 
x '+  1  =  ±  11. 
.-.  a;  =  10  or -12. 
Hence,  the  numbers  are  10,  11,  12. 

2.  Three  times  the  product  of   two  consecutive  numbers 
exceeds  four  times  their  sum  by  8.    Find  the  numbers. 

Let  x  =  first  number, 

and  x  + 1  =  second  number. 

3  a*  -f  3  s  =  three  times  product, 
8  a;  +  4  =  four  times  sum. 
.-.  3a?  +  3a; -(8a; +  4)  =  8, 
3a?-5a;=12, 
36a*-()  +  25  =  169, 
6x-5  =  ±13, 
6a;=18or-8. 
.-.  a;  =  3or-f 
Hence,  the  numbers  are  3,  4. 

3.  The  product  of  three  consecutive  numbers  is  equal  to  three 
times  the  middle  number.    Fiud  the  numbers. 

Let  x  =  first  number. 

Then  x  4- 1  =  second  number, 

and  x  +  2  =  third  number. 

,\x(x  +  l)(x  +  2)  =  3(07  +  1), 
x*+Sx*'+2x  =  3a;  +  3, 
a*  +  3a?-a;-3  =  0, 
(*  +  l)(a-l)(s  +  3)-0. 

.•.s  =  l,~l,-3. 
Hence,  the  numbers  are  1,  2,  3. 


teachers'  edition.  289 

4.  A  boy  bought  a  number  of  apples  for  16  cents.  Had  he 
bought  4  more  for  the  same  money  he  would  have  paid  J  of  a 
cent  less  for  each  apple.    How  many  did  he  buy? 

Let  x  =  number  of  apples  bought. 

■I  n 

Then        —  =  number  of  cents  one  apple  costs, 

1  ft 

and =  number  of  cents  one  apple  costs  when  he  gets 

x  +  ^         four  more. 

...  16"       16    =lt 
x      x  +  4     3* 
48s  +  192  -48  a?  =  a*  +  4a;f 
x2  +  4s  =  192, 
a;'  +  ()  +  4  =  196, 
a; +  2- ±14. 
.\  a;  =  12or-16. 
Hence,  12  =  number  of  apples  bought. 


5*  For  building  108  rods  of  stone-wall,  6  days  less  would  have 
been  required  if  3  rods  more  a  day  had  been  built.  How  many 
rods  a  day  were  built? 

Let  x  =  number  of  rods  built  in  a  day, 

108 

—  =  number  of  days  in  which  the  whole  wall  was 
*  built, 

108 
=  number  of  days  it  would  have  taken  to  build 

x  +  ^         the  whole  wall  if  3  rods  more  a  day  had 

been  built. 

Then  12§_J08_  =  6. 

X         3  +  3 

108a;  +  324  -  108  a;  =  6a;2  +  18a?, 
6a?  +  18a;  =  324, 
a?  +  3a;  =54, 
4a?+  ()  +  ^=225, 
2a; +  3=  ±15, 

2a;  =  12  or -18. 
.\  a;  =  6  or -9. 
Hence,  6  =*  number  of  rods  built  in  a  day. 


290  ALGEBRA. 


6.  A  merchaut  bought  some  pieces  of  silk  for  #900.  Had  he 
bought  three  pieces  more  for  the  same  money,  he  would  have 
paid  $  15  less  for  each  piece.    How  many  did  he  buy? 

Let  x  =  number  of  pieces  bought. 

900 
Then       —  =  number  of  dollars  each  piece  cost, 
x 

900 
and  =  number  of  dollars  each  piece  would  have  cost 

as  +  3         if  be  had  received  three  more  for  $900. 

Then  *X>-*KL  =  i5t 

x       x  +  3 

900a;  4-  2700  -900  a;  =  15a*  +  45s, 
15a*  +  45a;  =  2700, 
a*  +  3x=180, 
4a*  +  () +  9  =  729, 
2a?  +  3  =  ±  27. 
.*.  x  =  12  or  -15. 
Hence,  12  =  number  of  pieces  bought. 


7.  A  merchant  bought  some  pieces  of  cloth  for  .$163. 75.  He 
sold  the  cloth  for  $  12  a  piece,  and  gained  as  much  as  1  piece  cost 
him.    How  much  did  he  pay  for  each  piece? 

Let  x  =  number  of  pieces, 

12  a;  =  number  of  dollars  received  for  all, 

'■ —  =  number  of  dollars  paid  for  one  piece. 

Then  12  a;  — 168.75  —  number  of  dollars  gained. 

io        icQKK     168.75 

.-.  12  a;  — 168.75  = , 

x 

12a2-  168.75  a;  =  168.75. 

Multiply  by  j,  16  x*  -  225  a;  =  225, 

1024  a?  -  (  )  +  (225)*  =  65025, 

32a; -225  =  ±255, 

32  a;  =  480  or -30. 

.-.a;  =  15  or -1|, 


and  15^5=11.25. 

15 

Hence,  one  piece  cost  $11.25. 


16 


teachers'  edition.  291 

8.  Find  the  price  of  eggs  per  score  when  10  more  in  62J  cents' 
worth  lowers  the  price  3IJ  cents  per  hundred. 

Let  x  =  number  of  eggs  at  62J  cents. 

Then      — —  =  cost  of  one  egg  in  cents, 
x 

62  5 

and        : —  =  cost  of  one  egg  in  cents,  if  he  had  received  ten 

x  + 10         more. 

6250       6250       ,«•  v     j     i 

,\ =  difference  in  price  per  hundred. 

x        a?  +  10  r         r 

.   6250      6250  _  125 

x        x  + 10       4  ' 

Divide  by  125,   52 — -t» 

J  x      a; +  10     4 

200a;  +  2000  -  200a;  =  x*  +  10a;, 
a?  +  10a;  =  2000, 
a*  +  (  )  +  25  =  2025, 
x  +  5  =  ±  45. 
.-.a;  =  40. 

on  e  on e 

Hence,  one  egg  cost  — —  cents,  and  20  eggs  cost  — —  X  20 
&e>  40  °°  ■  40 

=  31 J  cents. 

9.  The  area  of  a  square  may  be  doubled  by  increasing  its 
length  by  6  inches  and  its  breadth  by  4  inches.  Determine  its 
side. 

Let  x  =  the  side  of  the  square. 

(a?  +  4)(a?  +  6)-2*\ 
a;a  +  10a;  +  24  =  2arl, 
a;2  -  10s  =  24, 
a*-()  +  25  =  49, 
x  —  5  =  ±  7. 
.-.  a;  =12  or -2. 
Hence,  the  side  of  the  square  is  12  inches. 

10.  The  length  of  a  rectangular  field  exceeds  the  breadth  by  1 
yard,  and  the  area  is  3  acres.     Determine  its  dimensions. 

Let  x  =  number  of  yards  in  breadth, 

x  +  1  =  number  of  yards  in  length, 
and  x(x  -f  1)  =  number  of  square  yards  in  area. 

But  area  is  3  k,  or  14,520  square  yards. 
.-.  x2  +  x  =  14,520. 
4a?  +  ( )  +  1  =  58,081, 
2a;  +  1  =  ±  241. 

.-.a;  =120  or -121. 
Hence,  the  field  is  121  yards  long  by  120  broad. 


292 


ALGEBRA. 


11.  There  are  three  lines  of  which  two  are  each  $  of  the  third, 
and  the  sum  of  the  squares  described  on  them  is  equal  to  a 
square  yard.    Determine  the  lengths  of  the  lines  in  inches. 
Let  x  —  number  of  inches  in  third  line, 


and 


—  =  number  of  inches  in  each  of  the  others. 


^+   49  +  49 


1fi  T*         lftfT* 

Then  a*  +  -^-+— —  =  the  sum  of  the  squares. 
49        49  * 

1  square  yard  =  1296  square  inches. 

=  1296, 

81  x2 
49 
9* 

7  " 
.-.  s  =  ±28. 
Hence,  the  lengths  are  16, 16,  and  28  inches. 


■  - 1296, 
=  ±36. 


12.  A  grass  plot  9  yards  long  and  6  yards  broad  has  a  path 
round  it.  The  area  of  the  path  is  equal  to  that  of  the  plot. 
Determine  the  width  of  the  path. 


Let  x  =  number  of  yards  in  width  of  path. 

Then  (9  +  2x)  2  +  6  X  2  =  entire  length  of  path  in  yards. 
Also,  [(9+2a)  2+6x2]  x 
or  (30  +  4  a:)  x  =  area  of  path  in  square  yards, 

and  9x6  =  area  of  grass  plot  in  square  yards. 

But  area  of  path  equals  area  of  grass  plot. 
.-.  (30 +  4*)*  =  54. 
4s* +  30  a?  =  54, 
16s* +  ()  +  225  =  441, 
4* +  15 -±21. 
.-.  »=l}or-9. 
Hence,  the  width  of  the  path  is  1 J  yards. 


teachers'  edition.  293 


13.  Find  the  radius  of  a  circle  the  area  of  which  would  be 
doubled  by  increasing  its  radius  by  1  inch. 

Let  x  =  radius  of  circle, 

and  x  +  1  =  radius  increased. 

The  ratio  of  the  circles  is  the  same  as  the  ratio  of  the  squares  on 
the  radii.  ...  2a;2  =  a*  +  2x  +  1, 

a*-2a;=l, 
x*-()  +  l  =  2, 

«-l  =  ±V2, 
s«l±V2, 
•  a:  =2.4142. 

14.  Divide  a  line  20  inches  long  into  two  parts  so  that  the 
rectangle  contained  by  the  whole  and  one  part  may  be  equal  to 
the  square  on  the  other  part. 

Let  x  =  one  part. 

Then  20  —  x  =  the  other  part. 

.-.  20(20-30  =  0?, 
400-203  =  0* 
a?2 +  20  a?  =  400, 
a*  +  ()  +  (10)2  =  500, 

.T  +  10=±\/oXJb, 

x  =  -  10  ±  22.36, 
ar=12.36. 
Hence,  one  part  is  12.36  inches,  and  the  other  is  7.64  inches. 

15.  A  can  do  some  work  in  9  hours  less  time  than  B  can  do  it, 
and  together  they  can  do  it  in  20  hours.  How  long  will  it  take 
each  alone  to  do  it? 

Let  x  =  number  of  hours  it  takes  B. 

Then  x  —  9  =  number  of  hours  it  takes  A, 

and  -  =  part  B  could  do  in  1  hour. 

x 

=  part  A  could  do  in  1  hour. 

x  —  y 

x     x-9     20 
20a;  -  180  +  20x  =  x*-  9x, 
a* -49  a?  =  -180, 
4a?-()  +  (49)2  =  1681, 
2a: -49  =  ±41, 
2a  =  90  or  8. 
.*.  a;  =  45  or  4. 
Hence,  B  can  do  the  work  in  45  hours  and  A  in  36  hours. 


294  ALGEBRA. 

16.  A  vessel  which  has  two  pipes  can  be  filled  in  2  hours  less 
time  by  one  than  by  the  other,  and  by  both  together  in  2  hours 
55  minutes.  How  long  will  it  take  each  pipe  aloue  to  fill  the 
vessel? 

Let  s  =  number  of  hours  it  takes  first  pipe, 

x  —  2  =  number  of  hours  it  takes  second  pipe. 
2  hours  55  minutes  equals  f  f  hours. 
.   1         1     _12 
"x      3-2      35* 
35*  _  70  +  35s  -  12s*  -  24x, 
12a2 -94s  =  -70, 
144  s2 -()  +  (47)2  =  1369. 
Extract  the  root,        12  s  -  47  -  ±  37, 

12s  =  84  or  10. 
.-.  s=  7  or  f. 
Hence,  one  pipe  will  fill  it  in  7  hours,  the  other  in  5  hours. 

17.  A  vessel  which  has  two  pipes  can  be  filled  in  2  hours  less 
time  by  one  than  by  the  other,  and  by  both  together  in  1  hour 
52  minutes  30  seconds.  How  long  will  it  take  each  pipe  alone  to 
fill  the  vessel? 

Let  s  =  number  of  hours  it  takes  first  pipe. 

Then       s  +  2  =  number  of  hours  it  takes  second  pipe. 

-  =  part  first  pipe  fills  in  1  hour, 

and =  part  second  pipe  fills  in  1  hour. 

s  +  2 
1  hour  52  minutes  30  seconds  equals  -^  hours. 

...1  +  -1— L 

s     x  +  2      15 
15s  +  30  +  15s  =  8s2  +  16s, 
8s2 -14s  =  30, 
64s2 -112s  =  240, 
64s2 -0  +  49  =  289, 
8s -7- ±17, 

8s  =  24  or -10. 
.-.  s  =  3  or  -1J. 
Hence,  one  pipe  will  fill  it  in  3  hours,  the  other  in  5  hours. 

18.  An  iron  bar  weighs  36  pounds.  If  it  had  been  1  foot 
longer,  each  foot  would  have  weighed  J  a  pound  less.  Find  the 
length  and  the  weight  per  foot. 


teachers'  edition.  295 


Let  x  =  number  of  feet  in  length. 

36 
Then  —  =  weight  in  pounds  per  foot, 

and        =  weight  per  foot  if  it  had  been  1  foot  longer. 

But  =  weight  per  foot  if  it  had  been  1  foot  longer. 

.36     1  =    36 
x      2     x  +  l' 
72a:  +  72 -a;2- a:  =  72a:, 
x*  +  x  =  72, 
4s* +  ()  +  1  =  289, 
2a:  +  l=±17. 
.-.  x  =  8  or -9, 

25-4J. 

%        x 
Hence,  the  bar  is  8  feet  long,  and  weighs  4i  pounds  per  foot. 

19.  A  number  is  expressed  by  two  digits,  one  of  which  is  the 
square  of  the  other,  and  when  54  is  added  its  digits  are  inter- 
changed.   Find  the  number. 

Let  x  =  digit  in  tens'  place, 

Then  a?  =  digit  in  units'  place, 

10  a?  +  x9  =  number, 
and  10  x8  +  x  =  number  with  digits  reversed. 

.-.  10s  +  x2  +  54  =  10a?2  +  a;, 
-  9a2  +  9a:  =  -54, 
x2  -  x  =  6, 
4a*-()  +  l  =  25, 
2s-l  =  ±5. 
.'.x  =  3. 
Hence,  the  number  is  39. 

20.  Divide  35  into  two  parts  so  that  the  sum  of  the  two  frac 
tions  formed  by  dividing  each  part  by  the  other  may  be  2^. 

Let  x  =  one  part. 

Then  35  —  x  =  the  other  part. 

x         35-a?_25t 

'"'35 -a         x         12 
12a?  + 14700  -840  a:  +  12a?  =  875*  -25  a2 
49a? -1715a:  =  -14700, 
x2  -35  x  =  -300, 
4a?-()  +  (35)2  =  25> 
2x-36  =  ±5. 
.-.  a:  =  20  or  15. 
Hence,  the  parts  are  20  and  15. 


296  ALGEBRA. 


21.  A  boat's  crew  row  3}  miles  down  a  river  and  back  again 
in  1  hour  40  minutes.  If  the  current  of  the  river  is  2  miles  per 
hour,  determine  their  rate  of  rowing  in  still  water. 

Let  x  =  rate  in  still  water, 

x  +  2  =  rate  down  stream. 

1  hour  40  minutes  equals  J  hours. 

— *—  =  number  of  hours  going  down  stream, 
x  +  2 


2 
7,75 


•-  number  of  hours  going  up  stream. 


2(x  +  2)     2(a-2)     3 
21x-  42  +21  x  +  42  =  10  a?  -40, 
10a? -42s  =  40, 
400a?  -  ( )  +  (42)2  =  3364, 
20aj-42  =  ±58, 

20  a  =  100  or  -16. 
.*.  x  =  5  or  —  J. 
Hence,  the  rate  of  rowing  in  still  water  is  5  miles  an  hour. 

22.  A  detachment  from  an  army  was  marching  in  regular 
column  with  5  men  more  in  depth  than  in  front.  On  approach- 
ing the  enemy  the  front  was  increased  by  845  men,  and  the  whole 
was  thus  drawn  up  in  5  lines.    Find  the  number  of  men. 

Let  x  =  number  of  men  in  front, 

and  x  +  5  =  number  of  men  in  depth. 

Then  a?  +  5x  =  number  of  men  in  all. 

But  x  +  845  =  number  of  men  in  front, 

and  5  =  number  of  men  in  depth. 

Then  5x  +  4225  =  number  of  men  in  all. 

.-.  ic2  +  5a;  =  5a;  +  4225, 
a:2  =  4225. 
.*.  x  =  ±65. 
Hence,  the  whole  number  of  men  is  4550. 

23.  A  jockey  sold  a  horse  for  $144,  and  gained  as  much  per 
cent  as  the  horse  cost.     What  did  the  horse  cost? 


teachers'  edition.  297 


Let  x  =  number  of  dollars  the  horse  cost. 

Then  -^-~  =  gain  per  cent, 

100     &       r 

of  x  =  whole  gain, 

100  6 

and  x  +  —  =  amount  received. 

100 

....  +  JL..1* 

x*  +  100  a;  =  14400, 
x7  +  ( )  +  2500  =  16900, 
x  +  50  =  ±  130. 
.\  a;  =  80  or  -180. 
Hence,  the  horse  cost  $80. 


24.  A  merchant  expended  a  certain  sum  of  money  in  goods, 
which  he  sold  again  for  $24,  and  lost  as  much  per  cent  as  the 
goods  cost  him.    How  much  did  he  pay  for  the  goods? 

Let  x  =  number  of  dollars  paid  for  goods. 

Then  —  =  per  cent  lost, 

100     r 

and  -^-  of  x  =  whole  loss. 

100 

100 
100*  -x*  =  2400, 
x2-  100  a;  =  -2400, 
aa_()  +  (50)»  =  100, 
a;-50  =  ±10, 

x  =  60  or  40. 
Hence,  the  goods  cost  either  $60  or  $40. 

25.  A  broker  bought  a  number  of  bank  shares  (f  100  each), 
when  they  were  at  a  certain  per  cent  discount,  for  $  7500 ;  and 
afterwards  when  they  were  at  the  same  per  cent  premium,  sold  all 
but  60  for  $5000.  How  many  shares  did  he  buy,  and  at  what 
price? 


298  ALGEBRA. 


Let  x  =  number  of  shares  bought, 

7500 

— —  =  number  of  dollars  each  share 

x  cost, 


and 
Then 
100- 

7500 

X 

loo-irrr 

X 

100*^-7500 

100      Ui 
Also, 

Then 

100a; 
x-QO 
5000 

=  number  of  dollars  discount  on 
each  share. 


■  rate  of  discount. 

■■  number  of  shares  sold. 

--  number  of  dollars  received  for 


5000  x~  eacn  Bhare» 

and  100  =  number  of  dollars  premium  on 

K/y^  05  —  60  eacn  share. 

ouuu     -100 


x-60  11000 -100a:         .  ,    * 

— ___  or =  rate  per  cent  of  premium. 

100  100  x  -6000  r  r 

But  rate  percent  discount  was  equal  to  rate  percent  premium. 
.  100a?  -  7500  _  11000  -  100a; 
100a;  100a;-  6000 ' 

x*  -135  a;  +  4500  =  110a?  -a* 

2a?2 -245  a?  =  -4500, 
16a*-()  +  (245)2  =  24025. 
Extract  the  root,     4  a;  —  245  =  ±  155, 

4a?  =  400or90. 
.-.  a?=100or22£. 
Hence,  the  broker  bought  100  shares  at  75. 

26.  The  thickness  of  a  rectangular  solid  is  }  of  its  width,  and 
its  length  is  equal  to  the  sum  of  its  width  and  thickness ;  also, 
the  number  of  cubic  yards  in  its  volume  added  to  the  number  of 
linear  yards  in  its  edges  is  $  of  the  number  of  square  yards  in  its 
surface.    Determine  its  dimensions. 

Let  3  a?  =  number  of  yards  in  width, 

2  a?  =  number  of  yards  in  thickness, 
and  5  x  =  number  of  yards  in  length. 

303?  + 40  a;  =  1(622?*), 
90  a?8  -  310  a?2  =  -  120  a?. 
Divide  by  10  x,      9  a?2  -  31  a?  =  - 12, 
9^-()  +  (W  =  W. 
3a;-V  =  ±V. 
3a?  =  9  or*. 
.\  a?  =  3  or  J. 
Hence,  the  dimensions  are  15  X  9  X  6  yards, 
or  2}  X  \\  X  f  yards. 


TEACHERS     EDITION. 


299 


27.  If  a  carriage-wheel  16J  feet  round  took  1  second  more  to 
revolve,  the  rate  of  the  carriage  per  hour  would  be  1 J  miles  less. 
At  what  rate  is  the  carriage  travelling? 


Let 


59400 


3600 


»  number  of  seconds  it  takes  the  wheel  to 
revolve; 


=  number  of  revolutions  it  makes  per  hour, 


or  16$  X 


x 
3600 

x 
59400 


=  number  of  feet  it  goes  per  hour, 

■  number  of  feet  it  would  go,  if  it  took  one 
second  more  to  revolve, 


Then 


x  +  1 
59400_59400  =  99(X)j  number  of  feet  in  x ,  miles> 
x         x  +  1 


59400a;  +  59400  -  59400a;  =  9900x2  +  9900x,  " 
9900  a*  +  9900  x  =  59400, 
a2  +  x  =  6, 

*  +  ()  +  *-¥. 
re  +  i  =  ±  J. 

.-.  x  =  2  or -3. 
ai  £lq.  =  29700. 

Since  29700  feet  equal  5$  miles,  the  carriage  is  travelling  at  the 
rate  of  5f  miles  per  hour. 


Exercise  XCI. 


3 

(3) 
(4) 


1.  x  +  y  =  13 

ay  =  36 
Square  (1), 

x*  +  2xy  +  y2  =  l69 
(2)x4is        ±xy         =144 

Subt.,rc2-2icy  +  y2=    25 

Extract  root,   x  —  y  =  ±  5       (5) 
Add  (1)  and  (5),  2  s  =  18  or  8. 
.•.  x  =  9  or  4. 

Subtract  (5)  from  (1), 

2y  =  8  or  18. 
f\  y  =  4or  9, 


2.  a?  +  y  =  29 

ay  =  100 
Square  (1), 

a*+2a#+y2  =  841 
(2)x4is        4  ay  =400 


(1) 
(2) 

(2) 


Subt.,  x2  -  2xy  +  y2  =  441 
Extract  root,    x  —  y  =  ±  21      (5) 
Add  (1)  and  (5),  2a;  =  50  or  8. 

.  \  x  =  25  or  4. 
Subtract  (5)  from  (1), 

2y  =  8  or  50. 

•\  y  =  4or25. 


300 


ALGEBRA. 


3.  s-y  =  19 

xy=*  66 
Square  (1), 

xi-2xy  +  y2  =  361 
(2)x4is        4zy  =264 


Add,   x2  +  2:ry  +  y,  =  625 
Extract  root,  x  +  y  =  ±  25 
Add  (5)  and  (1),  2x  =  44  or 
.-.  a;  =  22  or 

Subtract  (1)  from  (5), 

2y  =  6  or  - 
.-.  y  =  3or- 


4.  x  —  y  =  45 

sy  =  250 
Square  (1), 

x*-2xy+yi 
(2)x4is        4sy 


(1) 
(2) 

(3) 
(4) 

(5) 
-6. 
-3. 

-44. 
-22. 


5.  x  —  y  =  . 

x*+y2  =  178     (2) 
Square  (1), 

a?  —  2:cy  +  y*  = 


(1) 
(2) 


Add,   x*  +  2xy  +  y*  = 

Extract  root,  x  +  y  = 
Add  (5)  and  (1),  2x  = 

.*.  x  = 

Subtract  (1)  from  (5), 

2y  = 

.-.  y  = 


2025  (3) 
1000     (4) 

3025 

±55  (5) 
100or-10. 
50  or -5. 

10or-100. 
5  or  -50. 


=  10      (1) 

(2) 


(2)  is  x2  +  ya  = 


Subt., 


100 

178 


(2)  is   a* 


2xy 


-fyV 


■    78 

=   178 


(3) 


Add,  x*  +  2xy  +  y*  = 
Extract  root,  x  +  y  = 
Add  (5)  and  (1),  2x  = 

.*.  x  = 
Subtract  (1)  from  (5), 
2y  = 

.-.  y  = 

6.  x  —  y  = 

s*  +  y»  = 
Square  (1), 

x*—2xy+y*  = 
Subtract  (2)  from  (3), 
-2xy  = 
Subtract  (4)  from  (2), 
x1  +  2xy  +  y*  = 
Extract  root,  x  +  y  = 
Add  (1)  and  (5),  2x  = 

Subtract  (1)  from  (5), 
2y  = 


256 

±16  (5) 
26  or  -6. 
13  or -3. 

6  or -26. 
3  or -13. 


i1' 
(2) 


196     (3) 

-240  (4) 

676. 

±2Q  (5) 
40  or -12. 
20  or -6. 

12  or -40. 
6  or  -20. 


Square  (1), 
Subtract  (2)  from  (3), 
Subtract  (4)  from  (2), 
Extract  root, 
Add  (1)  and  (5), 

Subtract  (5)  from  (1), 


s  +  y  =  12  (1) 

xa  +  y2  =  104  (2) 

a?  +  2&y  +  y2  =  144  (3) 

2sy  =  40  (4) 

x*-2xy  +  yi  =  64:. 

a;-y  =  ±8  (5) 

2x  =  20  or  4. 

.-.  x  =  10  or  2. 

2y  =  4  or  20. 

.\  y  =  2or  10. 


TEACHERS     EDITION. 


301 


8.  1  +  1. 

x     y 

X*       V* 

Square  (1),  9 

1       2       1 

—  + —  +  —  = 

3*        Xy       y* 

Subtract  (2)  from  (3), 
2_  = 
xy 

Subtract  (4)  from  (2), 

i_A+I= 
x2     xy     y* 

Extract  root, = 

x    y 

Add  (1)  and  (5),     ?  = 
x 

Subtract  (5)  from  (1), 
2 

y~ 

.-.  y  = 


3 
4 
_5 
16 

$_ 

16 

4 
16 

J_ 
16* 

-\ 

lor}. 
2  or  4. 

Jorl. 
4  or  2. 


(1) 
(2) 

(3) 
(4) 

(5) 


H=5    (1) 

Square  (1),  * 

-.  +  -  +  -.  =  25        (3)         n. 


xy    y 
Subtract  (2)  from  (3), 
2 


Subtract  (4)  from  (2), 

L-l-  +  ±-  = 
x2     xy     y2 

Extract  root, = 

x    y 

Add  (1)  and  (5),     -~ 

.•.  x  = 
Subtract  (5)  from  (1), 

y 
.-.  y  = 


^=12    w 


1. 

±1        (5) 

6  or  4. 
JorJ. 

4  or  6. 
J  or  J. 


10.         7x2-8zy  =  159       (1) 
5a;  +  2y=7  (2) 

2y  =  7-5x. 

•'         2 
Substitute  in  (1), 

7aa-8s(^i^  =  159, 

14^-56a;  +  40a;a  =  318, 
54aJ-56x  =  318. 
Divide  by  6, 

9x*-*£x  =  53. 
Complete  the  square, 

»*-<  )  +  (W  =  J4f»- 

Extract  root, 

3a;-V=±V. 

3»  =  ^or-^. 

.-.  z-3or-lff. 

Substitute  value  of  a;  in  (2). 

.-.  y=--4or8if 


s  +  y  =  49        (1) 
a^  +  yJ  =  1681     (2) 
Square  (1), 

3a  +  2ajy+y»  =  2401.    (3) 
Subtract  (2)  from  (3), 

2ay  =  720       (4) 
Subtract  (4)  from  (2), 

a?-2zy  +  ya  =  961. 
Extract  root,  x—y  =  ±  31     (5) 
Add(5)and(l),2a>  =  80orl8. 
.-.  x  =  40  or  9. 
Substitute  value  of  x  in  (1). 
.\y  =  9  or  40. 


302 


ALGEBRA. 


12.  7?  +  f  = 

x  +  y  = 
Divide  (1)  by  (2), 

x2-  ay+y2  = 
Sq.  (2),  s»+ 2ay+y2  = 
Subt.,        — 3ay 

.\-ay  = 
Add  (3)  and  (5), 

a*-2ay  +  y2  = 

Extract  root,  x  —  y  = 

Add  (2)  and  (6),  2a;  = 

.•.  x  = 

Subtract  (6)  from  (2), 

2y  = 

.-.  y  = 


341       (1) 

ii      (r 


2) 


31 
121 


-90 
-30 


(5) 


1. 
±1 

12  or  10. 
6  or  5. 


(6) 


10  or  12. 
5  or  6. 


8! 
8 


(5) 


13.  ^+^=1008 

s  +  y  =  12 
Divide  (1)  by  (2), 

x*—  xy+y*  =      84 

Sq.(2),    x*+2xy+y*=    144 

Subt.,       -  Sxy       =  -  60 

...  -xy  =  -  20 

Add  (3)  and  (5), 

aa-2»y  +  y,  =  64. 
Extract  root,  re  —  y  =  ±  8       (6) 
Add  (2)  and  (6),  2  a;  =  20  or  4. 
.\  x  =  10  or  2. 
Subtract  (6)  from  (2), 

2y  =  4or  20. 
.\y  =  2or  10. 


14.  x*-tf  =  98 

x-y  =  2 
Divide  (1)  by  (2), 

x2+  ay+y2  =  49 

Sq.  (2),  a?-2sy-fy2  =    4 

Subt.,  3ay        -  45 

,\  xy  =  15 

Add  (3)  and  (5), 

a*  +  2sy  +  y2  =  64. 
Extract  root,  x  +  y  =  ±  8 
Add  (2)  and  (6),  2  s  =  10  or  - 
.  \  x  =    5  or  - 
Subtract  (2)  from  (6), 

2y  =  6or- 
•  \  y  =  3  or  — 


(3) 
(4) 

(5) 

(6) 
-6. 
-3. 

10. 
5. 


15.  s8-y3  =  279 

a;-y  =  3 

Divide  (1)  by  (2), 

x*+    xy  +  y2  =  93 
Sq.(2),  x*-  2  xy  +  y*=   9 

Subt.,         3  ay         =84 
,\  ay  =  28 
Add  (5)  and  (3), 

a*  +  2ay  +  y2  =  121. 
Extract  root,  x  -f  y  =  ±  11 
Add  (6)  and  (2),  2a;  =  14  or 
.\  a;  =  7  or  - 
Subtract  (2)  from  (6), 

2y  =  8  or  - 
.  \  y  =  4  or  - 


8 


(6) 
-8. 
4. 

14. 

7. 


16. 


-3y  =  l 


(1) 
ay  +  y2  =  5         (2) 

Transpose  (1),       x  =  1  +  3y. 
Substitute  m  (2), 

y(l+3y)  +  y2  =  5, 

y  +  3y2  +  y2  =  5, 

4y2  -f  y  =  5, 

2y  +  i  =  ±f. 
.-.  y  =  lor-l}. 
Substitute  value  of  y  in  (1), 

a;  =  4or-2f. 


17. 


4y  = 
2xy  = 

y  = 


5s+l  (1) 
-**  (2) 


33 

5s  +  l 

4 
in  (2), 

33 -x2, 


Substitute  value  of  y 
10a?  +  2a?_ 

4 

14a2  +  2a;  =132. 
Divide  by  2, 

7^  +  x  =  66, 
196  a? +  ()  +  1  =  1849. 
Extract  root, 

14a;  +  l  =  ±43, 
14  a;  =  42  or -44. 
.-.  a;  =  3or-3f 
Substitute  value  of  a;  in  (1). 

.-.  y  =  4or-3Jf 


TEACHERS     EDITION. 


303 


1    1 

18.                    ±_i  =  3 

(1)         19.                1-1  = 

=  2} 

(1) 

a;     y 

x     y 

1-1  =  21 
x2     y1 

(2)                  L-L  = 

■81 

(2) 

Divide  (2)  by  (1), 

Divide  (2)  by  (1), 

1  +  1-7 

(3)                             1  +  1- 

s7 

(3) 

x     y 

x     y 

2 

Add  (3)  and  (1),       ?  =  10. 

X 

Add  (1)  and  (3),-  = 
x 

. 12 
"  2 

•••*  =  *. 

.*.  05  = 

-h 

Subtract  (1)  from  (3), 

Subtract  (1)  from  (3), 

?-4. 

2  = 

=  1. 

y 

y 

•••  y  =  J. 

.-.  y  = 

20. 

=  2. 

x*-2xy-y*  =  l 

(1) 

x  +  y  =  2 

(2) 

Square  (2), 

x*  +  2xy  +  y2  =  4 

(3) 

Add  (3)  and  (1), 

2a?  =  5, 

a2  =  2J. 
.-.  x  =  ±V2}. 

Substitute  value  of  x  in  (2), 

y  =  2W2J. 

Exercise  XCII. 

a?  +    a;y-f2y2  =    74 

(1) 

2a?2  +  2ay  +    y2  =    73 

(2) 

Add, 

3a^  +  3ajy+3ya  =  147 

Divide  by  3, 

a;2  +  xy  +  y*  =  49 

(3) 

Subtract  (3)  from  (1), 

ya  -  25. 
,\y  =  ±5. 

Substitute  value  of  y  in  (3), 

a*±5a:  +  25  =  49, 
a?±5x  =  24, 
4a-2  ±  20a:  +  25  =  121. 

Extract  the  root, 

2a;  ±  5  «  ±  11, 

2a;  =  ±6or  ±16 
.*.  a:  =  ±3  or  ±8. 

304  ALGEBRA. 


a*  4-  xy  +  4y2  =  6  (1> 

3^+8^  =  14  (2) 

Substitute  vx  for  y  in  both  equations. 
From  (1),  x*  +  vx2  +  4vV  -  6. 

...  3*  = 5 (3) 

From  (2),  3  x2  +  8  tftc2  =  14. 

...a2-     14  (4) 

Equate  values  of  a2, 
H  6  14 


l  +  17+4v«     3  +  8v» 
18  +  48  v2  - 14  +  14v  +  56  V2, 
8a2 +  14*  =  4  (5) 

64t?  +  ()  +  49  =  81, 
8t;  +  7  =  ±9, 

8t>.=  2or-16. 
.•.  t;=«  J  or  —2. 

14  14 

Substitute  values  of  v  in  (4),  x*  -  — --  or  — — 

Then  a*  =  4orf 

.\  x-±2t  ±Vf 
From  (2),  y  =  ±},  ±2V|. 

3. 

a»-^  +  y2»2i  (l) 

y2-2sy  =  -15  (2) 

Substitute  vx  for  y  in  both  equations. 
From  (1),  a2  -  to1  +  v2*2  =  21. 

...*-_?L—  (3) 

1—  v  -{-v2 

From  (2),  vV  -  2  vz2  =  -  15. 

.,^-pi§-  (4) 

v2  —  2v 

Equate  values  of  a2, ;  =  — — —  (5) 

21  v*  -  42  v  =  - 15  +  15t;  -  15v2, 
36«2-57v  =  -15, 
5184^ -()  +  (57)2  =  1089, 
72V  _  57  =  ±  33. 


.-.  v  =  f  or  J. 

4),  z2  =  16       ' 


Substitute  values  of  v  in  (4),  x2  =  16  or  27. 

.•.  a;  =  ±4  or  ±3  VS. 
,\  t/  =  ±5  or  ±V3. 


teachers'  edition.  305 


a2-4y2-9  =  0. 

xy  +  2y2  -  3  =  0.  * 

a?  -  4y2  =  9  (1) 

Transpose,  xy  +  2y2  =  3  (2) 

Substitute  va?  for  y  in  both  equations. 
From(l),  x*-lv*x*  =  9. 

-"-nSsi  (3) 

From  (2),  3*t;  +  2^^  =  3. 

9  3 

Equate  values  of  S2,  — 


1-4^     v  +  2v* 
30^  +  9^  =  3, 
10*  +  3t>-l, 
400v2  +  ( )  +  9  -  49, 
20 1;  +  3  _  ±  7, 

20v  =  4or-10. 
.\t>  =  *or-J. 
Substitue  values  of  v  in  (3),    a?2  =  ^  or  oo. 

.-.  z  =  ±5Vf 

.•.y-±vf. 

5. 

0^-0^  =  35  (1) 

*y+y2  =  18  (2) 

Substitute  vx  for  y  in  both  equations. 

From(l),  aja-va:2  =  35. 

-*-T=i  (3) 

From  (2),  va^  +  ^a^=18. 

.•.s2=-^-  (4) 

Equate  values  of  s2,  -iL  =  -^-t 

1  —  v     v  +  v* 
35v2  +  53v  =  18, 
4900  v2  +  ()  +  (53)2  =  5329, 
70v  +  53  =  ±  73, 

70v  =  20or-126. 
.-.  v  =  |or-f 
Substitute  values  of  v  in  (3),  a?  =  49  or  4£. 

.\  &  =  ±  7  or  ±5V|. 
.\  y=.±2orT9V|. 


306  ALGEBRA. 


6. 

x2  -f  xy  +  2y2  =-  44 
2x*-xy  +  y2  =  16 

(1) 
(2) 

Substitute  vx 

for  y  in  both  equations. 

From  (1), 

a?  +  vx2  +  2v2x2  =  44. 

44 

(3) 

From  (2), 

2s*-irc2+irtc*  =  16. 

■  *'-        16      • 

2-v-\-v2 

Equate  value 

nofo?         44                  16      4 

'1+V  +  2V2     2-v  +  r2 

88  -  44v  +  44^  =  16  +  16v  +  32w», 

12^-60v  =  -72, 

4t?-()  +  25  =  l, 
2t>-5  =  ±l. 
.-.  v  =  3or  2. 
Substitute  values  of  v  in  (3),   x*  =  2  or  4. 
From  (3).  .-.  a:  =  ±V2  or  ±  2. 

.•.  y  =  ±  3  V2  or  ±  4. 


7. 


3»  +  ay=15  (1) 

xy-y2  =  2  (2) 

Substitute  vx  for  y  in  both  equations. 

From  (1),  x*  +  vx2  =  15. 

1  +v 
From  (2),  vx*  -  v*x*  =  2. 

'.a1 ^-r  (4) 

v—v* 

Equate  values  of  x*,  - = 


1+v     v—v2 
I5v-15v*  =  2  +  2v, 
15v»-13t;  =  -2, 
900t>*-()  +  169  =  49. 
Extract  the  root,        30 1;  - 1 3  =  ±  7,       . 
30v=20or6. 
.\  v  =  f  or  J. 
Substitute  values  of  v  in  (3),  x2  =  9  or  Af . 

.\  a  =  ±3  or  ±5\/j. 
.\  y  =*±2  or  ±VJ. 


TEACHERS*   EDITION. 


8. 

x2  —  xy  +  y2  =  7 

(1) 

3x2  +  13a?y  +  8?/2  =  162 

(2) 

Substitute  vx 

for  y  in  both  equations. 

From  (1), 

a*2  —  vx  +  iPx2  =  7. 

•   r2           7 

(3) 

l-v+v* 

From  (2), 

3x*  +  13v  +  8v2x*  =  162. 

■*»-             162 

- 

3  +  13v  + 

8v2 

-.nf*                      ^            „.             162 

U\ 

^  1-v  +  v2     3  +  13v  +  8i;2 

.-.  106V2- 253v  =  - 141, 
44944^ -()+(253)2  =  4225. 
Extract  the  root,        212v  -  253  =  ±  65. 

Substitute,  values  of  v  in  (3),     x2  =  4  or  ^ft». 

.-.  ar=±2or  ±2Jf 
.-.  y  =  ±3or  ±2^. 

9. 

2a2  +  3sy+y2  =  70  (1) 

6a^  +  ary-y2  =  50  (2) 

Substitute  vx  for  y  in  both  equations. 


From  (1), 

2&2+3trc,  +  v,a,= 

=  70. 
70 

2  +  3v 

+  v* 

From  (2), 

6^  +  1?^  —  v*x"  = 

=  50. 
50 

6  +  v- 

v2 

70 

50 

~~  '    2  +  3tf+va      6  +  v- 

-V2' 

420  +  70i;-70t>*  = 

=  100  +  1500  +  50^, 

12v*  +  8v  = 

=  32, 

36^  +  ()  +  (2)*  = 
6v  +  2  = 

aoo, 

.±10. 

.*.  v  = 

•  ljor- 

-2. 

Substitute  vali 

ae  oft?  in  (3),      a*  = 

» 9  or  oo. 

.\  «  = 

=  ±3. 

.-.y- 

=  ±4. 

(3) 
(*) 


S08  ALGEBRA. 


Substitute  vx 
From  (1), 

10. 

x*  —  xy  —  y*  =  5 

for  y  in  both  equations. 
a1  —  wx*  —  v*x*  —  5. 

-   *»             5 

a) 

(2) 
(3) 

From  (2), 

1-v-v8 
23*  +  3itt*  +  t*r*  =  28. 

.  ^            28 

2  +  3v  +  v* 

M 

^  l-v-vi      2  +  3v  +  v* 

10  +  15v  +  5v«  -  28  -  28v  -  28V8, 
33v*  +  43t>  =  18, 
4356v»  +  ()  +  (43)*  =  4225I 
66t>  +  43  =  ±  65. 

.'.•-!« -if- 
Substitute  values  of  v  in  (3),     x*  =  9  or  - 121. 

.-.  x  =  ±3or±llV^l 
.-.  y  =  ±l  or  =f18\^I. 

Exercise  XCIII. 

1. 

4sy  =  96-a:V  W 

*  +  y  =  6  (2) 

Let  »  =  («  +  v), 

and  y  =  (u-v). 

From  (2),  n  +  v+u-v-6, 

2«  =  6, 
u  =  3. 
From  (1),  4(u*  -  v8)  =  96  -  w*  +  2wV-^, 

4m»  -  4v»  =  96  -  u*  +  2mV-v*. 

Substitute  value  of  u,     v*  -  22v*  -  -  21, 
4r2-()+(22)*  =  400, 

2v»-22  =  ±20, 
2v*  -  22  ±  20, 
^  =  21  or_l. 
•\  v  =  ±V21  or  ±1. 
...  <c  =  m  +  v  =  3±V21,  4,  2, 
and  y  -  w  -  v  =  3  t  V21,  2,  4. 


teachers'  edition.  309 


a»+y»«18-*-y  (1) 

^ -6  (2) 

Put  u  +  v  for  ar,  and  w  —  v  for  y. 

(1)  becomes  (u  +  v)2  +  {u  —  v)2  =  18  —  2w, 

u%  +  v*  +  u  =  9  (3) 

(2)  becomes  (u  +  v)  (u  —  v)  =  6, 

or  u*  -  v2  =  6  (4) 

Add  (3)  and  (4),  2w2  +  u  =  15. 

Complete  tbe  square,  16w2  +  ()  +  l  =  121, 
4w+l  =  ±ll. 
.-.  w=2Jor-3. 
Substitute  value  of  u  in  (4),      —  v*  =  6  —  *£-  or  6  —  9. 
.\  i>  =  ±  J  or  ±V3. 
.-.  a:  =  u  +  i>  =  3,  2,  or  -3±V3, 
and    y  =  u  —  v=  2,  3,  or  —  3=fV3. 


3. 


2(s2  +  y2)  =  5xy 
4(x-y)  =  xy 
Put  u  +  v  for  a,  and  u  —  r  for  y, 

2(2u2+2v2)  =  5(m2-«2) 
4(2v)  =  u2-t;2 

(i) 

(2) 

(3) 
(4) 

Transpose  and  combine,"     9v2  —  u2  =  0 
+  w2-t?  =  8tf 

(5) 
(6) 

Add  (5)  and  (6),                          8i>*  =  8v 

8v2-8v  =  0. 

.•.  v  =  0  or  1. 

Substitute  value  of  v  in  (6),          wv  =  8 1;  +  v2, 

u2  =  0  or  9, 

u  =  0  or  ±  3. 

.\  a>  =  w  +  v  =  0,  4,  —  2, 

and  y  —  u  —  v  =  0,  2,  —  4. 


310  ALGEBRA. 


4. 

4(*4-y)  =  3*y  (1) 

x  +  y  +  x>+y*  =  26  (2) 

Put  u  -+-  v  for  a,  and  «  — v  for  y. 

(1)  becomes  8  u  =  3  u*  —  3  v2. 

.-.  8u-3u2  +  3v2  =  0  (3) 

(2)  becomes  2u  +  2^+2u2  =  26  (4) 

Multiply  (4)  by  3,  6w  +  6u2  +  6v2  =  78 

Multiply  (3)  by  2,  16u-6u2  +  6^  =    0 

Subtract,  12u2-10u  =  78 

Complete  the  square,     144u2  -  ( )  4-  25  =  961, 

12w-5  =  ±31, 

12u  =  36or-26. 
.-.  M  =  3or-2J. 
Substitute  value  of  u  in  (3),  3 1?  =  3. 

.*.  v  =  ±  1. 
Substitute  -  2J  for  u  in  (3),  3t?  =  i}fi. 

.-.  v  =  ±jV377. 
.-.  a  -  u  +  v  =  4,  2,  or  K-13  ±V377), 
and    y  =  m -  v  -  2,  4,  or  i(-l3  *  V577). 


5. 

4a2  +  ay  +  4y2  =  58  (1) 

5a2  +  5y2  =  65  (2) 

Multiply  (1)  by  5,     20a2  +  5w+  20y2  =  290 
Multiply  (2)  by  4,     20a2  +  20y2  =  260 

Subtract,  5  ay  =30 

.\  xy  =  6  (5) 

Divide  (2)  by  5,  a2  +  y2  -  13  (6) 

Substitute  w  +  v  for  a,  and  w  —  v  for  y  in  (5)  and  (6), 

u2  -  v2  =  6  fl) 

2u2  +  2v2  =  13  P9 

Multiply  (7)  by  2,  2  u2  -  2  v2  -  1 2  (9) 

Add,  4u2  -  25 

.'.   M  =  ±  }. 

Subtract  (9)  from  (8),  4v2  =  1, 

v2  =  }. 
.\  v  =  ±  J. 
.-.  a==w  +  v  =  ±|±J  =  ±3or±2, 
and    y  =  u  —  v  =  ±\t £  =  ±2  or  ±3. 


TEACHERS     EDITION. 


311 


6. 

ay(*  +  y)-30  (1) 

a*  +  y3  =  35  (2) 

Substitute  u  +  v  for  .r,  and  u  —  v  for  y. 

(1)  becomes    (u+v)  (u—v)  {(u+v)  +  (u— v)}  =»  30, 

or  2«s-2m?«30  (3) 

(2)  becomes  (?*  +  vf  +  (w  —  vf  =  35, 

ot  2w8+6uv2=35  (4) 

Multiply  (3)  by  3,  6w8-6uv2  =    90  # 

\4)is  2ws  +  6w^=    35 

A.dd,  8w8 


250 


Substitute  value  of  u  in  (3), 

w     8  2 


-125 
2tt  =  5, 
u  =  l 
10v2 


30, 

250-40v2  =  240, 
40v2  - 10. 
.-.  v  =  ±£. 

.•.  a?  =  w  + v  =  3  or  2, 
and    y  =  u  —  v  =»  2  or  3. 


Exercise  XCIV. 


1.  a:-y  =  7 

«*  +  ay  +  y*  =  13 

Square  (1), 

x*-2xy  +  y2  =  49 
Subtract  (2)  from  (3), 

-3ay  =  36. 
Divide  by  -  3,    xy  =  - 12 
Add  (4)  and  (2), 

a;2  +  2a;y  +  y2  =  1. 
Extract  root,  x  +  y  =  ±  1 
Add  (5)  and  (1),  2  a  =  8  or  6. 
.\  <r  =  4  or  3. 
Substitute  value  of  x  in  (1), 

y=*-3or-4. 


(1) 
(2) 

(3) 


W 


(5) 


2.  a?2  +  xy  = 

jry  -  y2  = 

Substitute  i>a;  for 
From  (1),      z2  = 

From  (2),      a2  = 

•      35 
1  +  v 
35v2-29v  = 
4900v2-(  )+(29)2  - 

70v  = 

Substitute  value 

z2  = 

a?  = 

y  =  va;  = 


i1) 

(2) 
(3) 


35 

6 

5l 

1  +  v 

6 
v  — v2 

6 
i>  — t^ 
-6, 
1, 

±1, 

30  or  28, 

forf 

of  v  in  (3), 
^or25. 
±7Vj  or  ±5. 
±3V|or±2. 


312 


ALGEBRA.. 


X- 

Transpoae  in  (1), 


-12  =  0 
2y  =  5 
«y  =12, 
^      12 


(1) 

(2) 


Substitute  value  of  y  in  (2), 

x =  5. 

x 

Simplify,  ars-24  =  5a;. 

Transpose,      #  x*  —  5  x  —  24. 
Complete  the  square, 

4a*- () +  25  =  121, 
2s  -5-  ±11, 
,  2a;  =16  or  -6. 

.-.  a:  =  8  or -3. 
Substitute  value  of  x  in  (2), 

y=lJor-4. 


4.  ay-7  =  0  (1) 

a?+y2  =  50        (2) 
Transpose  in  (1),  xy  =  7  (3) 

Multiply  (3)  by  2, 

2xy  =  14        (4) 
Add  (4)  and  (2), 

a*  +  2  ay  +  y*  =  64. 

.\a;+y  =  ±8       (5) 
Subtract  (4)  from  (2), 
a*-23y  +  y*  =  36. 

.-.  a--y  =  ±6       (6) 
Add  (5)  and  (6),  2a;  =  ±14or±2. 
.\  a?  =  ±7or±l. 
Subtract  (6)  from  (5), 

2y  =  ±2or±14. 
.•.  y  =  ±lor±7. 


5. 


2a;-5y  =  9 
x*-xy  +  y*=7 

From(l),  v-^T* 

Substitute  value  of  x  in  (2), 

Simplify,  19y*  +  72y  =  -  53. 

Complete  the  square, 

1444y»  +  ( )  +  (72)»  =  1156, 
38y  +  72  =  ±34, 

38y  =  -38or-106. 
.\y  =  -lor-2}J. 
Substitute  value  of  y  in  (1),  2  x  =  9  +  (—  5)  or  9  +  (- 

.-.  x  =  2or  -Zfg. 


0) 
(2) 

(3) 


imr 


Transpose  (2), 
Square  (1), 
Multiply  (3)  by  4, 
Add, 

Extract  root, 
Add  (1)  and  (4), 

Subtract  (1)  from  (4), 


6. 

x  —  y  =  9 
ajy +  8  =  0 
sy  =  -8 
a?-2sy  +  y2  =     81 
ixy         =-32 
a?  +  2ay  +  y»  =     49 

x+y=± 7 
2a?=16or  2. 
.\  x  =  8  or  1. 

2y  =  -  2  or  -  16. 
.\  y  =  —  1  or  —  8. 


(3) 


W 


TEACHERS     EDITION. 


313 


7.                  5a;-7y  =  0          (1) 

5a;3_13^  =  4_7y2(2) 
4 

9.         a*  +  4sy  =  3                (1) 

4sy+y*  =  2i              (2) 
Substitute  vx  for  y, 

2^  +  4^  =  3                (3) 

From(l),                 x=?£ 
5 

SimpLfy  (2), 

20x2-13ajy  +  28y2  =  16 

(3) 

4ta»  +  tW  =  2t              (4) 
From  (3)  and  (4), 

l  +  4v 

Substitute  value  of  x  in  (4), 

«g£    ^'  +  28y'  =  16, 

1225y*  =  400, 
35y  =  ±  20. 

a*  -       9          (6) 
16v+4v*    v  ; 
.       3               9 
1  +  4v     16v  +  4v2 
48i;  +12va  =  9  +  36t;, 
12v2  +  12t>  =  9, 

-•.y-±f 

4t?+4v  =  3, 

Substitute  value  of  y  in  (1), 

4i;a  +  ()  +  l=4, 
2v  +  1  =  ±  2. 
.*.  v  =  £  or  —  1£. 

.-.  *-±f 

Substitute  values  of  v  in  (5), 

x  =  ±  1  or±V— f, 
and  y  =  ±£or=Fj\/-$. 

8.  *-y-l  (1) 

s»  +  y'  =  8J         (2) 

Square  (1), 

x*-2xy  +  y*  =     1       (3) 
(2)  is     s2  +  y2  =     8} 

Subt.,         -2zy         =  -  7J     (4) 

Subtract  (4)  from  (2), 

a^  +  2a^  +  y2  =  16. 
Extract  root,      x  +  y  =  ±  4       (5) 
Add  (5)  and  (1),       2  a;  =  5  or -3. 
,\  ic  =  2Jor—  1$. 
Subtract  (5)  from  (1), 

-2y  =  -3  or 5. 
.-.y  =  lJor-2}. 


10.     s2-a;y  +  y2  =  48  (1) 

a-y-8=0  (2) 

(l)isa2—  xy+y*  =     48 
Sq.(2),  x*-2xy+y*  =     64 

Subt.,       xy       =  - 16 
Multiply  by  3, 

3sy  =  -48       (3) 
Add  (3)  and  (1), 

x*  +  2xy  +  ya  =  0. 
Extract  root, 

x  +  y  =  0  (4) 

Add  (4)  and  (2), 

2x  =  8. 
.*.  a?  =  4. 
Subtract  (2)  from  (4), 
2y  =  -8. 
.-.  y  =  -4. 


AlA7l!iI)n&. 

11. 

x2  +  Sxy  +  y2  =  1 

(1) 

3x*  +  xy  +  3y2=13 

(2) 

Subtract  (1)  from  (2), 

2sa-2xy  +  2y2  =  12. 

Divide  by  2, 

x7-    xy+y2  =  6 

(3) 

(l)i8 

x*  +  3xy  +  y*=l 

(4) 

Subtract, 

-4rcy         =5 

Add  4  x  (1)  to  (4), 

4s2  +  8zy  +  4y»  =  9 

(5) 

extract  the  root, 

2a:+2y  =  ±3. 

Divide  by  2, 

x+y=±$ 

(6) 

Add  J  of  (4)  to  (3), 

^-2rcy  +  y*  =  ^, 

a;-y  =  ±  JV29 

(7) 

Add  (6)  and  (7), 

2a:  =  ±}±J\/29. 
.-.  a;  =  J(±3±\/29).    " 

Subtract  (7)  from  (6), 

2y  =  ±}T  }V29. 
.-.  y  =  J(±3=FV29.) 

12. 

x*-2xy  +  3y2  =  lf 

(1) 

a£  +  ay-y*  =  i 

(2) 

Substitute  vx  for  y. 

From  (1), 

a*-2va*-f-3vV=yk 

From  (2), 

x2  +  vx2  —  v232  =  J. 

Whence 

9-18v+27i* 

(3) 

and 

*■_           1 

(4) 

9H  9v-  9V2 

11             r           1 

9-18V+27V1     9+9V-9V2 

99  +  99„  _  99  tf  .  9  - 18  v  +  27 1*. 

-126v2  +  117tf  =  -90. 

Divide  by  —  9, 

14v2-13i;  =  10. 

Complete  the  square, 

784v»-()  +  169  =  729. 

Extract  the  root, 

28v-13  =  ±27, 

28^=40  or -14. 
...  Vs=^L0r  -J. 

Substitute  value  of  v  in  (4),                  x2  =  ^T  or  $. 

.-.  »  =  ±J\/TVor±f 

and    y  =  ±-yV^or  tJ. 

TEACHERS     EDITION. 


315 


13. 


x  +  y  =  a  (1) 

4ay  =  a2-4&2(2) 
Square  (1), 

s*  +  2:ry  +  y2  =  a2  (3) 

(l)ia  4zy         =  a2-462 


14. 


*-y  =  l  (1) 
2*.  (2) 
y  +  1. 


-  +  * 
y     a; 


Subt.,  re2  —  2  ay  +  y2  = 


462  (4) 


Extract  root,    a?  —  y  =  ±  2  b       (5) 
Add  (5)  and  (1),   2x  =  a±2b. 


Subtract  (5)  from  (1), 

2y=*a*2b. 


In  (1),  * 

Substitute  in  (2), 

y  +  1        y        13 
y        y  +  1      6* 
6y2  +  12y  +  6  +  6y2  =  13y2+13y, 

y2  +  y  =  6. 
Complete  the  square, 

4y2  +  ( )  +  1  =  25, 
2y  +  1  =  ±5. 
.-.  y  =  2or-3. 
,\  x  =  3  or  -2. 


Subtract  (2)  from  (1), 


15. 

^  +  9^  =  340  (1) 

7zy-y2  =  171  (2) 

x2  +  2zy  +  y2  =  169, 

s  +  y  =  ±13  (3) 

.-.  a;  =  13-y  or-(13+y). 
Substitute  in  (1)  the  first  value  of  a, 

(13  -  y)2  +  9  (13 -y)y  =  340, 
169  -  26y  +  y2  +  117y  -  9y2  =  340, 
8y2-91y  =  171. 
Complete  the  square, 

256y2-()  +  (91)2  =  2809, 
16y  -  91  -  ±  53. 

.-.y  =  9or2f  (4) 

Substitute  in  (1)  the  second  value  of  x, 

(13  +  y)2  -  9  (13 +y)y  =  340, 
169  +  26y  +  y2  -  117y  -  9y2  =  340, 

8y2  +  91y  =  -171. 
Whence,  y  =  -  9  or  -  2|. 

Substitute  values  of  y  in  (3),  x  =  ±  4  or  ±  lOf . 


316 


ALGEBRA. 


16.  s  +  y-6 

*»  +  2?  =  72 

Divide  (2)  by  (1), 

a^-sy+y'-  12 
Sq.(l),  a*+2sy+y*  -  36 
Subtract,  -3sy        -  -  24 

.-.ay  =  8 
Subtract  (5)  from  (3), 

x*-2ay  +  y*  =  4. 
Extract  root,  x  —  y  =  ±  2 
Add  (6)  and  (1),  2x  =  8  or  4. 
.\  x  =  4  or  2. 

Subtract  (6)  from  (1), 

2y  =  4  or  8. 

.\y  «2  or  4. 


18. 


8 


(5) 


(6) 


17.    3xy  +  2z+y  = 
3x-2y- 

In(2),  ■- 

Substitute  value  of  s 

6y2  +  7y  - 

Complete  the  square, 

144y2  +  ()+49  = 

12y  +  7  = 

12y- 

.-.  y  = 

Substitute  value  of  y 

3  = 


485 
0 

2* 
3 

in  (1), 

485, 
1455. 


(1) 
(2) 

(3) 


34969, 

±187. 

180  or -194. 

15  or  -  16J. 

in  (3), 

lOor-lOJ. 


a?-y=-l 
s»-y»  =  19 
Divide  (2)  by  (1), 

x*+  xy+y2  =  19 

Sq.  (1),  x*-2xy+yi=   1 

Subt.,  3xy       =  18 

.\xy  =  6 

Add  (4)  and  (3), 

a:2  +  2ay  +  y*  =  25. 
Extract  root,  x  +  y  =  ±  5 
Add  (5)  and  (1),  2x  =  6  or  -  4. 
.-.  *  =  3  or  -2. 
Subtract  (1)  from  (5), 

2y  =  4  or  —  6. 
.-.  y  =  2or  —  3. 


19, 


(2) 
(3) 


W 


(5) 


ar^  +  y8' 
x*-xy+y*  = 
Divide  (1)  by  (2), 

x  +  y=* 
Square  (3), 

x*  +  2xy  -f  y*  = 

Subtract  (4)  from  (2), 

-3a?y  = 

Divide  by  3,   —xy  = 

Add  (5)  and  (2), 

3*-2sy  +  y,= 

Extract  root,  x—y  = 

Add  (3)  and  (6),  2  x  - 

.\  x  = 

Subtract  (6)  from  (3), 

2y  = 


20. 


Square  (1), 
Subtract  (2), 

Subtract  (4)  from  (2), 
Extract  root, 
From  (1)  and  (6), 


x  +  y  =*a 

x*+y'  =  6* 

x*  +  2xy  +  y2  =  a1 

x%  +  y*  =  b* 

2xy         =  a*  -  6* 
x2-2x.y  +  y2  =  262--q2 

x  — y  =  ±^/262  — a2 
2x  =  a±*/2d2-a2. 
.-.z  =  i(q±«/2ff-a?). 

2y  =  ag:vr2&2  —  a2. 
.•.y  =  4(a^v^262  — a2). 


2728    (1) 
124      (2) 

22       (3) 

484      (4) 

-360. 
-120  (5) 

±2       (6) 
24  or  20. 
12  or  10. 

'20  or  24. 
.  10  or  12. 

(1) 
(2) 
(3) 

(5) 
(6) 

teachers'  edition.  317 

21. 

a?-y2  =  0  (1) 

3a?-4zy  +  5y2  =  9  (2) 

From  (1),  a?  -  y2. 

Hence,  in  (2),  3a?  ± 4a2  +  5a?  =  9, 

'  12a;2  or  4a?  =  9. 

.\  »  =  ±  jV3  or  ±§, 
and  y  =  ±  }  V3  or  ±  }. 


X  +  V  j  x-y     10 

(1) 

x-y     a;  +  y      3 

x2  +  y2  =  45 

(2) 

From  (1),       3  (a?+2a#+y2)+3  (a?-2a?y +y2)  =  10  a?  -  10y2, 

3a?  +  6a^  +  3y2 

+  3a?  -  Gxy  +  3y2  =  10a;2  -  10y2, 
-4a?  +  16y2  =  0, 

-a?  +  4y2  =  0 

(3) 

Add  (2)  and  (3), 

5y2  -  45. 

.\  y  =  ±3. 

n  (2),             j?  +  9  =  45. 

Substitute  values  of  y  i 

.\  x  =  ±6. 

23. 

1      !      K 

a;     y 
1      (      1     ^17 
a;  +  l     y  +  l  =  12 

Clear  of  fractions  and  unite,               x  +  y  =  5a?y 

(2) 

5a;  +  5y  =  7  —  I7xy 

Divide  (2)  by  (1), 

5  =  7-17ay 
5ay 
25a^  =  7-  17  xy 

(3) 

W 

42xy  =  7, 

sy  =  J 

(5) 

From  (1), 

a  +  y  =  f 

(6) 

Square  (6), 
Multiply  (5)  by  4, 

a?  +  2xy  +  y2  =  f$ 

Subtract, 

a?  -  2xy  +  y2  =  ^ 

a;-y  =  ±i 

(7) 

Add  (7)  and  (6), 

2a;  =1  or  J. 

.-.  a;  =  J  or  J. 

2y  -  J  or  1. 

Subtract  (7)  from  (6). 

-•-J/-Jor|. 

318  ALGEBRA. 


8 


24. 

ar»-sy  +  y2  =  7  m 

.T*+a^»  +  y*  =  133  (2) 

Divide  (2)  by  (1),  x*  +  xy  +  y2  =  19 

Subtract  (1 )  from  (3),  2  zy  =  12. 

.\zy  =  6  (4) 

Add  (4)  to  (3),  and  Bubtract  (4)  from  (1), 

x*  +  2xy  +  y2  =  25, 
a*-2asy +y*«l. 
Whence  as  +  y  =  ±  5, 

a?  — y  =  ±  1. 

.•.  jt  =  ±3  or  ±  2, 
and  y  =  ±  2  or  ±  3. 


25. 

*  +  y  =  4  (1) 

z*  +  y*  =  82  (2) 

Put  u  +  v  for  a,  and  u  —  v  for  y . 

(1)  becomes  2u  =  4. 

.-.  u  =  2. 

(2)  becomes  u*  +  6uV  +  tr*  =  41  (3) 
Substitute  2  for  u  in  (3),       16  +  24  v8  +  «•  =  41, 

v* +  24^  =  25. 
Complete  the  square,  *>*  +  ()  +  144  =  169, 

*  +  12-±13, 

*  =  1  or  -  25. 
.\  v  =  ±  1  or  ±V-25. 
.-.  a;  =  3,  1,  or  2±V^25, 
and    y  =  l,  3,  or  2  tV-  25. 


Divide  (1)  by  (2), 
Square  (2), 

Subtract, 

Add  (3)  and  (4), 
Extract  the  root, 
Subtract  (2)  from  (5), 


Add  (2)  and  (5), 


x*  —  y8  —  a* 

(2) 

x-y^a 

x*  +    xy  +  y2  =  a2 

(3) 

x2  —  2ay  +  y2  =  a2 

3ay         =0 

<cy  =  0 

W 

x2  +  2sy  +  y2  =  a2. 

x  +  y  =  ±a 

(5) 

2y  =  0or-2a. 

.«.  y  =  0  or  — a. 

2a;  =  2a  or  0. 

,%x  =  aor  0. 

TEACHERS     EDITION. 


319 


27. 

x2  —  xy  = 
xy  —  yl  = 
Subtract  (2)  from  (1),  x2  -  2  xy  +  y%  = 

Extract  root,  %  —  y  = 

(l)is  .        x(x-y)  = 

Substitute  value  of  x  —  y  in  (1),  ±  x[a  —  o)  = 

,\  x  = 
y(x-y)  = 

±y(a-b)  = 
.-.  y  = 

28. 


a5 


(2)  is 

Substitute  value  of  (a;  —  y)  in  (2), 


*  +  & 

2ab  (2 

a2-2ab  +  b2. 
±(a-b). 
a2  +  62. 
.  a2  +  &>. 

a  — 6 
2a&. 

2a6. 
.   2ab 


x*-y*- 
xy-- 

y- 

(a2-b2)2 


In  (2), 

Substitute  value  of  y  in  (1),     x2  - 

x*-a*  +  2a2b*"-b*  = 

x*  —  4  afor2  = 

Complete  the  square,  «*  —  ()  +  4a2£>2  = 

Extract  root,  a2  — 2a&  = 

z2  = 

.\  x  = 

Substitute  value  of  x  in  (1),     [a  +  bf  —  y2  = 

y2  = 

.-.  y- 


■Aab 
a2-b2 
a2-b2 


(2) 


=  4a&, 

4a&r2, 

a*_2a2&2  +  64. 

a*  +  2a2&2  +  6*. 

±(a2  +  62), 

±(a2  +  2a&  +  62). 
-  ±  (a  +  6). 

4  aft, 

a2-2a&  +  62. 
.±(a-6). 


29.  ajy  = 

s2+y2: 
Multiply  (1)  by  2,  2a!y  = 
Add  (3)  and  (2), 

x2  +  2xy  +  y2  = 
Extract  root,  a;  +  y  = 
Multiply  (1)  by  4,  4zy  = 
Subtract  (6)  from  (4), 

x2  —  2xy  +  y2  = 

Extract  root,      x  —  y- 

Add(5)and(7),      2x- 

.\  x-- 

Subtract(7)from(5), 

2y  = 


=  0 
=  16 
=  0 

=  16 
=  ±4 
=  0 


(1) 
(2) 

(3) 

(5) 
(6) 


30. 


•y- 


=  16 

s±4       (7) 
■  ±  8  or  0. 
*  ±  4  or  0. 

=  0  or  ±  8. 
0  or  ±  4. 


a:2  +  xy  +  y2  = 
e4  -f  £2y2  +  y*  = 
Divide  (2)  by  (1), 

x2  -  xy  +  y2  = 

(1)  is    a2  +  xy  +  y*  = 

Subt.,     -2xy         = 

.\  —xy  = 

Add  (3)  and  (4), 

x2  —  2xy  +  y2  = 
Extract  root,  x—y  = 
Subtract  (4)  from  (1), 
x2  +  2zy  +  y2  = 
Extract  root,  x  +  y  = 
Add  (5)  and  (6),  2s  = 

Subt.(5)fr.(6),*2y  = 
.-.  y  = 


1. 
±1 


37 

481 


13     (3) 

37 
-24 
-12     (4) 


(5) 


49. 

±7  (6) 
±  8  or  ±  6. 
±  4  or  ±  3. 
±  6  or  ±  8. 
±  3  or  ±  4 


820  ALGEBRA. 


31. 

x*  =  ax  +  by  (1) 

y*  =  ay  +  bx  (2) 

If  x  =  0,  y  must  equal  0. 

If  a:  —  y,  and  does  not  equal  0,  then  x=*a  +  b,  and  y  =  a  +  b. 

If  a;  does  not  equal  y ,  subtract  (2)  from  (1),  and  divide  by  x  -  y. 
x  +  y  =  a  —  b  (3) 

Add  (1)  and  (2),  x*  +  f  =  a(x  +  y)  +  b(x  +  y). 

Substitute  a-  6  for  a;  +  y,  =  a(a -  b)  +  6(a  -  b). 

Thatis,  a8  +  ya  =  «2  -    fc2  (4) 

Square  (3),  x*  +  2sy  +  y»  =  a'-2a6  +    6*  (5) 

Subtract  (4)  from  (5),  2sy         -     -2aZ>  +  2&»  (6) 

Subtract  (6)  from  (4),  x*-  2xy  +  y»  =  a*  +2a6-3&«. 

Extract  root,  as  -  y  =  ±  y/a%  +  2.ab-$W  (7) 

Add  (7)  and  (3),  2  jc  =  a  -  b  ±  Va*  +  2  aft  HSff. 

.-.  x -  }(a~ & ±VaJ  +  2a&-3P). 
Subtract  (7)  from  (3),  2y  -  a  -  6  t  Vaa  +  2a6-3y. 

.'.  y  =  \{a-b*y/a*  +  2ab-ZV). 


32. 

a._y_2  =  0  (1) 

15(rf-y")-16xy  (2) 

Transpose  (1),  a>-y  =  2  (3) 

Divide  (2)  by  (3),  15  (x  +  y)  =  8  ay, 

15a;  +  15y-8;ry  =  0  (4) 

From(l),  z  =  y+2. 

Substitute  value  of  x  in  (4), 

15y  +  30  +  15y  -  8y*  -  16y  -  0, 
8y*-14y  =  30. 
Complete  the  square,  64y2-  ( )  +  (7)*  =  289, 

8y-7  =  ±17, 

8y  =  24or-10. 
.-.  y  =  3or-lJ. 
Substitute  value  of  y  in  (1).  .'.  x  =  5  or  f. 


teachers'  edition. 

321 

33. 

x  +  y     jc-y_89 
x  —  y     x  +  y     40 

(1) 

6a;  =  20y+9 

(2) 

Simplify  (1), 

9a*-169y2  =  0, 

9z*  =  169y2 

(3) 

Extract  the  root, 

3s  =  ±13y, 
3a?Tl3y  =  0. 

Multiply  by  2, 

ex*26y  =  0 

W 

Transpose  in  (2), 

6x-20y  =  9 

(5) 

Subtract  (4)  from 

(5), 

6y  =  9, 
or  -46y  =  9. 

.-.  y  =  lJor-A. 

Substitute  values  of  y  in  (2),  x  =  6}  or  -  2^. 

34. 


a     b 

a) 

a     b      . 

-  +  -  =  4 
x     y 

(2) 

Simplify  (1), 

bx  +  ay  —  ab 

(3) 

Simplify  (2), 

bx  +  ay  =  4ay 
.\  4a?y  =  a&, 

andy  =  -^- 
y     4a; 

(4) 

Substitute  value  of  y  in 

(3). 

4a; 

Simplify, 

4a£  +  a*  =  4  ax. 

Transpose, 

4a?  — 4aa?  =  — a2. 

Complete  the  square, 

4a»-()  +  a»  =  0. 

Extract  the  root, 

2a; -a  =  0. 

a 

.•.  a;  =  -- 

2 

Substitute  value  of  x  in 

(3), 

-i 

322  ALGEBRA. 

35. 

s2  +  y2  =  7  +  xy  (1) 

a»  +  y»-6xy-l  (2) 

Transpose  ay  in  (1),  a2  —  ay  +  y*  =  7  (3) 

Divide  (2)  by  (3),  <c  +  y  -  6ay"""1- 

Simplify,  7a?  +  7y  =  6sy  -  1  (4) 

Put  u  +  v  for  re,  and  u  —  v  for  y,  in  (4), 

7(u  +  t>)  +  7(u-v)  =  6(u*-T*)-l, 
6m»-6^-14w  =  1  (5) 

Put  u  + 1>  for  a\  and  w  —  v  for  y,  in  (3), 

(u +  vf -(&-<*)  + (u-vf-7, 

u*  +  3v»  -  7  (6) 

Multiply  (6)  by  2,  2u»  +  6t*  =  14  (7) 

Add  (5)  and  (7),  8  m*  -  14  u  =  15. 

Complete  tbe  square,     256  u»  -  (  )  4-  (14)*  =  676. 
Extract  the  root,  16  w  -  14  =  ±  26, 

16w  =  40or-12, 
u  =  {  or  —  }. 
Substitute  J  for  u  in  (6),  ^  +  3  v*  =  7, 

3*-f, 

+  -i- 

Extract  the  root,  v  =  ±  }. 

Substitute  -  f  for  u  in  (6),  -ft  +  3  v2  =  7, 

Extract  the  root,  v  =  ±  J  V1}*. 

Since  x  =  m  + 1>,  substitute  J  for  ti  and  ±  J  for  i>, 

»-§  +  (*». 

x  =  3  or  2. 
Substitute  value  of  —  }  for  w,  and  ijV1}1  for  v,         

ar  =  l(-3±Vi^). 
Since  y  =  u  —  v,  substitute  f  for  u,  and  ±  J  for  v} 

y-i-(±». 

y=2or  3. 
Substitute  —  f  for  w,  and  ±  jV^J*  for  v, 

y  —  i-(±y/m, 
y-*(-«TVSj»). 


TEACHERS1    EDITION.  323 


36. 

a*-y5  =  3093 

(1) 

x-y  =  S 

(2) 

Let  x  =  u  +  vt  and  y  =  u  —  v. 
From  (2),  u  +  v  —  u  +  v=3, 

2v  =  3. 

From  (1),   u5+5w4i;+10ttV+10tt2ys+5ttv4+t£ 

-(m5-5u*i;+10wV-10mV+5mv*-^)  =  3093. 
Transpose  and  combine,      10u*i>+20uV+2v*  =  3093. 

Substitute  value  oft,,      5^  +  5^3?  +  ?*5  _  3098. 

2  8  16 

Simplify,  240u4  +  1080u2  =  49245. 

Divide  by  15,  16u*  +  72u2  =  3283. 

Complete  the  square,  16  w4  +  ( )  +  81  =  3364. 

Extract  the  root,  4  u2  +  9  =  ±  58, 

u2  -  ^  or  -  3£ . 
.-.  u  =  ±}or  ±  JV-67, 
x  =  u  +  v  -  5,  -2,  or  £(3±V^67), 
y  -  u  -  v  =  2,  -5,  or  J(-3±  V^67). 

37. 

l(.-l)-*(*  +  l)(y-l)--ll  (1) 

i(y  +  2)  =  J(*  +  2)  (2) 

From  (1),    9  a;  -  9  - 10 xy  -  lOy  +  10a;  + 10  -  -  165, 

or  19a;  -  lOay  -  lOy  -  - 166  (3) 

From  (2),  4y  +  8  =  3a>  +  6. 

3a;-2 


4 
Substitute  value  of  y  in  (3), 

19,-10  J§f^-io(5til)--ie6b 

76a;  -  30a?  +  20a;  -  30  s  +  20  =  -  664, 
-303?  + 66  a;  =  -684, 
5a? -11a;  =144. 
Complete  the  square,  100a2  -  (  )  +  (ll)2  =  2401, 

10a; -11  =  ±49, 

10a;  =  60  or -38. 
.-.  g»6or-3f 
Substitute  values  of  x  in  (4),  y  =  4  or  —  3^. 


(4) 


324  ALGEBRA. 


38. 

10a*+15*y  =  3aft-2a» 

(1) 

10y*  +  15xy  =  3ab-2b* 

(2) 

Let 

ux  =  y. 

(1)  becomes 

10x*+  15a*i  =  3aft-2a*f 

^     3aft-2a* 

10  +  15u 

(3) 

(2)  becomes 

lOxV  +  15x*u  =  3  aft  -  26s, 

10u»+15u 

W 

Eau&te  values  o 

f^           3aft-2a*_  3aft-2ft» 

n  10+15U      10u»+15u 

Simplify, 

30aftu*  -  20a»u*  -  30o«u  +  30ft*u  =  30  aft  -  20  ft*. 
Divide  by  10, 

3aftu*  -  2a*u*  -  3a*u  +  Situ  =  3  aft  -  26*, 
or  uVSaft^a'J-Sttfa'-ft^Saft^ft*. 
Complete  the  square, 

4m,(3o6  -  2a*f  -  (  )  +  9(a*  -  ft')1  =  9a*-24asft+34a*ft*-24afts+9&« 
Extract  the  root, 

2u(3oft  -  2a*)  -  3(a*  -  5*)  =  ±(3a*  -4aft  +  3ft8), 

2w(3aft  -  2a8)  -  6a*  -  4aft  or  4aft  -  66*. 
3a -2ft  ft 

3ft-2a  a 

Substitute  value  of3a~~2&  for  u  in  (3), 
3ft -2a 


Extract  the  root, 


25 


Substitute  -  -  for  w  in  (3),  a1  =  -  -• 

a  5 


s  =  ±a 


^1 


Q.  3a-2ft  v  A  /3ft-2a\ 

Since  u*  =  y,  y  =  ___  x  ±  ^_ __j. 


3ft-2a 
3a-2ft 


.\  y  =  ±- 

ory  =  --x(±aV=~|). 
.•.  y  =  =FftV-J. 


teachers'  edition.  325 


Exercise  XCV. 

1.  If  the  length  and  breadth  of  a  rectangle  were  each  in- 
creased by  1,  the  area  would  be  48 ;  if  they  were  each  diminished 
by  1,  the  area  would  be  24.    Find  the  length  and  breadth. 

Let    x  =  length  of  rectangle,  r 

and        y  =  width  of  rectangle. 


Then                               (a  +  l)(y  +  1)  =  48 

(i) 

and                                    (a;-l)(y-l)  =  24 

(2) 

Simplify  (1),                 sy  +  a;  +  y  +  l=48 

(3) 

Simplify  (2),                  xy-x-y  +  1  =  24 

Add,                            2xy               +2  =  72 

xy  =  35 

(4) 

Substitute  value  of  xy  in  (3), 

35  +  x  +  y  +  1  =  48, 

a  +  y=12 

(5) 

Square  (5),                      a2  +  2xy  +  y2  =  144 

Subtract  4  X  (4),                     4xy         =  140 

x2  —  2xy  +  y2  =     4 

Extract  the  root,                          x  —  y  =  ±  2 

(6) 

From  (5)  and  (6),                               x  =  7  or  5, 

y  =  5  or  7. 

2.  The  sum  of  the  squares  of  the  two  digits  of  a 

number  is 

25,  and  the  product  of  the  digits  is  12.    Find  the  number. 

Let    x  =  digit  in  tens'  place, 

and       y  =  digit  in  units'  place. 

x*  +  y8  =  25 

a) 

ay  =  12 

(2) 

Multiply  (2)  by  2,                         2xy  =  24 

(3) 

Add  (3)  and  (1),              a:2  +  2sy  +  y*  =  49. 

Extract  the  root,                         x  +  y  =  ±  7 

W 

Subtract  (3)  from  (1),      a;2  -  2a;y  +  y2  =  1. 

Extract  the  root,                         x  —  y  =  ±  1 

(5) 

From  (4)  and  (5),                              2  a;  =  ±  8  or  ±  6. 

.\  a?  =  ±4  or  ±3, 

y  =  ±  3  or  ±  4. 

Hence,  the  required  number  is  43  or  34. 

326  ALGEBRA. 


8 


3.  The  sum,  the  product,  and  the  difference  of  the  squares  of 
two  numbers  are  all  equal.    Find  the  numbers. 

Let  x  +  y  =  one  number, 

and  x  —  y  =  the  other  number. 

Then  2x  =  the  sum  of  the  numbers, 

a*  —  y*  —  the  product  of  the  numbers, 
and  4xy»=  the  difference  of  the  squares. 

2x  =  a?  -  y2 
z*-y*  =  4ay 

Transpose  in  (1),    xt  —  2x  —  y2  =  0 

Transpose  in  (2),    a*  —  4xy  —  y2  =  0 

Subtract,  2x  — 4xy  =0 

l-2y  =  0, 
2y  =  l. 

Substitute  value  of  y  in  (1),  2x  =  x%  —  \1 

x*-2x  =  \. 
Complete  the  square,  x*  —  2  a;  +  1  =  $. 
Extract  the  root,  x  —  1  =  ±  J  V5. 

.-.  «=l±iV5, 
a;  +  y  -  f  ±  £  V5  or  J  (3  ±V5), 
*-y-J±jV5or  }(1±VB). 


4.  The  difference  of  two  numbers  is  f  of  the  greater,  and  the 
sum  of  their  squares  is  356.    What  are  the  numbers? 

Let  x  =  greater  number, 

y «-  lesser  number, 
and  x  —  y  =  difference  of  the  numbers. 

Then  *-y  =  T  W 

and  sc'+y'-SSe  (2) 

Simplify  (1),  8*-8y  =  3*. 

.•.«-2*  (3) 

5 
Substitute  value  of  x  in  (2),  ^1  +  y*  =  356. 

Simplify,  64y2  +  25y2  =  8900, 

89y2  =  8900, 
y2  =  100. 

Extract  the  root,  y  =  ±  10. 

Substitute  value  of  y  in  (3),  bx  =  ±  80. 

.-.  a;  =  ±16. 


I 

i 


teachers'  edition.  327 

5.  The  numerator  and  denominator  of  one  fraction  are  each 
greater  by  1  than  those  of  another,  and  the  sum  of  the  two  frac- 
tions is  1^ ;  if  the  numerators  were  interchanged  the  sum  of  the 
fractions  would  be  1J.    Find  the  fractions. 

Let  -  =  one  fraction, 

y 

r  4-  1 

and  =  the  other  fraction. 

y  +  1 

Then  -  +  ^  =  T?  « 

y    y  + 1    12 

and  £±i  +  _^.  =  §  (2) 

y      y  +  i    2 

Simplify  (1),  I2xy+\2x+\2xy+\2y  =  17y2  +  17y. 
Simplify  (2),      2xy+2y+2x+2+2xy  =  3y2  +  3y. 
Transpose  and  combine, 

_17y2  _  5y  +  24ary  +  12a;  =  0  (3) 

-  3y2  -  y  +  4a;y  +  2a;  =  -  2  (4) 

Multiply  (4)  by  6, 

-  18y2  -  6y  +  24sy  +  12a  -  -  12  (5) 

Subtract  (5)  from  (3),  y2  +  y  - 12  (6) 

4y2  +  ()+l  =  49, 
2y  +  1  =  ±  7, 
2y  =  6or-8. 
.*.  y  =  3  or  —4. 

Substitute  3  for  y  in  (1),    ~  +  £±i  -  i~ 

Simplify,  ,  4a;  +  3a;  +  3  =  17, 

7a;  =  14. 

.-.  a;  =  2. 

Hence,  the  fractions  are  §  and  J. 


6.  A  man  starts  from  the  foot  of  a  mountain  to  walk  to  its 
summit.  His  rate  of  walking  during  the  second  half  of  the  dis- 
tance is  £  mile  per  hour  less  than  his  rate  during  the  first  half, 
and  he  reaches  the  summit  in  5}  hours.  He  descends  in  3| 
hours,  by  walking  1  mile  more  per  hour  than  during  the  first 
half  of  the  ascent.  Find  the  distance  to  the  top  and  the  rates  of 
walking. 


328  ALGEBRA. 


Let  2x  =  distance, 

and  y  =  rate  at  first. 

Then  -  =  number  of  hours  he  was  walking  1st  half, 

y 

and  — —  =  number  of  hours  he  was  walking  2d  half. 

y-i 

Hence,  £  +  _£_  =  5}.  (1) 

y    y-i 

Also,  -^r=3}  (2) 

y  +  i 

Clear  (1)  of  fractions,  4  xy  —  2  x  +  4  xy  =  22  y*  — 11  y, 

22y*-8ay+2a?-lly  =  0  (3) 

Clear  (2)  of  fractions,  8  x  =  15y  +  15. 

.  r  -  lgy  +  15     m 

Substitute  value  of  a;  in  (3),  "  " x  g  W 

22y*_8y(^^y2(l^)-lly  =  0, 

176y»  -  120ys  -  120y  +  30y  +  30  -  88y  =  0, 

56y*-178y  =  -30. 
Complete  the  square,  3136y8  -  (  )  +  (89)»  =  6241. 
Extract  the  root,  56y  -  89  -  ±  79. 

.-.y-s. 

Substitute  value  of  y  in  (2),  ^  =  — . 

4       4 

2a:  =  15. 

Hence,  the  distance  is  15  miles ;  and  the  rates  of  walking,  3, 

2},  and  4  miles. 

7.  The  sum  of  two  numbers  which  are  formed  by  the  same 
two  digits  in  reverse  order  is  $f  of  their  difference ;  and  the  dif- 
ference of  the  squares  of  the  numbers  is  3960.  Determine  the 
numbers. 

Let  x  =  digit  in  ten's  place, 

and  y  =  digit  in  unit's  place. 

Then  10  a;  +  y  =  first  number, 

10  y  -f  x  =  second  number, 
llaj-flly^  sum  of  the  numbers, 
9x  —  9y  =  difference  of  the  numbers. 
(10a:  +  y)2  -  (x  -f  10y)3  =  difference  of  the  squares. 

.*.  lla:  +  lly  =  ft(9a;-9y)  (1) 


teachers'  edition.  329 

and  (10a  +  yf  -  {x  +  10y)2  =  3960  (2) 

Simplify  (1),  s+y  =  -5a;~5y, 

7y-3s  =  0  (3) 

Substitute  value  of  x  in  (2), 

#  «!  =  3960, 

y*  =  9. 

.-.  y  =  ±3. 
From  (3),  3.u  =  7y. 

.■.  x  =  ±7. 
Hence,  the  numbers  are  73  and  37. 


8.  The  hypotenuse  of  a  right  triangle  is  20,  and  the  area  of 
the  triangle  is  96.    Determine  the  sides. 

Let  x  =  longer  side, 

and  y  =  shorter  side. 

Since  sum  of  squares  on  sides  equals  square  on  hypotenuse, 
x2+ya  =  400  (1) 

Since  area  of  triangle  equals  one-half  product  of  sides, 

^  -  96  (2) 

xy  =  192. 
Multiply  (2)  by  2,  2xy  -  384  (3) 

Add  (1)  and  (3),      x2  +  2xy  +  y*  =  784. 
Extract  the  root,  x  +  y  —  ±  28  (4) 

Subtract  (3)  from  (1),  x2  -  2xy  +  y2  =  16. 
Extract  the  root,  x  —  y  =  ±  4  (5) 

From  (5)  and  (4),  2x  -  ±  32  or  ±  24. 

.-.  a;  =  ±16  or  ±12. 
2y  =  ±24or  ±32. 

.-.  y  =  ±12or  ±16. 

Hence,  the  sides  are  16  and  12. 


330  ALGEBRA. 


9.  Two  boys  run  in  opposite  directions  round  a  rectangular 
field  the  area  of  which  is  an  acre ;  they  start  from  one  corner 
and  meet  13  yards  from  the  opposite  corner ;  and  the  rate  of  one 
is  $  of  the  rate  of  the  other.  Determine  the  dimensions  of  the 
field. 

Let  x  =■  length  of  first  side, 

and  y  —  length  of  second  side. 

x  +  y  +  13  =  number  of  yards  one  boy  runs, 
x  +  y  —  13  «=  number  of  yards  the  other  boy  runs. 
s  +  y-13  =  $(a  +  y  +  13). 
.\  6x  +  6y  -  78  =  5x  4-  5y  +  65,  • 

and  x  +  y  =  143  (1) 

xy  =  area  of  field  of  one  acre. 
(Since  4840  Bq.  yds.  =  1  acre), 

xy  -  4840  (2) 

Square  (1),        x*  +  2xy  +  y*  =  20449 
(2)  x  4  is  4ay      19360 

a*-2xy+y*=    1089 

x  -  y  =  ±  33  (3) 

From  (1)  and  (3),  2a? «  176  or  110. 

.-.  a;  =  88  or  55. 

2y=110or  176. 
.-.  y  =  55or  88. 
Hence,  the  dimensions  are  88  yds.  by  55  yds. 


10.  A,  in  running  a  race  with  B,  to  a  post  and  back,  met  him 
10  yards  from  the  post.  To  make  it  a  dead  heat,  B  must  have 
increased  his  rate  from  this  point  41  f  yards  per  minute ;  and  if, 
without  changing  his  pace,  he  had  turned  back  on  meeting  A,  he 
would  have  come  4  seconds  after  him.  How  far  was  it  to  the 
post? 

Let  x  —  number  of  yards  to  the  post. 

Then        2x  =  number  of  yards  to  the  post  and  back. 

Let  y  =  number  of  yards  A  runs  per  minute. 

Then       —  =  number  of  minutes  A  is  running  the  race. 

y 

B  runs  (x  —  10)  yards  while  A  is  running  (a;  +  10)  yards. 

Hence,  B  runs  ^-^ —  of  v  yards  **  %2Lz — U  yards  per  minute. 
x  +  10      *  J  aj  +  10'  * 


teachers'  edition.  331 

A  has  (a;  — 10)  yards  to  run  when  B  meets  him ;  and,  as  he  runs 
y  yards  per  minute,  it  will  take  him  ?-Z —  minutes  to  finish  the 
race. 

B  has  (x  +  10)  yards  to  run;  and,  if  he  increases  his  pace  41f 
yds.  per  min.,  he  will  be  running  at  the  rate  of  I  **&  ~  &  +  41f  J 
yards  per  minute ;  and,  as  he  has  (x  +  10)  yards  to  run,  it  will 

take  him  (x  +  10)  -s-  (  xV~l  V  +  41*  J  minutes  to  finish  the  race. 

\   x  +  10  J 

But  this  change  of  rate  will  make  it  a  dead  heat ;  therefore, 

(*+M^+41*)=*-f        (1) 

Since  4  seconds  =  ^  minute,  B,  without  changing  his  rate, 
will  be  -j^  of  a  minute  longer  than  A  in  running  the  (jc— 10) 
yards  which  A  has  to  run  when  he  meets  B  ;  therefore, 

(g-io)  +  fo-10yV*-10-JL  (2) 

V  '     \   x  + 10   )  y         15  K) 

a-       vr    /o\  a;+10      2-10        1  /oX 

Simplify  (2), -  —  (3) 

y  y  lb 

.-.y  =  300. 

Simplify  (1), 

(      iqn      /7y(»-10)+290ap  +  2900\     ac-10 

{X+     >'{  7(»  +  10)  )         y    ' 

7  (a;  +  10)' a- 10 

7y(a;-10)  +  290a;  +  2900  =      y 

Substitute  300  for  y,     ^*  +  ™Z-*=™ 
y      2390a; -18100        300 

210a?  +  4200a;  +  21000  -  239  x«-  4200  x  4- 18100, 

29  a?-  8400  a;  =  2900, 

^_  8400a  =1(X) 

29 

*-(  )  +  (*«*)■- (M**)1. 

.-.  x  =  290  or  -£$. 
Hence,  the  distance  to  the  post  was  290  yards. 


332  ALGEBRA. 


11.  The  fore  wheel  of  a  carnage  turns  in  a  mile  132  times 
more  than  the  hind  wheel ;  but  if  the  circumferences  were  each 
increased  by  2  feet  it  would  turn  only  88  times  more.  Find  the 
circumference  of  each. 

Let         x  =  circumference  in  feet  of  the  fore  wheel, 
and  y  —  circumference  in  feet  of  the  hind  wheel. 

Then  5*2-5*2_im  (1) 

x  y 

52§P_5280  =  88  (2) 

x  +  2     y  +  2  w 

Simplify  (1),  5280y  -  5280a;  =  132xy. 

Divide  by  132,  40y  -  40a:  =  xy  (3) 

Simplify  (2),  5280y+10560-5280a;-10560  -  88xy+176a;+176y+352. 

Divide  by  88,  60y  -  60a:  =  xy  +  2a:  +  2y  +4. 

Transpose  and  combine,  58  y  —  62  a:  =  xy  +  4  (4) 

(3)  is  40y-40x  =  zy 

Subtract,  18y-22x=  4 

2  +  11* 


y  =  - 


9 


Substitute  value  of  y  in  (3), 

Simplify,  80  +  440a;  -  360a;  =  2a:  +  11  a* 

11a? -78  a?  =80. 
Multiply  by  11,  121  x*  -  858  x  -  880. 

Complete  the  square,    121  x*  -  ( )  +  (39)*  =  2401. 
Extract  the  root,  1 1  x  -  39  =  ±  49, 

11a;  =  88  or  -10. 
.-.  a;  =  8or  -|f 
Substitute  8  for  x  in  (3).  .  \  y  =  10. 


12.  A  person  has  $  6500,  which  he  divides  into  two  parts  and 
loans  at  different  rates  of  interest,  so  that  the  two  parts  produce 
equal  returns.  If  trie  first  part  had  been  loaned  at  the  second 
rate  of  interest,  it  would  have  produced  $  180 ;  and  if  the  second 
part  had  been  loaned  at  the  first  rate  of  interest,  it  would  have 
produced  $  245.    Find  the  rates  of  interest. 


teachers'  edition.  333 

Let  x  =  number  of  dollars  in  one  part  of  the  capital, 

6500  —  z  =  number  of  dollars  in  the  other  part, 
and  y  =  return  from  each  part. 

y 

Then  -  =  rate  of  interest  on  first  part. 

Also.  „,.j[ —  =  rate  of  interest  on  second  part, 

6500  —  x  r 

x  (       3| J  =«  return  of  first  part  when  loaned  at  second  rate. 

•"(ssfc ,)  =  180  (1> 

y 
(6500  —  x)-  =  return  of  second  part  when  loaned  at  first  rate. 

.\  (6500-.x)|  =  245  (2) 

Simplify  both  equations  and  add, 

xy  =  1170000  -180a?  (3) 

-ay  =  +  245a;-6500y    (4) 

0  =  1170000+    65a;-6500y 

Transpose  and  divide  by  65, 

100y-a  =  18000  (5) 

1170000-180o: 
From  (3)  y  = 

Substitute  value  of  y  in  (5), 

100  (UTOOOO-IM^.,  18(X)0 

Simplify,  x1  +  36000*,  •=  117000000, 

x*  +  ( )  +  (18000)2  -  441000000. 
Extract  the  root,  x  +  18000  =  ±  21000. 

.-.  x  =  3000, 
and  6500  -  x  =  3500. 
From  (5),  y  -  210. 

•••1-0.07. 

and65bfc  =  0.06. 

Hence,  the  rates  of  interest  are  7%  and  6%. 


334 


ALGEBRA. 


Exercise  XCVI. 


1.        2*+lly  =  49. 
Transpose,     2  x  =  49  —  1 1  y . 

.-.  a:-24-5y  +  i^. 

Let  *-^  =  m, 

2 

l-y  =  2m. 
.-.  y  =  l  — 2m. 

Substitute  value  of  y  in  original 

equation, 
2* +  11 -22m  =  49. 

.-.  x  =19  +  11m. 
If  m  =  0,  x  -  19,  y  -  1. 
If     m  =  -l,  3  =  8,    y  =  3. 


2.  7s  +  3y  =  40. 

Transpose,     3y  =  40  —  7  a;. 
:13-2jb  + 


Let 


Lz£. 
3 

1-x* 


1-x 


-3  m. 
,\  x=  1  —  3m. 

Substitute  value  of  *  in  original 
equation, 

7-21m  +  3y  =  40, 

3y  =  21m  +  33. 

.*.  y  =  7m  +  11. 

If     m  =  0,      y  =  ll,a;=l. 

If      m  =  —  1,  y  =  4,    x  =  4. 


3.       5s  +  7y  =  53. 
Transpose,  5x  —  53  —  7y. 

...,«io-y+l^a! 

*_10  +  y  =  3^. 

5 

Multiply  by  3, 
3x-30  +  3y  =  2^^ 


=  l-y+izX 


Let 


i^a 


<m, 


4  — y  =  5m. 
.\  y  =  4  — 5m. 
From  given  equation, 

x  =  5  +  7m. 
If   m  =  0,      a;  =  5,    y  =  4. 


4.        a;  +  lOy  =  29. 
Transpose,     x  =  29  —  lOy. 
If  y  =  l,a;  =  19. 

If  y  =  2,  a:  =  9. 

If  y  =  3,a;  =  -l. 

.  \  y  can  only  =  1  or  2, 
x  can  only  =  19  or  9. 


TEACHERS     EDITION. 


335 


5.         3*+8y  =  61. 

3*  =  61-8y. 

.-.  *  =  20-2y+*^. 
.3 

*_20  +  2y  =  1-^i& 
*  3 

Multiply  by  2, 

2*-40  +  4y  =  ^i# 
y  3 

._,  +  Sjl. 

Let  ^  =  m, 

3 
2-y  =  3m. 
.-.  y  =  2-3m. 
Substitute  in  original  equation, 
3s  +  16  -24m  =  61, 

3a  =  45  +  24  m. 
.\  a;  =15  + 8m? 
If    ?n  =  0,       *=15,  y  =  2. 
If    m  =  — 1,  *=7,    y  =  5. 


7:      16*  +  7y  =  110. 

7y  =  110-16*. 

.•.y«15-2*  +  ?=??. 


Transpose, 

y  +  2*-15  = 
Multiply  by  4, 


5-2* 


4y  +  8*-60  = 


20-8* 


Let 


6- 


=2-*+^ 


=  m, 


6  —  *  =  7m. 
.\  *  =  6  —  7m. 
Substitute  in  original  equation, 
96~112m  +  7y  =  110, 

7y  =  14  + 112m. 
.\y  =  2  +  16?n. 
If         m  =  0,  x  =  6,  y  =  2. 


8*  +  5y  =  97. 

5y  =  97-8*. 

.-.  y  =  19— *  + 
2-2* 


2-3* 


y-19  +  *  = 
Multiply  by  2, 
2y-38  +  2*: 


4-6* 


=  -*  + 


4-* 


Let 


4-* 


.*.  *  =  4  — 5m. 

Substitute  in  original  equation, 
32-40m  +  5y  =  97, 

5y  =  65  +  40m. 
.\y  =  13  +  8m. 
If    m  =  0,      *  =  4,  y  =  13. 
If    m  =  -  1,  *  =  9,  y  =  5. 


8.     7*  +  10y  =  206. 

7*  =  206-10y. 


=  29- 


y+ 


3-3y 


Let 


*_29+y  =  ^l_^). 

l^  =  m. 

7 

.-.  y  =  1-  7m. 
Substitute  in  original  equation, 
7* +  10 -70m  =  206, 

7*  =196  + 70m. 
.-.  *  =  28  +  10m. 
If     m  ^0,       *  =  28,  y  =  1. 
If     ?n  =  -l,  *=18,  y  =  8. 
If     m  =  -  2,  *  =  8,     y  -  15. 


336 


ALGEBRA. 


9.  12x-7y-l. 

Transpose,        7y  =  12x-l 

Multiply  by  3, 


5s  -1 


11.    23y-13a;=3. 

Transpose,  13  a;  =  23  y  -  3. 
10y-3 

.*.  X  —  V  = *- 


3y-3a;  =  2a;  +  ^— -• 
Let  2L=J*  =  m. 

.-.  x  -7m  +  3. 


Multiply  by  4, 


13 


Let 


4a:  — 4y  =«3y  +  * 
V-12     _ 


-12 


13 


13 

.-.y- 13m +  12. 


Substitute  this  value  of  x  in  origi-  Substitute  this  value  of  y  in  orig- 
nal  equation,  inal  equation, 

84m  +  36  -  7y  -  1,  23(13m  +  12) -13*-  3, 

7y  -  35  +  84m.  13a.  =  299m  +  273. 

.-.  y  «  5  +  12m.  ^  x  _  23m  +  21. 


If 


m  =  0,  x  =  3,  y  =  5.  If        *»_<),  *-21,  y-12. 


10.         5a>-17y  =  23. 

5x  =  23  +  17y. 


,-.  s  =  4  +  3y+' 


.3±& 


3  +  2S 
5 


a._4_3y  = 
Multiply  by  3, 

3*-12-9y  =  l+y+^- 

Let  ±P*=rn. 

o 
Then  y  =  5m-4. 

Substitute  this  value  of  y  in  origi- 
nal equation, 
5a;-17(5m-4)  =  23, 
5a;-85?»  +  68  =  23, 

5a;  =  85m -45. 
.*.  x«=  17m  — 9. 
I!  «i-lf  *-8,  y=l. 


12.      23s-9y  =  929. 

9y  =  23a;-929. 

.-.y  =  2x-103+^. 

y-2*  +  103  =  ^p. 

Multiply  by  2, 
2y-4a;  +  206  =  a;+^— 

Let  =  m. 

9 
Then         x-4  =  9m. 

,\  a  — 9m +  4. 

Substitute  this  value  of  y  in  orig 
inal  equation, 

207m+92-9y  -  929, 

9y  =  207m -837. 

.-.  y  =  23m-93. 

Jf        m  =  5,  x-49,  y-22. 


teachers'  edition.  337 

13. 

23  a; -332/ =  43. 

23a:  =  33y +43. 

y  23 


Let  £-±_2  =  m. 

23 
Then  y  =  23m-2. 

Substitute  this  value  of  y  in  original  equation, 
23*-33(23m-2)  =  43, 
23  a; -759m +  66 -43, 

23jc  =  759m-23. 
.-.  a;=33m-l. 
If  m--lf  a  =  32,  y  =  21. 


14. 

555a:-22y  =  73. 

22y  =  555s-73. 

...y  =  25s-3+^l2. 

9  22 

Transpose.  y  -  25  a:  +  3  =  5a?~^ 

Multiply  by  9,  9y-225<c  +  27  =  2a;  +  2  +  ^=i?. 

Let  *=l?-m. 

22 

Then  *  -19  =  22m. 

.    .\a;=19  +  22m. 

Substitute  value  of  x  in  original  equation, 

555(19  +  22m)  -  22y  =  73, 

10545  +  12210m  -  22y  =  73, 

22y  =  10472  + 12210m. 

.\  y=»  476  +  555m. 

If  m-0,  x  =  19,  y  =  476. 


338  ALGEBRA. 

15.  How  many  fractions  are  there  with  denominators  12  and 

18  whose  sum  is  {§? 

Let  *.  +  .£_* 

12      18      36 

Simplify,  3a:  +  2y  =  25, 

2y  =  25-3s. 

Let  — — —  =  m. 

2 
Then  l-ar  =  2m. 

.-.  x=l-2m. 
Substitute  value  of  x  in  original  equation, 
3-6m  +  2y  =  25. 

.\y  =  ll  +  3m. 
If  to  =  0,      x  =  1,  y  =-  11. 

If  m  =  -l,  x  =  3,  y  =  8. 

If  to  ="-  2,  a;  =  5,  y  =  5. 

If  to  =  -3,  a?=7,  y  =  2. 

Hence,  the  pairs  of  fractions  are 

AH;  A.  A;  A,  A;  A  A 

16.  What  is  the  least  number  which,  when  divided  by  3  and 
*.  leaves  remainders  2  and  3  respectively? 

Let  n  =  number, 

^-2-*  (i) 

n-=^-y  (2) 

From  (1)  and  (2),  n  =  Sx  +  2  and  5y  +  3. 

.-.  3z  +  2  =  5y  +  3, 

3*  =  5y  +  l  (3) 


Transpose,  x  —  1 


y-f  2 


Multiply  by  2,  2a;  -  2  =  y  + 

Let  ^  =  m-    ^ 

Then  y  =  3m-2. 

From  (3),  3a -15  m -9. 

,\  a;  =  5m-3. 
If  *  m-1,  ar  =  2,  y=L 

But  n  =  3a;  +  2. 

.•.  n-8. 


teachers'  editiok.  339 

17.  A  person  counting  a  basket  of  eggs,  which  he  knows  are 

between  50  and  60,  finds  that  when  he  counts  them  3  at  a  time 

there  are  2  over ;  but  when  he  counts  them  5  at  a  time  there 

are  4  over.     How  many  are  there  in  all? 

•  .  "  t   *  n  — 2 

Let  =  x. 

3 

and  — -y. 

Then  n  =  2  4-  3a:  or  4  +  5y. 

.*.  2  +  3a;  =  4  +  5y, 

3aj  =  2  +  5y  (1) 

*        -*  +  *G±A 

Let  i-^-2  =  m. 

3 
Then  y-3m-l. 

Substitute  value  of  y  in  (1),  3  a;  =  2  +  5  (3  m  —  1), 
3a;  =  15m- 3. 
.*.  a;  =  5m  —  1. 
If  m  =  4,  a;  =  19,  y-11. 

Hence,  the  number  of  eggs  is  59. 

18.  A  person  bought  40  animals,  consisting  of  pigs,  geese, 
and  chickens,  for  $40.  The  pigs  cost  #5  apiece,  the  geese  $1, 
and  the  chickens  25  cents  each.  Find  the  number  he  bought  of 
each. 

Let  x  =  number  of  pigs, 

and  y  =  number  of  geese. 

Then  40  —  x  —  y  =  number  of  chickens. 

5*  +  y  +  10*- 2f-40  (1) 

4      4 
or  20a;  +  4y  +  40  -  x  -y  =  160, 
or  19a;  +  3y  =  120, 

3y  =  120-19a;  (2) 

y  =  40-6s-5. 
y  3 

Let  -  =»  m. 

3 
.*.  ic  =  3m. 
Substitute  value  of  x  in  (2),  3y  =  120  —  57  m. 

.-.  y  =  40-19m. 
If  m  =  1,  x  =  3,  y  =  21. 

If  m  =  2,  x  =  6,  y  =  2.  . 

Hence,  he  bought  3  pigs,  21  geese,  and  16  chickens ;  or  6  pigs, 
2  geese,  and  32  chickens. 


340  ALGEBRA. 


19.  Find  the  least  multiple  of  7  which,  when  divided  by  2,  3, 
4,  5,  6,  leaves  in  each  case  1  for  a  remainder. 

Let  7x  =  least  multiple  of  7, 

and  y  =  sum  of  quotients. 

Then 
7x-l  ,  7s-l     7x-l  ,  7x-l  .  7a; -1 
2~+~3~  +  ~lT+~5~+_6~=y- 
Simplify, 
210*  -  30  +  140*  -  20  +  105a:  -  15 

+  84*  -  12  +  70a:  -  10  =  60y, 
609x-60y  =  87. 
(1) 


Divide  by  3, 
Transpose, 

203s-20y  =  29 

-20y  =  -203x  +  29. 

.-.y-10*     l+3*~9. 
20 

y                            20 

Let 

20 
x-  3  =  20m. 
.-.  x  =  20m  +  3. 
n(l), 
+  609-20y  =  29, 

Then 

Substitute  value  of  x  i 
4060971 

20y  =  - 4060m -580    (2) 
.\y  =  203m  +  29  (3) 

If  m  =  2,  x  =  43,  y  =  435. 

Hence,  the  number  is  301. 

20.  In  how  many  ways  may  100  be  divided  into  two  parts,  one 
of  which  shall  be  a  multiple  of  7  and  the  other  of  9? 

Let  7ar  =  one  part, 

and  9y  =  the  other  part. 

.-.  7x  +  9y=100. 

7x  =  100-9y. 

■-.«-14-y  +  2(1-y). 

Let  "~y  =  m. 

7 
Then  1— y  =  7m. 

.-.  y  =  l  -7m. 
Substitute  value  of  y  in  the  original  equation, 
7a: +  9(1 -7m)  =  100, 

7a:  =  100 -9(1 -7m), 
7a;=91+63m. 
.-.  x  =13  +  9m. 
If  m  =  0,      s=13,  y  =  l. 

If  m«  —  1,  a:  =  4,     y  =  8. 

Hence,  the  parts  are  91  and  9,  or  28  and  72. 


teachers'  edition.  341 


21.  Solve  18  a; 

-5y 

~  70  so 

that  y  may  be  a  multiple  of  x,  and 

both  positive. 

18a 

-  by  -  70. 

Let 

y  =  mx. 

Substitute  value 

of  y  in  1 

this  equation, 

18s- 

-5ma;  =  70. 

ar(18- 

-5m)  =  70. 

18-5m 

and 

^70m_. 
y      18  -5m 

Now,  if  m  = 

■2, 

rr  =  ^  or  8}, 

and 

y«l^orl7J. 

And,  if  m  =- 

'3, 

*-  V  or23i. 

and 

y  -  *f*  or  70. 

22.  Solve  8ac+12y  =  23so  that  ac  and  y  may  be  positive,  and 
their  sum  an  integer. 

8a;  +  12y  =  23  (1) 

Let  x  +  y  =  m. 

Transpose,  at  =  m  —  y  (2) 

Substitute  value  of  x  in  (1), 

8m-8y  +  12y-23, 

4y  =  23-8m. 
.  23-8m 

••y        4 

Substitute  value  of  y  in  (1), 

8a;  +  69  -24m  =  23, 

8a;  =  24m -46. 

24m -46 
..... —  • 

Let  m  -  2. 

48-46     1 


Then  a;  = 

and  y  =- 


8  4 

23-16     7 


342  ALGEBRA. 

23.  Divide  70  iuto  three  parts  which  shall  give  integral  quo- 
tients when  divided  by  6,  7,  8,  respectively,  and  the  sum  of  the 
quotients  shall  be  10. 

Let  x  =  first  part, 

y  =  second  part, 
and  70  —  x  —  y  =  third  part. 

6      7  8  W 

Simplify, 

24a:  +  24y  +  1470  -  21  x  -  21y  =  1680, 

7*  +  3y  =  210  (2) 

3y  =  210-7a;. 

.\  y  =  70-2z--- 
y  3 

Let  §  =  w-   * 

3 

.*.  x  =  3m. 

Substitute  value  of  m  in  (2), 

21m  +  3y  =  210, 

3y  =  210-21m. 

.\y  =  70-7m. 

If  m  =  2,  4,  6,  8, 

(the  lowest  values  that  will  produce  multiples  of  the  numbers), 

a=   6,  12,  18,  24, 

y  -  56,  42,  28,  14, 

70-a;-y  =    8,  16,  24,  32. 

24.  Divide  200  into  three  parts  which  shall  give  integral  quo- 
tients when  divided  by  6,  7,  11,  respectively,  and  the  sum  of  the 
quotients  shall  be  20. 


Let 

a;  = 

<=  first  part, 

and 

y  = 

=  second  part. 

Then 

200-a:-y  = 

=  third  part, 

E  +  3U 

200-ar-.y_ 

=  20. 

5     7 

11 

Simplify, 

77  a:  +  55y 

+  7000 

-35x  —  35y  = 

=  7700, 

42a;  +  20y  = 

=  700, 

21a;  +  10y  = 

=  350 

.-.  y  = 

=  35_2a:-A. 
10 

(1) 


Let  —  =  m, 

10 

a:  =  10m. 
Substitute  value  of  a;  in  (1),    y  =  35  —  21  m. 
If  m  =  1,  x  =  10,  y  =  14. 

200-a:-y=176. 


teachers'  edition.  343 


25.  A  number  consisting  of  three  digits,  of  which  the  middle 
one  is  4,  has  the  digits  in  the  units'  and  hundreds'  places  inter- 
changed by  adding  792.    i1ind  the  number. 

Let  x  =  digit  in  hundreds'  place, 

and  y  =  digit  in  units'  place. 

.•.  100 x  -f  40  +  y  —  the  number. 

lOOy  +  40  +  x  =  792  +  100a;  +  40  +  y. 
Transpose  and  combine, 

99y-99a:=792. 
Divide  by  99,  y  -  x  =  8  (1) 

y  =  x  +  8. 
Let  x  +  8  =  m, 

x  =  m-S  (2) 

and  y  =  m. 

From  (2),  m  must  be  equal  to  9,  in  order  to  make  x  positive. 
Then  x  =  1, 

Hence,  the  number  is  149. 


26.  Some  men  earning  each  $2.50  a  day,  and  some  women 
earning  each  $1.76  a  day,  receive  all  together  for  their  daily 
wages  $44.75.  Determine  the  number  of  men  and  the  number 
of  women. 

Let  x  =  number  of  men, 

and  y  =  number  of  women. 

Then  5£  +  l2  =  I7§, 

2        4        4 
10a+7y-179, 

y  =  25-x  +  l=l£ 

4  _  Q.B 

Transpose,  y  —  25  +  x  =  - — — • 

Multiply  by  5,  5y-  125  +  5z  =  2-  2a?  +  6~x 


Let  6  — 


7 
x 

-  =  m 

7 


x  =  6  —  7m. 

Substitute  6-  7 m  for  x  in  the  original  equation, 
60-70m  +  7y  =  179, 

7y  =  119+70m. 
.-.  y=  17  +  10m. 
If  w  =  0,      x  =  6,    y  =  17. 

If  m  =  - 1,  x  =  13,  y  =  7. 


344  ALGEBRA. 


27.  A  wishes  to  pay  B  a  debt  of  £1  12  s.,  but  has  only  half- 
crowns  in  his  pocket,  while  B  has  only  four-penny  pieces.  How 
may  they  settle  the  matter  most  simply? 

Let  x  =  number  of  half-crowns, 

and  y  =  number  of  four-penny  pieces. 

Then  half-crowns  =  30  x  pence, 

and  four-penny  pieces  =  4y  pence. 

£1  +  \2b.  =  384  pence. 
But  £1  +  12«.  =  30a:  -  4y. 

.-.30x-4y  =  384  (1) 

4y  =  30a:-384. 
30  s  -384 

••*- — i — 

ory  =  7a;-»96-|-|. 

Let  -  =  m. 

2 
Then  x  =  2m. 

Substitute  value  of  a?  in  (1), 

60m-4y  =  384. 

.\  y  =  15m-96. 
If  m  -  7,  x  =  14,  y  =  9. 

Hence,  A  can  give  B  14  half-crowns,  and  receive  from  B 
9  four-penny  pieces. 


29.  A  farmer  buys  oxen,  sheep,  and  hens.  The  whole  num- 
ber bought  is  100,  and  the  whole  price  £100.  If  the  oxen  cost 
£5,  the  sheep  £1,  and  the  hens  1  s.  each,  how  many  of  each  did 
he  buy? 

Let  x  —  number  of  oxen, 

and  y  =  number  of  sheep. 

Then  100  —  x  —  y  =  number  of  hens. 

5*  +  y  +  100~20~'V  =  100  (1) 

100a:  +  20y  +  100  -  x  -  y  =  2000, 

99a; +19y  =  1900  (2) 

Transpose,  19  y  =  1900  -99  a;. 

Divide  by  19,  y  =  100  -  5  x  -  —  • 

9  19 


teachers'  edition.  345 

Transpose,  100  —  5  x  —  y  =  — • 

Multiply  by  5, 

600-25  x-5y  =  x+~ 
*  19 

Let  —  =  m. 

19 

Then  a;=19m. 

Substitute  value  of  x  in  (2), 

1881m  +  19y  =  1900. 

Transpose,  19  y  =  1900  - 1881  m, 

y  =  100 -99m. 

If  m  =  l,  3=19,  y=l, 

and  100-x-y  =  80. 

Hence,  he  buys  19  oxen,  1  sheep,  and  80  hens. 

30.  A  number  of  lengths  3  feet,  5  feet,  and  8  feet  are  cut ; 
how  may  48  of  them  be  taken  so  as  to  measure  175  feet  all 
together? 

Let  x  =  number  8  feet  long, 

y  —  number  5  feet  long, 

and  48  —  x  —  y  =  number  3  feet  long. 


%x  +  by 

+  3(48 -a- y)  =  175                                (1) 

Simplify, 

5a  +  2y  =  31. 

Transpose, 

2y  =  31-5a;                         (2) 
.•.y-16-2»  +  !=5. 

Let 

l-s_TO 
2 

Then 

1  —  x  =  2m. 
.•.  x—  1  —  2m. 

Substitute  value  of  x  in  (2),  2y  =  31  —  5  +  10m. 

y  =  13  +  5m                        (3) 

If 

m  =  0,  -1,  -2, 

x  = 

»    1,    3,    5  —  number  of  8-ft.  lengths, 

y- 

=  13,    8,    3  =  number  of  5-ft.  lengths, 

48-a- 

-y  = 

-  34,  37,  40  =  number  of  3-ft.  lengths. 

346  ALGEBRA. 


31.  A  field  containing  an  integral  number  of  acres  less  than 
10  is  divided  into  8  lots  of  one  size,  and  7  of  4  times  that  size ; 
and  has  also  a  road  passing  through  it  containing  1300  square 
yards.    Find  the  size  of  the  lots  in  square  yards. 

Let  x  =  number  of  acres. 

.•.  4840  x  =  number  of  square  yards. 

y  =  number  of  square  yards  in  1  lot, 
8y  =  number  of  square  yards  in  8  lots, 
28  y  =  number  of  square  yards  in  7  lots  of 
second  kind. 
Sy  +  28y  +  1300  =  4840a;  (1) 

9y  +  325  =  1210*. 

9y  =  1210a;  -  325  (2) 

y  =  134a?-36  +  4z~1. 

y-134x  +  36  =  i^— i- 
*  9 

Multiply  by  7, 

7y  -  938  a;  +  252  =  3x  +  ^?. 

9  9 

Let  ^?  =  m. 

9 

Then  x-  7  =  9m. 

.*.  a:  =  9m  +  7. 

Substitute  value  of  x  in  (2), 

9y  =  10890m  +  8470  -  325, 

9y  -  10890m  +  8145. 

.-.  y  =  1210m  +  905. 

If  m  =  0, 

x  =  7       =  number  of  acres. 

y  =  905    =  number  of  sq.  yds.  in  1st  lot. 

4y  =  3620  =  number  of  sq.  yds.  in  2d  lot 


32.  Two  wheels  are  to  be  made,  the  circumference  of  one  of 
which  is  to  be  a  multiple  of  the  other.  What  circumferences 
may  be  taken  so  that  when  the  first  has  gone  round  three  times 
and  the  other  five,  the  difference  in  the  length  of  rope  coiled  on 
them  may  be  17  feet? 


teachers'  edition.  347 

Let  x  =  circumference  of  the  first  wheel, 

and  y  =  circumference  of  the  second  wheel. 

Zx  —  by  =  17,  difference  of  length  of  rope  coiled  on  them 
when  first  wheel  goes  round  three  times,  and 
second  five  times. 
Let  x  =  my, 

3a:-5y  =  17, 
3  my  —  by  =  17, 
y(3m-5)  =  17. 

y      3m-5 
If  m  =  2,  y  =  17,  s  =  34. 

33.  In  how  many  ways  can  a  person  pay  a  sum  of  £  15  in 
half-crowns,  shillings,  and  sixpences,  so  that  the  number  of 
shillings  and  sixpences  together  shall  be  equal  to  the  number  of 
half-crowns? 

Let  x  =  number  of  shillings, 

and  y  —  number  of  sixpences. 

Then  x  +  y  =  number  of  half-crowns, 

y  sixpences  =  J  y  shillings. 
(x  +  y)  half-crowns  =  J(a?  +  y)  shillings. 
Then  a;  +  £y  +  J (x  +  y)  =  whole  number  of  shillings. 

But  300  =  whole  number  of  shillings. 

.*.  z  +  iy  +  i(a;  +  y)=300. 
2x  +  y  -f  5a  +  5y  =  600, 
7a?  +  6y  =  600. 

a;  =  85  + *• 

7 
Transpose,  and  multiply  by  6, 

6z~510  =  4-5y  +  ^=^. 
7 

Let  —^  «=  m. 

7 

.\  y  =  2  —  7m. 
Substitute  value  of  y, 

7x+12-42m  =  600. 

.•.  a;=6m  +  84. 
Ifm  =  0,  -2,-2,-3,-4,  -5,  -6,-7,  -8,-9,  -10,-11, 
- 12,  — 13,  then  x  and  y  would  each  have  14  positive  values. 
Hence,  there  are  14  ways. 


348  ALGEBRA. 


Exercise  XCVII. 


1. 

a*  +  Zb*iB>2b(a  +  b). 

If 

a2  +  3o2is>2ao  +  2&2, 

if  (transposing), 

a* +  fi,is>2a&. 

But 

a*  +  b*is>2ab. 

.\  a2  +  3o2is>2i(<M  b). 

§249. 


a»o  +  aft*  is  >  2  a2&*. 
If  (dividing  both  sides  by  ab), 

a2  +  o2is>2ao. 
But  a*  +  o*is>2ao.  §249. 

.-.  a86  +  a6»is>2aW 


3. 

(a*  +  6*)(a*  +  64)is>(a8  +  &8)*. 
If  (simplifying), 

a«  +  a4^  +  a«j4+  J«  i8  >a«  +  2a86s  +  6« 
if  (transposing),        a46*  +  o*64  is  >  2a868, 
if  (dividing  by  a262),      a2  +  6*  is  >  2a6. 
But  a2  +  &2is>2a&.  §249. 

.-.  (a2  +  6*)(a*  +  b*)  is  >(a8  +  6s)8. 


4. 

o*6  +  a2c  4-  a&2  +  o*c  +  ac2  +  ic2  is  >  6aoc. 
a(6*  +  c2)  +  6(a2  +  c2)  +  c(a*  +  i2)  is  >  Qabc. 
Since  (62  +  c2)  is  >  2  oc,  §249. 

.-.  a^  +  c2)  is  >2abc. 

Since  (a*  +  c2)  is  >  2ac, 

.-.  6(a*  +  c2)is>2a6c. 

Since  (a2  +  62)  is  >2ab, 

.\  c(as  +  62)is>2a6c. 

Therefore  (by  adding), 

a(b*  +  c2)  +  6(<z2  +  c2)  +  c(o2  +  &2)  is  >  6abc. 


teachers'  edition.  349 

5.  The  sum  of  any  fraction  and  its  reciprocal  is  >  2. 

Let  -  =  the  fraction. 

6 

Then  -  =  the  reciprocal, 

and  ?  +  ^is>2, 

6     a 

if  (multiplying  by  a6),a2  +  62  is  >  2ab. 

But  a2  +  62is>2a6.  J  249. 

..  «  +  *i8>2. 
b     a 


If  32=a2+62,  and  y2=c*+cP,  xy  is  not  less  than  oc+6o*,  or  ad+bc. 
Now,  if  xy  equals  or  is  >  ac  +  bd, 

then  x*y*  equals  or  is  >  (ac  +  5c?)2, 

and  (by  substituting  the  values  of  a3  and  y2), 

(a2  +  ft2)  (c2  +  d2)  equals  or  is  >  (ac  +  6cJ)2, 
or  (simplifying), 

o2^  +  a2^2  +  c2^2  +  o2^2  equals  or  is  >  aV  +  2a6cd  +  62d2, 
and  a2cP  +  62c2  equals  or  is  >  2a6co*. 

But  a*&  +  62c2  equals  or  is  >  2abcd.  J  249. 

.\  xy  equals  or  is  >  ac  +  6a\ 

7. 

o6  +  ac  +  6c  is  <  (a+6-c)2+(a+c-6)2+(6+c— a)*, 
if  (by  expanding  and  combining), 

oft  +  ac  +  5c  is  <3a2+362+3c2-2a6-2ac-26c, 
if  3a6  +  3ac  +  36c  is  <3  a2  +  362  +  3c2, 

if  a6  +  ac  +  be  is  <  a2  +  62  +  c2. 

But  2a6  is<    a2  +    62 

2ac  is<    a2  +    c2 

26c  is  <  ¥  4-    c2 


2a6  +  2ac  +  26c  is  < 2a2  +  26s  +  2c2 

a6  +  ac  +  6c  is  <  a2  +  62  +  c2. 
.•.  a6  +  ac  +  6c  is  <  (a+6— c)2+(a+c-6)2+(6+c-a)2. 


350  ALGEBRA. 

8.  Which  is  the  greater, 

(a8  +  6»)(c*  +  d8)  or  (ac  +  bdyi 

Simplify,  aV  +  6*c*  +  a*d*  +  ^d8  is  >  aV  +  2a6cd  +  68d8. 
If  W  +  (W»  is  >  2  a&cd 

But  68c8  +  a«d»  is  >  2  aicrf,  J  249. 

.-.  (a8  +  &)(<?  +  d8)  is  >  (ac  +  6d)». 


9.  Which  is  the  greater, 

m8  +  m  or  m8  +  1  ? 

m8  +  m  is  >  or  <  wi8  +  1, 
b  m(m  + 1)  is  >  or  <(m8  —  m  + l)(m  + 1), 

b  m  >  or  <  m8  —  to  +  1, 

b  2m  is  >  or  <  m8  +  1. 

But  m8  +  1  is  >  2m, 

•  \  to8  +  1  is  >  m8  +  to. 


10.  Which  is  the  greater, 

a*  —  b*  or  4a8(a  —  6),  when  a  is  >  6? 

4 (^(a  —  6)  is  >  or  <  a4  —  b\ 
as  (dividing  by  a — 6),        4  a8  is  >  or  <  a8  +  a86  4-  a&8  +  ft8, 
as  (subtracting  a8  from  both  sides), 

$a*\B>or<a2b  +  ab*  +  V, 
as  (transposing  a*6),  3  a8  —  a86  is  >  or  <  a&2  +  6s, 
as  a8(3a  -  b)  is  >  or  <  68(a  +  6), 

if  the  factor  a8  be  taken  out  from  the -left  side,  and  the  factor  6*  from 
the  right  side,  since  a  is  >  bt  the  left  side  will  have  been  divided  by 
a  greater  number  than  the  right ;  so  that,  if  the  left  is  greater  than 
the  right,  after  both  factors  have  been  taken  out,  it  must  have  been 
greater  before. 

If,  therefore,  3a  —  6  is  >  a  +  6, 

if  (by  adding  6  —  a  to  both  sides), 

2a  is  >  26. 
But  2a  is  >  2b. 

.-.  4a8(a-6)is>a*-&*. 


teachers'  edition.  351 


11.  Which  is  the  greater, 

•  or  Va+Vb? 


U  +  \      is>or<vS+V5, 

a8(squaring),        T  -\-V2ab  +  —  is  >or  <a  +V2ab  +  6, 
o  a 

as  (transposing),  ^-  +  —  is  >  or  <  a  +  6, 

b      a 

as  (multiplying  by  ab\     a8  +  6s  is  >  or  <  a26  +  a&2, 

as  (a  +  6)(a2-a6  +  62)  is>or<a&  (a  +  6), 

as  (a2  +  62)is>or<2a&. 

But  (a2  +  62)  is  >  2  a6,  §  249. 


.^+^£»>Va+V5. 


12.  Which  is  the  greater, 

«±*or-?«*-? 
2         a  +  6 

"Jt*is>or<2a\ 
2                   a  +  6 

as 

a2  +  2ab  +  62  is  >or <4a6, 

as  * 

a2  +  62  is>or<2a6. 

But 

a2  +  62is>2a6, 

•    °  +  ^  in  ^  2o^ 
2            a  +  6* 

13.  Which  is  the  greater, 

6*     a2      6     a 

a      6  •   ^       ^  1     1 
62     a2                 6     a 

J249. 


b  a8  +  68  is  >  or  <  a2b  +  a62, 

b  (a  +  6)(a2  -  a6  +  62)  is  >  or  <  (a  +  b)ab, 
9  a2  +  62is>or<2a6. 

But  a2  +  62  is  >  2a6,  J  249 

6J     a2         6     a 


352 


ALGEBRA. 


Exercise  XCVHI. 


1.  >/?   =  x* 

(x/x)*  =x«. 

^   -A 


^iy?       =  x*y*2. 
Vx*y2z4     =*  ary*2*. 


3.   a* 


aH* 


4x*y~*=4\/xy~6. 


4.  a 


a* 

1 
a1' 

3x"1y-» 
6x_8y 

xy8 

X8 

x*y5 
2a~1x 

X* 

V 
6x7s 

S^y"1 

1      a6» 

=  3xy2~8. 

z 

=  x-*y_42. 

a 
6c 

=  ab~lc~l. 

=  alV<?. 

x"« 

=  ^§yl 

x"» 

=  a:-2f/-*. 

6.  a*xa*  =o*. 
6*X6*  -ft*. 

d*x  <**  =  <*'. 


7.  m*x  m~*  =  m*. 
n*  X  w"  *  =  fA 
a°xal     =  a*. 

cfixa-i  -i- 
a* 


8.  a*  xVa   =a*xa*  -a. 
c-ixVc    4xc»  -1. 

y*  x^y   =  y*xy*  -jf1- 
x*  xVxri  =  a*xx"*=»'. 


teachers'  editiok.  358 


9.  ab*cxa-lbc*  «a*&M.       14.  Qr*)-i        -1>*. 

aiblc-*xalb-*cld  =  aJd.  (?*)*  =  ?*• 

(*"*V)'*     -*Vf- 
(alxa*)-tt-(d»)-M-.a-« 

10.  3*yMxaf*3f*z'*  =  x~M2~*. 

xiyMxaT«yiz~*=ajV*-  15.  (4a"V*    =  4"*a=~ 

8 

11.  a»x«-»X«-*x.-»-«-*  IW*  -64"*C"V 


(fN^W 


1 


X"TXaHi 


~fo  16.  /w^V'-iS^ 

U^8/        81-*V* 

^81*  a8  ft* 

12.  a*  4- a*    =a*.  "     ^i 

c«  ^c*    =  c*\  27  a8  ft* 

n&.±-  a*    =  n~*.  8 

\\6b-'J       i6-»6* 
16» 

13.  (a«)U(a«)»  =  (a«)-*  =  a-»  9,<*61 

=  1.  64 

=  a  27  a6  6*' 

l**t  =c"*=7»  (3la-»)-l=3-^  =  J 

(m"ty  -m"1-™  f^Y1    _  256"*  =  625* 

1  \625/  625"*     256* 

(n*)-»  =w-1=--  ^125 


354  ALGEBRA. 


Exercise  XCIX. 
.       1.  4. 

x*  +  xPyP  +  y*r  8a*  +  4a1 6*  +  5a*  6*  +  9&* 

x»-g*y*  +  y»  2a*- 6* 

x+  +  x»yp+x**y*  16a  +  8a*&*  +  10a*6*  +  18a*5* 
-  **y*-  x**y*-  x*y»  _8a*6*-4aW-5aW-9i 


a^y^+ar'y^+y4* 
a4*  -fs^y2*  +y* 


5. 


2.  l+oi-'  +  a^-2 


2*  _j_  y**~ 


l+abl+a*b* 


+  x~»-»y~«-*  +  fl»J>  +  a86-s  +  a*6-4 
-V*        1  +  a'&»  +a46-« 


gum  —  xH W*  -f-  x^11*-*  y"1 


*i_2**+i  '  oif:+'+o";s 

1 .  a*6-*  +  2aJ6",  +  l 

x-2x*+    x*  _2a,?r1-4-2a-,5I 

-    s*  +  2x*-l  -l-2a-8y-a-464 

z  _ 3xt  +  3«* - 1  a46"4  -4-4aJ6,-a*54 


4x-s  +  3x"s  +  2a;-1  +  l 
x-»_    x^  +  l 


4ar6  +  3x-*  +  2ar8+    x~* 

-4aT4-3x-8-2aT2-    x'1 

+4x-8  +  3aT2+2x-1  +  l 

4a:"6-    x-4  +  3x-8  +  2x"2+    x~l  + 1 


TEACHERS*    EDITION.     '  355 


8. 

a;4n  _  yAn  |  #»*  _  yn 


xAn__x9nyn  x9n  +  x2nyH  +  ^yin  +  y3 


xSnyti  _ 

xSn yn _ 

-y4n 

-  jc^y^* 

x2ny2n  . 
x2ny2n  _ 

-  a^*y** 

9. 

|a?*  +  y*+z* 


a;  +  x'y* 

+«••» 

a£  —  a;*y*  —  z*2* 

+  yl 

-yM 

+  ** 

—  aj*y* 

—  x*y* 

-«m 

-x*y* 

-3a:*yM  +  y +  z 
—    afly*z' 

—  x*z*  ■ 

f  x*y* 

-2s*yM+y  +  z 
—    a:*y*z* 

a:*y* 
jc*y* 

+  je*  2*  —  x*yM  +  y  +  s 
+  yM                  +  y 

—  x*y*z*  +  a"  2*  — 

—  jc*y»2»             — 

yl2* 
yM 

+  2 

-yM 

+  yM 
+  yM 

+  2 

+  2 

10. 


3  +  y  |  a:*  -  x*yl  +  x^y*  —  sM  +  y* 

aj  —  a£y*  +  ajf  yf  _  ^1  yi  +  &*y*  x*  +  y* 

g4yi  _  x% yl  -f  #1  y$  _  #4y*  +  y 

a;tyi  _  jpf  yl  +  aj$yf  —  aj*y*  +  y 


856  ALGEBRA. 


11.  x*y-*  +  2  +  x-iy*     \xy~l  +xly 
rr*y-4  +  l  xyl+x~ly 

1+*"V 

12.  a-* +  a-*&-*  +  &-*  |a-»-g-i6-i  +  &-« 
a"*  +  a*b*  -  a"8*"1  a"8  +  a1 6"1  +  6"* 

<r26-2- a1  &-»  +  &-* 


a" 


13.  ^aft'1)?  =16a«6*. 

(a*-&*)2  =a-2a*5*+6. 

(a  +  a"1)*  -  a2  +  2  +  a"1. 

(2a*6*-a-*6ty  =  4aZ>*  -  46  +  a"1^ 


14.   aH  =4*x2      =4. 

Soft"1       =5x4x}  =  10. 
2(a6)*      =2^8  =  4. 

a-^c^JxJXl      -J. 
12<r'&-s  =  12xAxi  =  A. 


15.  (a*-2>ty        =  a*-3a&*  +  3ai&*-6. 

(2a;-1  +  x)*     -  (2a:-1)*  +  4(2aT1)»(aO  +  6(2aTlJ»(*)P 
+  4(2aT1)(a:)s  +  tf* 
-  16aT*  +  32aT2  +  24  +  8x*  +  x*. 

(aJr1-^1)6  =a66-«-6(a56-6x5y-1)+15(a*6-4xJ2y-1) 
-20(a86-sx6sy8)  +  15(a26-2xJV4) 
-6(ai-1x&5y-5)  +  J6y"6 
«a«5-6_6a56-4y-i+15a46-.y-._20ay8 
+  15a2  6V4  -  6a6*y6  +  &V« 


teachers'  edition. 


357 


16. 

9aT4  -18g'8y*-{-  15aT2y  -  63;"^* +  y2|3aTa- 3ar'y*  +  y 
9a;-* 


Qx^-dx^y^ 
6aT* 


-lSx-*yk  +  15  x'%y 
-18ar8y*  +    9aT2y 


■Gar^vi 


y*  +  2/ 


6rc~2y  —  6a?_1y*  +  y2 
CaT2y  —  GaT^  +  y2 


3(2a;)»  =  12x2 
(6*+l)(l)=__+6^+l 


17. 

^g  +  l-Sg-1 
8a?8  +  12a*  -  30a?  -  35  +  45a--"1  +  27aT2  -  27 ar* 
8ar> 

~  12a* -30  a; -35 


12a*+6aH-l 
12a?+12x+  3 


12a*  +    6a;  +    1 


-18+9a;-1+9a;-4 


12a^+12a;-15^9a;-1+9a;-a 


-  36a;  -  36  +  45a;"1  +  27 a;"8  -  27  a;"8 

-  36a;  -  36  +  45a;1  +  27  a;'2  -  27a;-8 


18.   v'l2=v'3x2x2  =  3*x2». 

v/72=v^3«x^      =  3*x2* 

</m=</3x¥       =  3*x2*. 

^64«^2*  =2* 

=  2»x3  = 


19.  [(a**)ix(a»)-»]»«-2 
=  [(a«)  x  (a?-10*)^ 

24.         ~a*». 


^) 


20.  (x***  X  a;-11)**-2 

l 

=  (*18.-12)8.-2 

21.  3  (a*  +  &*)»  -  4(a*  +  6*)  (a*  -  &*)  +  (a*  -  2 &ty 

=  3a+6ai&i  +  3&-4a  +  46  +  a-4aM  +  4& 
=  2a*&*  +  116. 


{(am)m~m},"+1 
=  am_1. 


23.  ^Y+f— lT'f 


358  ALGEBRA. 


24.  [«a--)-}i»]t  +  [{(*»H-»]-t 
-1. 


25. 


a-SKt-D  _y*(|,-i) 

ajjKff-l)  -fy«(l»-l) 

By  factoring  the  numerator, 

(arKf-i)  +  y«(j»-D)(a;J*«-i)_?/«(j>-i)) 
"*  ar^'-1)  +  y«(p-D 

=  x*«-i)  -  y^F"1). 

Exercise  C. 

1.    3V5   -\/5x(3J»      =V55.  2.  Sy'v^       =^8liy. 

3V2l=V21x(3)»    =Vl89.  2x\fejj          =^32^. 

5V32=V32x(5)2    -V800.  o8^5?         =v^. 

a*6  V£-  -Vfcc  x  (a*&)»  =Vtf&c.  3<*\fcbc       =\^27Sc7. 

*^y*=^*yOc)»      =v^iy.  SabcVcficr*  =V25aWc. 

3.  *>/£     - VAX^_  -  VJ^ 
lfiVgf  -  V256  X  Hi  =V244f 

'os  +  2ay  +  y8       *  *  xa  +  2sy  +  y8        ' 

4.  VaPifz        =  xy*Vz. 

y/&a*b         =VWx2a&       =  2aV2a£ 
v^54a^y  =v^27ay  x  2  ax*  -  3ay  v^?. 
V24  =V4x6  -  2V6. 

\/l25a*5»  =  \/25a*d*x5a*    =  5a2aV5d. 

5.  vTOOO^"        =  v'10xl0xl0xa  =  10v'a". 
v^l60x*y7      -v^20ayx8ay        =2o^2^20^. 
\^108  m9n10    =v^27m97i9x4n        -SflAi'Mn] 
^1372  au&w  =v^343a16615  x  46      =  IcP&Vlb. 

6.  \V-  3as6  +  3aW  -  ab*  =  Va(a*-Za?b  +  Zab*--V)  =  (a-6)W> 
V50a2-100a&  +  50&2     -  V2  X  25(a2  -  2a6  +  62)  =  5 (a -6) VS. 


TEACHERS*    EDITION.  359 

7.  2^80 aW<*  =  2v'l6a*c4  X  5aiV  =  iacy/ba^?. 
7\/396^      =  7V36xlla;  -  42VIT¥. 

9\/sTtftf~z  -  9^27  y»X  3&*«        =  27y  #TSg. 
5 V726         =  5V121  X  6  -  55 V6. 


|V^0f=tVlF^i^2xy29»tV58. 
2y/\     -^xl=^4. 


,.  ^  .^  .i^. 


WcPbx       /3  cPbcxy        a      ._ 


V5         5 

2 


O701 


-vs-vii-*^ 


11.  (ax)x(M»)*-(aW)*-a&.rv'5. 

(2a»&*)  x  (8V)*  =  2(a«6Mx8)*  =  2a*64j?v^. 


360 


ALGE&ttA. 


12.   Show  that  V20,  V45,  V$  are  similar  surds. 

V20=V4~x~5  =  2\/5. 
Vi5=>/5x^  =  3V5. 

Therefore,  they  are  similar  surds  because  they  have  the  same  sard 
factor. 


13.  Show  that  2\faWt  \/$¥,  JVft 

2Vtfb*  =  2aW. 
#8P    =  2bW. 


are  similar  surds. 


•v? 


Since  they  all  have  the  same  surd  factor,  they  are  similar  surds. 


14.  If  y/2  -  1.414213,  find  the  values  of 

V50;    4^288;  -7=;   -; ' 

*  V2'   V450 


V60 


=  5\/2 

=  5x1.414213 

-  7.071065. 


1 
V2 


f  V288  =  JV144X2 
=  30V2 
=  30  x  1.414213 
-  42.426390. 


=  JV2 
_^  1.414213 

2 
=  0.707107. 


V450 


=  VT^X2 
=  TVV2 

-^X  1.414213 
-0.1414213. 


TEACHERS     EDITION. 


361 


Exercise  CI. 


1.  Which  is  the  greater, 

3>/7or2\/l5? 

3V7    =V9~X7   =\/63. 


2Vl5=V4xl5=V60. 
Since  63  is  >  60, 
.-.  3V7>2VIB. 

2.  Arrange  in  order  of  magni- 
tude 

9V3,  6V7,  5>/l0. 

9V5  =>/243. 
6V7  =V252. 
5VI5-V250. 

Since  V252  >  V250  >  V243, 
.-.  6\/7>5Vl0>9V3. 


3.  Arrange  in  order  of  magni- 
tude 
4^4,  3^5,  5^3. 

4^4-^256. 
3^5=^135. 
5^3  =    v^376. 
Since  #135  <  #256  <  #375, 
.\  3#5<4#4<5#3. 


4.  3V2X4V6       =12Vl2 
=  24V3. 

-VS. 


5.  5VjxfVT62. 

5Vf     =  *Vl4; 
$Vl62  =  ^V2. 
$  Vl4  x  V- V2  =  W  V28 
=  J#V7. 
t#4x2#2      =  #8  =  2. 

6.  2V5-*-3Vl5    =  jVj 

=  $V3. 
tVa  +  AV5-|V85^ 
-  1*  Vl5. 

7.  J\/3x$V5-^>/2 
=  ^Vi6H-fV2 

-t^Vso. 


8    2VJ0     7VJ8 


4vT5 


3V27     5VT4      15V21 

-9xixV)Vtfxttx« 

9.    2#4x5#32-s-#l08. 

2#4x5#32    =  10#l28. 
10#128  -s-v^T08  =  10  #^f 
=  10#|f 
-10^x4 
=  6J#4. 


962 


ALGEBRA. 


Exercise  CII 


1.  2v^- 2(3)*  =  2(3)*  =^576. 
3V2  -  3(2)*  -  3(2)*  =v^58|_ 
§  y/l  -  §  (2)*  -  $  (2)*  -  v^l953j. 

.-.  the  order  of  magnitude  is  3V2,  |v^i,  2^3. 

2.  VJ     -(f)*    =  (!)*   -J'ffi. 

3.  2^^-^l76-v^l76*=v/30976. 
3^7  -^189  =^189*  =  ^35721. 
4V2   -V32   «v^3?   =#32768. 

.-.  the  order  of  magnitude  is  3v^7,    4>/2,   2^22. 


4.  3 Vl9  - Vm  -  171*  =- 171*  - vT71»  =#5000211. 
5#2   -#^  =  250*  =  250*=#^=#625W. 
3#3   =#81   -81*   -81*   =#81»   =v6561. 
.-.  the  order  of  magnitude  is  3\/l9,    5  #2,   3  #3. 

5.  2Vax  x #3a*6  x  V26x;   -&a*xy*  xy/a*xy. 
2\/as     =2aM. 

#3^&  =  3*a*6*. 
V26i  =  2*6***. 
^26x3*x28xa*x6*Xa:  =  2ax  #72^. 

#S^»  =  (aV)*  -  (as*y»)*  -vWey* 
y/a*xy  =  (atxyfi  =  (a'sy)^  —  y/ahfy*. 
v/5s^Xv^y=v/a»xa*xyu=  a^Vy5. 


TEACHERS*    EDITION.  363 

6.  3(4a&2)»-*-(2a8&)*. 

3  (4  a&2)*  -  3  (4  a&2)*  =  3  y/WaW. 
(2a36)*  =  (2a86)*  =v^8^P  -  a^W. 

a  a* 

(2a»ft>)*x(aW)*-i-(eiW)* 
(2W)x(a*&)  =  2*a**&*. 

=  (2a6)*x(3a62)^(5a6»)»  #       '    W  VW  U^V 

=  ^(2^8x^^+v/5^55  '           _  aty  6M  aM 

=  ^Wxv/3W-j-v/5^                      x*  v1  y* 
=  v'28x32xa5x&7  +V$a& 


-*F* 


a6» 
32  x  PaW  a&M 


5« 
=  ^225000  a*5*. 

4Vl2^-2V3  =  ^^V- 

=  2V5=2x2  =  4. 


a# 


9.  (7V2-5\/6-3V8+4\/20)x3\/2. 

7V2-5V^-3V8  +  4V20 

3v^2 

42T30\/3-36  +  24Vl0 
=  6-30V3  +  24VlO. 

=  (f)7x(f)6  =  a8&^2. 


dHir 


^(l^S2?  X  V^(2a26)«  =  a'1  ft*  cl*  (H. 

=  ^(8a3ft8)* 

-(2oft)" 


364  ALGEBRA. 


Exercise  CUT. 

1.   V27  -    V9x3 

=   3\/3 

2V48   -2V16X3 

=   8V3 

3V108-3V36X3 

=  18V3 

Sum 

=  29V3 

3V1000-   3VlOOxlO=   30V10 

4V50     -   4V25x2     -   20V2 

12V288   =12Vl44x2   =  144V2 

Sum         -164V2  +  30V10 

2.   ^128=^64x2   =   4v^2 

4.  2V3+3VIJ-V5 

^686=^343x2  =    7v^2 

-2V3  +  3V|-Vy 

y/16   -v^     -   2v^2 

:=2V3  +  $V3-|V3 

Sum                      - 13^2 

=  fV3. 

2VJ+Veo->/i5-V| 

7^54  -7^27x2  =  21^2 

=  }Vl5  +  2Vi5-Vl5 

3^16   =3v^8x2     -   6^2 

-JV15 

=  f|Vl5. 

^432-    ^216x2=   6v^2 

Sum                           -33^2 

5-  ^-gv* 

Vg-g^ 

la*cd*  _ad    .— 

3.    12V72   =12(2»x32)4  =  72>/2 
3  Vl28  -  3(2*x42x2)*=24V2 

.^IK    ^/aV    ^/H? 

Difference                   =  48V2 

7^/81     =7(3*)*        =21^3 

3V|  +  2v^S-4VA 

3  #1029  =  3(3  x  78)*  -  21  #3 

-SVIo  +  jVIo-jVio 

Difference                   *-  0 

-|VTa 

teachers'  edition. 


365 


6.  V±M+V2&ab*-(a-5b)\/ab 
=  2aVab  +  5 6\/o6  —  (a  -  5b) Va& 
=  (a  +  106)Va6. 

7.  cv^V-a\/^F?  +  6#aW 

=  abcVab2<?  —  abcVab'W  +  abcVabW 
=  abcy/ab'W. 

8.  2#40     =  4#5. 

3#l08    ~3#33~x~22*  =  9#4. 

#500  -  #5rx~2*  =  5#4. 

-#320   =  -#28~x5  =»-4#5. 
-  2#l37~2  -  -  2#2*x78  -  -  14  #4. 
And  4#5  +  9#4  +  5#4  -4#5  -14#4  =  0. 


9.  (2#3a*&)»    =  [2(3a*6)*]8 
=  28(3a46)*  =  8a*V3l 

(3#3)2   =  [3(3)*]» 
-3*(3)*  =  9#3. 


-(f)' -(t)'-»^- 

(V27)*=(27i)i  =  27*  =  #27. 


11.   (#81)*     =  (#3*)*  =#3. 

(#512)*   =  (#*)*  =  2*        -#8. 

#16        -(2*)*    =#2. 
#27        =(33)A    =#3. 


12.  #4  =  (2*)*=#2. 
#36  =  (62)*=#6. 
#32  -(25)*=#2. 
#243=,(35)*=V3. 
#125  =  (58)»  =#5. 
#49    -(7»)*  =#7. 


13.    #8?    =  (23a*)*     =  #2^". 
#9a2&*  =  (32d*6*)*  =  #3^5*. 
#l6aIi=#2V7i    =  2*a* 

=  a#2a. 
#320^-  (2*a10)*   -  2a2. 


366  ALGEBRA. 

14.  (y/S)*  =(v/2»)*  =  2*-2^2. 
(^27)*  =  ( #38)«  =  3*  -  3v^3. 
(^6l)»  =  (^26)s  =  2*  =  4v/2. 
(\/iy   =-(\/^)J-2*=2v/2. 

15.  (av^)"8      =(a*)-»         -a"*. 

(a-'v^78)-*    -(v'St*)-*-^"*) -*-a*«. 

16.  (Va+Vb?  =  (at  +  blf 

-  a*  +  5a26*  +  10a*6  +  lOai*  +  5a*&*  +  61 

-  a2Va  +  5a2 V5  +  lOai Va  +  lOabVb  +  5^  +  b*Vb. 

(-Vtd+Vity-lmi  +  df 

-  (m*)8  +  3(mtyx*  +  3m*(z*)2  +  (aty 
=  m2  +  Smxy/mta?  +  3a?Vm*  -f  a^Vx. 

(Va^-2V&)6  =  (a*-2&ty 

=  (ai)5-5(ai)*(26*)+  10(ai)»(26i)2-10(ai)J  (26*)8 

+  5(a*)(2ftty-(2&4)5 

-  a2 VS -  10a2\/5  +  40 obVa-  80 a&V&  +  80 62Va"  -  324* V8. 

17.  (2a*-lVaf 

=  (2a2)8  -  6(2a2)5(J Va)  +  15  (2  a1)4  (J  Va)2 -  20(2a2)8(jVa")8 

+  15(2a^(JvG)4-6(2a»)(}vG)H(JV,a)1 

-  64a12  -  96al0Va  +  60a» -  20a7  Va  +  3fa8  -  f  a*Va  +  ~ 

64 

=  (2^  -  4(2,*)'  (*)  +  6(2,*)»  (*)'-  4(2**)  (£)*+  (f )' 

16 


TEACHERS*   EDITION.  367 

(~^)'=(2*y'-y»)' 

-  (2*V)«  -  6(2a!y  7>(y»)  +  15(2a!»y-1)«(yl),  -  20(2*y  >)»(y»)» 
+  15(2^-')J(y1)4-  6(2*y ')(y»)»  +  (3/1)4 
=  teas'y*-  ma^V^y1  +  240 a»y  'v^T' - leOafy1 

+  60**^ -12ay  v'y+y*. 

18.  fv^6 S_V 

V  2Vb) 

(«i_viy 

-(S)(T)'-(¥y 

.W_"4W(g)!W(0_4(^(0+(*)* 

=  a«6*  -  2a*6*  +  §**-£!&?  +  ?'* 


2  16 


19. 


d^-viy 
=(!^y-(!^y(Vf)+3(!^)(^y-(v?y 

-  a*b*cid-*  -  3a*6-«  +  $ab~lc'*d*  -  cH\ 


368  ALGEBRA. 


(n'-a"5)* 

-(a»)*-4(a5)»(a"*)  +  6(a5)'(a'*),-4(a^)(a"*)»  +  (a"7 


=  a**-4a*  +  6  — 4a-»+a-*1. 


'(t)K-H!W;)' 


16a4     32a*     8  a*     8  a*     a«6« 
ft8         36*      36*       27       81 


a26* 


81 


20. 


a.__  VcV 
be      Sab) 


\(6c)V         \(6e)V\3a6/        \(&c)V\3aJ      UaJ 
(6c)«        Uc     3a67        \bW\9aWj 


a*        3ac*        3a*c  c* 

'  (6c)*  ~  3a6*c     9  a*&M     27  a*V 

a*         1  c*  c* 

+  - 


&M     62c*     3a*6*     27a86» 


(^H^^w^-^' 


86*      46*  2 


TEACHERS     EDITION. 


869 


(aVa_Vb\* 


U*      2a) 


6*      26**   '  4a*6*      8aS 


a*       3a2  +      3 


21. 


a**  +  6a*»y»»  +  lla^y2*1  +  6xmySH  -hy4*^*2"1  +  3a*»yn  +  ySn 


2xSw+3a-»y» 


2aA»  +  Qxmyn  +  y2n 


2a^"*y2n  +  6a?*y3B  +  y4* 
2a£"ys,t  +  6a*lySM  +  y4" 


22. 


|l  +  2x"*  -3or*  +  4arl 


1  +  4a;-*  _  2x'i  -   Ax'1  +  25a:-*  -  24aT*  +  16a;-2 
1 


2  +  2aT* 


4a;-*  4-  4.t"" 


2  +  4ar*-3ar* 


_8a;-*-    4a;1  +  25  aT* 
-6ar*-12arl  +    9  a?-* 


2  +  4a;"*- 6a;-*  +  4a;"1 


8a;"1  +  16a;"*- 24a;"*  +  16a;-2 
8a;"1  +  16a;"*  -24aT*  + 16  aT2 


370  ALGEBRA. 


Exercise  CIV. 

2. 


(i)  -^—  0) 


V7+V5  V6-v^ 

3(V7-V5)  g(V6+V^) 

"  (V7  +  V5)(V7  -  V5)  (V^  +  Vc)(V6  -Vf) 

3V7-3V5  -«(^+^). 

"           2  6~c 

-}(V7-V5).  (2)      a  +  * 

a-Vb 

7  a  +  6       a+Vft 

(2)    - 7?X y= 

2V5-V6  a-V6     a+V6 

7             2V5+V6  _(a+&)(a+V6) 


:X 


2V5-V6     2V5+V6  a  ~° 

_7(2>/5+>/6)  (3)    2x-VwxV^  +  2y 

14  Vxy  —  2y      Vxy  +  2y 

=  i  (2  V5  +  V&).  =  2{x-y)Vxy  +  3xy 

xy-Af 

(3)    ±=*  '  3. 


1+V2 
(4-\/2)(l-\/2)  (1) 


2(V5) 


(1+V2)(1-V2)  *     <^<^ 

v  v  '  2\/3     3.463 


4-5V2  +  2 
-1 
=  5>/2-6. 


(2) 


3  3 

- 1.154 

1 


(4) 


VE-V2 

-2V6  _  1(V5+V2) 


6 


5  +  2\/§  (V5-V2)(V5+V2) 


5-2V6     5  +  2V6  V5+V2 

3 


30  +  12>/6 


3.649.. 


1  3 

=  30  +  12V6.  -  1.216.. 


teachers'  edition.                           371 

(3)        ^                                 '  (4)    I+2VI0 

V7+V3  7-2VI0 

7V5(V7-V3)  (7-f2VJ0)(7  +  2Vl0) 

(V7  +  V3)(v^7-V3)  (7-2VT0)(7  +  2Vl0) 

-/  /**       /Tin  _49  +  28VT0+40 

=  7(V35-Vl5)  j^^ 

7(5.91 'o -3.8729 )  =  S^VW 

--* 4  9 


89+28x3.162.. 
=  3.576 9 

- 19.726 


Exercise  CV. 

1.  4+V=3  =  4  +  3*V=T. 
4_V^3  =  4-3*\/^T. 

(4  +  3*V^T)(4  -  3*  V^T)  =  16  -  3(-  1)  -  19. 

V3  -  2V=^2  - V3  -  2  x  2*\£T 

V3  +  2V^2  -  VS  +  2  X  2* y/-T. 

( V3  -  2  x  2*V^1)(V3  +  2  x  2*V^I)  =  3  -  8(-  1)  =  11. 

2.  V54x  V^^  =  3>/6xV=l  =  3v^T2  =  6\/^3. 
aV^b  x  sv^~y  =  (a&*\/^l)  X  (aj^V^T)  =  a&*xy*(-  1) 
=  —  axVby. 

3.  V^S+vcrs^aiv^rT  +  ftiv^T. 

V=^  - V^6  -  a* V^l  -  b*V^i. 

(a*V=7  +  &*V=T)(a*V=n[  -  6*\/^)  =  a(-l)- &(-l)«6-a. 

02ft2 VZ1  x  a26*  V^l  -  a*6*(-  1)  =  -  a4M  VS. 

4.  ( V^IO)  (  V=^2)  -  (10*  V=T)(2*  V=3)  =  20* (-  1)  -  -  2  V5. 
(2V3-6V^5)(4V3-V^~5) 

=  (2  X  3*  -  6  X  5*  V-l)  (4  X  3*  -  5*  V^T) 

-  24  -  (24  X  3*  x  5* V^l)  -  [2  X  3*  x  5* V^l  +  30(-  1)] 

.  _  6  -  26(3*  X  6*  V-T)  -  -  6  -  26  V=HJ5  -  -  2(3  +  13  V^15). 


370 


1. 


(1) 


Vl  +V5 

3(V7-VE) 
(a/7+V">)(VT- V5) 

3>/7-3V5 
2 
-}(>/7-V5). 


Exercise  CIV 

(1) 


(2) 


2V5-V6 
7 


2\^5-V6     2V5+V6 

_1(2V5+y/6) 

14 
■l(2V5+v/G). 


2. 


(2) 


g(V6+Ve) 
(V5+Vc)(V5-Vf) 
_g(\/6+\/c) 

a  +  b 


a-y/b 

a  +  b      a  +  Vfe 
a—  Vb     a+Vh 
_(a+b)(a+Vb^ 
a* -6 

(3)    2x-ViyyV^ 
Vxy-2y      v^ 
_  2(x  —  y)Vxij 


(3,    *^ 


1  fV2 

4-nV2  +  2 


-i 


<*) 


5 
*  5^2"V^  *  5  +  2v4 
.30+  IfVfi 


-30  +  J2>/tf 


(1)   -L=^(v 
V3      (V3) 
_  2n^  = 

3 
-  L154L... 


(2) 


1 


Vs~V5 

,  !_ 


'.   '  V  ■  -   * 


-\¥>: 


i 


1 


I 


rrnra-,  ^ 


-20. 


13-2V30. 
13  +  2  V30. 

V169-120. 
7. 
-13. 
-10, 
y  =  3. 
y=\/l0-V3. 


10. 

c_v^=Vll-6>/2. 

/i+Vy-Vii  +  eV2. 

iltiplying,       

x-y=Vm-12. 
.:x-y  =  l. 
x  +  y  —  11. 
.-.  a:  =  9, 
and  y  =  2. 
.Vx-Vy  =  3-V2. 


374 


ALGEBRA. 


11. 

Let       Vx  -  Vy  -Vu"-4V6. 

Then    VS  +Vy  =Vl4  +  4V<3. 
By  multiplying, 

x-y-Vl96-96. 
.\x-y-10. 
But  x  +  y  —  14. 

.\x=12, 
and  y  =»  2. 
.\  Vx  -Vy  =  2V3  -V5. 


14. 


Let     Vx-V^=V57-12Vl5. 

Then  Vi  +  Vy  =V57  +  12V15. 
By  multiplying, 

x-y=V3249-2160. 

...  x-y  =  33. 

But  x  +  y  =  57. 

.-.  x  =  45, 

and  y  =  12. 

.\V£-Vy-3V5~2V3. 


12. 


15. 


Let      Vx"-Vy=V38-12VT6.  Let     VJ-Vy  =V|-VlO. 

Then    Vi"+Vy  =V38  +  12Vl0.  Then  VS  +  Vy  =  Vj  + VIO. 

By  multiplying,  By  multiplying,      

x-y=Vl444-1440.  x-y=*  V^-10. 

.\s-y  =  2.  .-.  x-y=f. 

But  x  +  y  =  38.  But  *  +  y  =  j. 

.\x  =  20,  .-.  x  =  J, 

and  y  =  18.  and  y  —  1. 

.\Vx-Vy  =  2V5-3V2.  .\Vx-Vy  =  JVI6-1. 


13. 


16. 


Let       Vx-Vy=Vl03-12VlT.  Let     VS  +  Vy  =V2a+2VSM*. 

Then    Vx  +  Vy  =Vl03+12VIT.  Then  VS  -  Vy  = V2a-2Va^P. 

By  multiplying,  By  multiplying, 

x  -  y  =  V10609-1584.  x  -  y  =  V4a2-4a*+4P. 

.\x-y  =  95.  .••  x  —  y  =  26. 

But  x  +  y  =  103.  But  x  +  y  =  2a. 

.\x  =  99,  .\  x  =  a  +  6, 

and  y  =  4.  and  y  =  a  —  6. 

.•.Vx-Vy-3VII  —  2.  .•.Vx+V^=Va+6+Va^6. 


teachers'  edition.  375 

17. 

Let  Vx  -Vy  =  Va2  -  2bVa2~^T\ 

Then  Vx  +  Vy  =  Va2  +  26  Va2  -  b*. 

By  multiplying, 

a;  —  y  =Va4  -  4a262  +  46*. 
.\a?-y  =  a2-262. 
But  a;  +  y  =  a2. 

.-.a;  =  a2-62, 
and  y  —  62. 
.;Vx—Vy  = Va2  —  b*  —  b. 

18. 

Let  v^  -  V£  =  V87  -  12  V42. 

Then  x-y  =V7569-6048. 

.\#-y  =  39. 

But  a;  +  y  =  87. 

.-.a?  =  63, 

and  y  =  24. 

.-.v^-Vy=>/63-V24 

=  3V7-2\/6. 

19. 

Let  Vx  -Vy  = V(a  +  b)2  -  4(a  -  6) Vab. 

Then  Vx+Vy  =V(a  +  6)2  +  4(a  -  6) Vab. 


.'.x-y  =  Va*  -  12a86  +  38a262  -  12a6s  +  b* 
=  a2-6a&  +  62. 
But  x  +  y  =  a*  +  2ab  +  b*. 

.:x  =  (a-b)\ 
and  y  =  4  aft. 
,\Vx  —  Vy  =  a  —  b  —  2Vab. 


37G 


ALGEBRA. 


Exercise  CVTL 


1.  Vx~=5  =  2. 

Squaring,  x  —  5  =  4, 

*  =  9. 


2V3x  +  4-ar  +  4. 
Squaring, 

12x  +  16  =  a^  +  8a?  +  16, 
a*-4a;  =  0, 
x(*-4)  =  0. 
,\x  =  4or0. 


3.     'S-VxT^i  =  2x, 


3-2a?=Va^-T. 
Squaring, 
9  -12*  +  4a*  =  a*-l, 
3x»-12x=-10, 
9a* -36a:  =  -30, 
9**-()  +  36  =  6, 

3x-6  =  ±V6, 
3a?  =  6±V6. 
.-.a;-2±iV6. 


4.         V3x-2  =  2(a;-4). 

Squaring, 

3a?-2  =  4x«-32x  +  64 

4a* -35  a;  =  -66, 

64xa-()  +  (35)»  =  169, 

8a;-,  35  =  ±13. 

.-.a;  =  6  or  2}. 


5.     4a;-12Vx  =  16. 

Divide  by  4, 

x-3Vx  =  4, 
ZVi  =  x-i. 
Squaring,      9  a;  =  a*  —  8  a;  + 16, 

a*- 17s  =  -16, 
4aJ-68a;  +  289  =  225, 
2* -17  =  ±15. 
.•.  a;  =  16  or  1. 


6. 


Squaring, 


VaT+l+V2a;-l  =  6. 


x  +  4  +  2 V2x2  +  7z-4  +  2 re  -  1  -  36, 


Squaring, 


2\/2a*  +  7a;-4  =  33  -  3ar. 
8  a*  +  28a;  -  16  =  1089  -  198a;  +  9a* 

a* -226  a;  =  -1105, 
a*-()  + (113)'  =  (113)* -1105, 
x  -  113  -  ±  108. 
.-.a;  =  221  or  5. 


TEACHERS     EDITION. 


377 


7. 

Vl3x-1  -V2Z-1  =  5. 
Squaring, 

13a;-  l-2V26a?-  15a;  +  l  +  2a;  -1  =  25  

15a;  -  27  -  2V26a?  -  15  a;  +  1. 
Squaring,  225a?  -  810a;  +  729  =  104a;2  -  60a;  +  4, 

121  a? -750  a; --725, 
58564a?  -  ( )  +  (750)2  =  211600, 
242  a; -750 -±460, 

242a;=1210or290. 
.\a;  =  5orl12^:. 


8. 


Squaring, 
Squaring, 


V4  +  a?  +  Vx  =  3. 
A  +  x  +  2V4a?  +  a?  +  a;  —  9, 

2V4a?  +  a?  =  5-2a;. 
16a:  +  4a?  =  25- 20  a;  +  4a?, 
36  x  =25. 

9. 


V25  +  g  +V25-a;  =  8. 
Squaring,  25 +a?  +  2  V625  -  a?  +  25  -  x  =  64, 
2V625-a?  =  14, 
2500 -4a?  =  196, 
4a?  =  2304, 
2a;  =  ±48. 
.\  a;  =  ±24. 


10. 

a?  =  21-fVs2^T9. 
a?-2t=Vi?^9, 
a?- 42  a;2 +  441  =  a? -9, 
x*-  43a?  =  -450, 
4a? -172a?  =  -1800, 
4a?-()  +  (43)2  =  49, 
2a?-43  =  ±7, 

a?  =  25  or  18. 
•\  a;«*±5  or  ±  3V2. 


378  ALGEBRA. 


11. 

2x-#S*»~+26  +  2  =  0. 

^8**  + 26  =  2* +  2, 

Sx*  +  26  =  8** +  24**  +'24* +8, 
24*a  +  24*=18, 
4*»  +  4*  =  3, 

2x  +  1  =  ±  2. 
.\*  =  J  or  —  f. 

12. 


V*  +  l  +V*  +  16  =  V*  +  25. 


*  +  l  +  2V*a  +  17*  +  16  +  *  +  16  =  *  +  25, 
2V**  +  17*  +  16  =  8-*, 
4**  +  68a?  +  64  -  64  -  16*  +  **, 
3**  +  84*  =  0, 
**  +  28*  =  0, 
*(*  +  28)  =  0. 

.\  a;  =  0  or -28. 

13. 

V2xTl  -  VxTl  =  i  V*  -  3. 
2*+l-2V2*»  +  9*  +  4+*  +  4  =  *~~3 


9 
27*  +  45  -  18V2**  +  9*T4  =  *  -  3, 

18\/2*»  +  9*+4  =  26*  +  48, 
9V2*«  +  9*  +  4-13*  +  24, 
162*2  +  729*  +  324  -  169*2  +  624*  +  576, 
-7** +  105*  =  252, 
3»_15*  =  -36, 
4**-()+225  =  81, 
2* -15  =  ±9. 
.\*  =  12or  3. 

14. 

V*  +  3  +V*  +  8  =»  5V*. 
*  +  3  +  2\/*2  +  ll*  +  24  +  *  +  8  =  25*, 

2\/**  +  ll*  +  24  =  23*-ll, 
4**  +  44*  +  96  =  529**  -  506*  + 1215 
525*2-550*  =  -25, 
21a*  -  22a- -1, 
1764 x2  -()  +  484  =  400, 
42* -22  =  ±20. 
.\  *=1  or  fa. 


TEACHERS      EDITION. 


379 


15. 

y/3+x+Vx* 


6 


Vs  +  ~i 


Clear  of  fractions,     3  +  x  +V3a  +  ;c2  =  6, 

y/3x  +  x*  =  3  -  a:, 
3a;  +  a;2  =  9-6a;  +  a:2, 
9a;  =9. 
.-.  «-l. 

16. 

V?^l 

a2  - 1  +  eVjF^l  =  16, 

6V^=l  =  17-a;a> 
36a*  -,36  =  289-343*  +  **, 
a?* -70  a;2  =  -325, 
tf*-()  +  (35)2  =  900, 
x*  -  35  =  ±  30, 
a2  =  65  or  5. 
.*.  a  =  ±V65  or  ±>/5. 


17. 

1 


1 


Vx  +  1      Va?  —  1       Va;2  —  1 
Va;  — 1  +Vx  +  1  =  1, 

/x  — 1  =  1  —  Va;  +  1, 


/x+  1  +  a?  +  1, 


Va;  +  2a-Va;  — 2a 


a; 
:2a" 


Vx  —  2a  +  VaT+2a 
2aVr  +  2a  —  2aVx  —  2a=*xVx  —  2a  +  ffVaT+lJa, 
(2a  -  a;)VaT2a  =  (2a  +  a)Va;-2a, 
(4a2  -  4aa;  +  a*)(x  +  2a)  =  (4a2  +  4aa;  +  a;2)(ar  -  2a), 
x9  -  2az2  -  4a2a;  +  8a8  =  x*  +  2a£  -  4a2x  -  8as, 
4aa;2  =  16a3. 
.*.  x  =  ±2a.- 


380  ALGEBRA. 


19. 


Zx+Vlx  —  x** 
Sx-Vlx-x* 

3s  +V4s-s2  =  6s  -  2V4s-s*, 

3>/4a-s,~  3s, 

V4s  —  a2  =•  x, 

4s  — a^  —  s2, 

2s2-4s  =  0, 

2s(s-2)  =  0. 

.-.  s  =  0or  2. 

20. 


V7s2  +  4  +  2V3s-l_7 

V7x2  +  4-2V3x-l       '   

V7s2  +  4  +  2Vj3T^T=  7V7s2  +  4  -  14V3s-l, 
16V3s-l  =  6V7s2+4, 
8V3a;-l  =  3\/7^  +  4f 
192x-64  =  63s2  +  36, 
63^ -192s- -100, 
4(63s)2-()  +  (192)2  =  11664, 
126  s  -192  =  ±108. 

.-.  s  =  2^  or}. 


21. 


V(s  —  a)2  +  2ab  +  6*  =  x  —  a  +  b. 
x1  -  2as  +  a2  +  2ab  +  6*  =  s2  +  a2  +  6*-2as— 2ab  +  2bx, 
26s==4a6, 
2s  =  4a. 
.-.  x  =  2a. 


22. 


V(s  +  a)2  +  2ab  +  62  =  6  -  a  -  s. 
a*  +  2as  +  a2  +  2ab  +  62  -  i2  +  a2  +  s2-2a6  +  2ax-26x; 
26s  =  -4a6. 
.-.  s 2a. 


teachers'  edition.  381 


23. 

VF3+a 

»-36  = 

-v? 

=VSx. 

V33+36+V3; 

3a;  +  36 

+  2V9a*-1296  +  3; 

r-36- 

=  8a;, 
-2* 
-a?, 

2V9JC2- 

-1296* 

>/9a*- 

- 1296  - 

9a*- 

-1296  = 

a*  = 

.\  a?  = 

24. 

- 162. 
=  ±9V2 

4ar*-3(a;i  +  l)(a 

r»  -  2)  - 

-a;*  (10 

-3a;*). 

4a;*-3aj+3j 

C*  +  6  = 

3ari- 
26. 

=  10a;*- 
=  6, 
-36. 
-4. 

-3a;, 

(a;1 -2)  (a; 

*-4)  = 

*x*(x*- 

-1)*-12. 

a;*-2a:,-r4: 

c*  +  8  = 

5a;*  = 

*•- 

=  a;*-2 
=  20, 
-4. 

a;§  +  a-!  _  ; 

12, 

Raise  to  third  power, 

3*  = 

»64. 

.\  a?  = 

■±8. 

26. 

a*-4a#  =  96. 

27. 

a;  +  x~l  ■- 

=  2.9. 

a*- 

-4a;*  +  4  =  100, 
x*-2  =  ±10, 

a; 

=  2.9, 

a;*  =  12  or  -  * 

a*  +  l  = 

=  2.9  a;, 

5, 

10a*- 29  a;  = 

=  -10, 

a;*  =  vT2or- 

-2-      400a* -| 

[)  +  (29)*  = 

=  441, 

.\  a;  =  (#144)  or  4 

20  a;--  29  = 

-±21. 

=  2^18  or 

4. 

.•.  a?  = 

=  Jor2}. 

382  ALGEBRA. 


x*  +  2a*ar*  =  3a. 
Multiply  by  re*,  a;  +  2  a*  =  3  ax* 

x-3ax*  =  -2a*, 
4x-()  +  9a*  =  a», 
y/±x  —  3a  =  ±a, 
V4x  =  3a±a, 
4x  =  16a*  or  4a2. 
.\  x  =  4a*  or  a* 


81^ +  -^  =  52*. 
Vx 

81a?*  +  81  =  52s*. 
Transpose,  52x*  -  81  x1  =  81, 

(104)»x*-()  +  (81)»  =  23409, 
104  x1-  81  =  ±153, 

104  a*  =  234  or -72, 
st-ftor-A. 


Exercise  CVIII. 
1. 


a-*_3x-6Vxa-3x-3  +  2  =  0. 
Add  —3  to  both  sides, 
(JB«_3x-3)-6(x8-3x-3)i  =  -5, 
(a4_3a._3)_(\  +  9  =  4, 
(x»-3x-3)*-3  =  ±2, 
(x»-3x-3)*  =  5orl, 
a-*--3x-3  =  25or  1, 
x*-3x  =  28or4, 
4x*-()  +  9  =  121or  25, 
2x -3  =  ±11  or  ±5, 
2x  =  14, -8,  8, -2. 
.-.  x  =  7,-4,  4,-1. 


TEACHERS     EDITION. 


883 


x      x2      36 


Subtract  2  from 
Since 


From 


H) 


i,  and  from 


36 


65 
36 


a(.-i)  +  8. 


16 


2a;- 


13 


s-*or- 
3  3 


6a;2-6  =  -5a;or-13a?, 
6a?  +  5a;  =  6, 
144  a? +  0  +  25  =  169, 
12a? +  5  =  ±13. 
.-.  a?  =  }or-li. 

6a?-6  =  -  13a?, 
v  6a;2  +  13a;  =  6, 

144  a? +  ()  +  169  =  31 3, 

12a;  +  13  =  ±V313. 

.•.*-A(-13±VSl3). 


(2a;2-  3a;)2-  2(2a?-3a;)  -  15. 

(2a?-3a;)2-0  +  l  =  16t 
(2a?-3a;)-l  =  ±4, 

2a,-2 -3a;  =  5  or -3, 
16a;2-()  +  9  =  49or-15, 

4a;  —  3  =  ±  7  or  ±V— 15, 
4a;  =10,  -4,  3  ±yClg. 
.-.x  =  2J,  -1,  J(3±V-15). 


384  ALGEBRA. 


(a*-6)»+4a(as-&)=^-. 
4 

4(ax  -  ft)P  +  16a(ax  -  &)  =  9as, 

4(ax  -  by  +  (  )  +  16a*  -  25a», 

2(ar  —  fc)  +  4a  =  ±5a, 

2(ax  —  &)  =  a  or  —  9a, 

2ax  =-a  +  2&  or  2&-9a. 

.  -     a  +  2b  M2b-9a 

•••  x  =  — or  — • 

2a  2a 

5. 

3(2c«-*)-(2zt-*)t-2. 

36(2^-»)-()  +  l-25, 
6(2**-a;)*-l  =  ±5t 
6(2x*-x)*  =  6or-4, 
(2a?-*)*- 1  or  -|, 

2x*-a;  =  l  orf, 
16x»-8x-8or  ^, 

16««-()  +  l  =  9orV, 

4«-l-±3or±jV41. 
.-.*-1,-*.«1±*VE)l 


6. 

15x-3s*  +  4(a*-5a;  +  5)*  =  16. 

Change  signs  and  add  15  to  both  sides, 

(3z,-15x  +  15)-4(*f-6*  +  5)*--l, 
3(a?-5x  +  5)-4(*,-5x  +  5)*  =  -l, 
36(s»-5a;  +  5)-()  +  16  =  4, 

6(a*-5a;  +  5)*  =  6or2, 

(x*-5*  +  5)*-lor  J, 

a£  —  5x  +  5  —  1  or  £, 

a?  —  5a;  =*  —  4  or  —  4$, 
4x«-()+25  =  9or^, 
2a;  —  5  =  ±3  or  ±  J. 
.\x  =  4,  1,  3f,  1}. 


teachers'  edition.  385 


• 

7. 

X2  +  X~2  +  X  +  0T1  = 

-4. 

(*+?)+(*+;)- 

-4, 

{x4j+ix+l)' 

=  6, 

4('+i),+4(*+-)" 

=  24, 

4^+iy+()+1= 

=  25. 

Extract  the  root,              2(  x  +  -  ]  = 

=  4  or -6, 

1 

=  2  or -3, 

•                                     X 

a»  +  l- 

=  2x  or  —  3a?. 

For  first  value,                       x2  +  1  = 

=  20?, 

a?-2a;  = 

•-1, 

a*-2»  +  l  =  0, 

a?-l  = 

=  0. 

.\  a?  = 

■  1. 

For  second  value,                   x2  +  1  = 

=  -3a>, 

aa  +  3a?  = 

■-1, 

4a»  +  ()  +  9- 

=  5, 

2a? +  3  = 

=  ±V5. 

.*.  <c  = 

=  J(-3±V5). 

8. 

x2  +Vx*  —  7  = 

-19. 

Subtract  7  from  each  side, 

(a*  _  7)  + (a* -7)* -12, 

4(s2-7)  +  ()  +  l  = 
2(a*-7)*  +  l  = 

-49, 

=  ±7, 

2(«"-7)*  = 

=  6  or -8, 

(a?_7)*  = 

=  3  or  -4, 

i»-7« 

=  9  or  16, 

a*. 

-16  or  23. 

.-.  a>  = 

-±4or±\/§3. 

J 


386  ALGEBRA. 


9. 

6(a? +  *)  +  (*■  +  *)*«  7. 
144(s»+aO  +  Q  +  l  =  169, 
12(^  +  ^  +  1  =  ^13, 

(0^  +  ^  =  1  or -{, 
(x8  +  x)  =  lorft, 
4a*  +  4a;  +  l  =  5or_W, 

2x  +  l  =  ±V5or  ±$V58. 

...  ^(-li^orJC-l**^) 

10. 

(s  +  l)*  +  (*-l)l  =  5- 
Squaring,     *  +  l  +  2(*»-l)*  +  *-l  =  25,    ' 

2(x*-l)4  =  25^2xf 

4a?  -  4  =  625  - 100*  +  4a2, 
100x  =  629. 
.•.a?  =  6^fiy. 

11. 

a;-l-2+2flri. 

a._l  =  2  +  4» 

2  x 

.-2-3, 

x* 

a;i_2  =  3a;i, 

<pi_3a-*  =  2. 

Squaring,  a»-6*»+9*-4l 

a*-6x2  +  9s-4  =  0, 

(s-l)(a*-5s  +  4)  =  0. 

...(a._l)or(x2-5a;  +  4)  =  0. 

If  (*-i)-o, 

*-l. 
If  a2- 5a? +  4  =  0, 

B*-5a:~-4, 
4a;2 -20  a?  =  -16, 
4aj2-20x  +  25  =  9, 
2x-5  =  ±3, 
2s  =  8  or  2. 
,\  x  =  4  or  1. 


TEACHERS*    EDITION.  *  387 


12. 


V3a;  +  5  W3a;-5  +  4,  

(3a  +  5)  =  (3  a; -5)  +  8V3aT^5  +  16, 
6  =  8V3rc-5, 
36=  192  a; -320, 
-192a  =  -356. 


13. 

(**  +  l)-»(^  +  1)--2a* 
Transpose,  a*  +  2a?  +  1  -  x(x*  +  1)  =  0, 

(a*  +  l)2-a;(a;2  +  l)  =  0. 
Multiply  by  4,  and  complete  the  square, 

Extract  the  root,  2  (a*  +  1)  -  x  =  ±  x, 

2x*  +  2  =  2x  or  0. 
For  first  value,  2a?  -  2x  -  -  2. 

Multiply  by  2,  and  complete  the  square, 

432_()  +  l  =  _3. 

Extract  the  root,  2a;-l  =  ±V-3.^ 

•••  &=  *(!*</- 3). 
For  second  value,  2  a2  =  —  2, 

a*--l. 
,\  a;  =  ±V^l. 


14. 


2a?2  -  2V2z2-5.r  =  5 (.r  +  3). 
2a*  _  5x  -  2V2a;2-5a;  =  15, 
(2a?  -  5a)  -  2(2a?  -  5s)*  +  1-16, 
(2a?-5a;)*-l.=  ±4, 
(2a;2 -5a;)*  =  5  or -3, 
2a?- bx  =  25  or  9, 
16a?-()+25  =  225or97, 

4s-5  =  ±15or  ±V97,  _ 

4  a;  =  20,  -10,  (5  +  V97),  (5-V97). 
.-.  x  -  5,  -2J,  J(5+  V97),  J  (5-  V97). 


388  ALGEBRA. 

15. 

x  +  2  -  4x>/x  +  2  =  12a* 
Complete  the  square, 

(x  +  2)  -  4xVx  +  2  -f  4x»  =  16x*, 
Vx  +  2-2x  =  ±  4x, 

y/x  +  2  =  6x  or  -2x, 
x  +  2  =  36X2  or  4x» 
36s«-x  =  2, 
5184x»-144x=s288, 
5184x2-()  +  l  =  289, 
72x-l  =  ±17, 

72x  =  18or-16. 
.\  x  =  J  or  —  f. 
Also,  x  +  2  =  4x*, 

4x2-x  =  2, 
64x*-16x  =  32, 
64x*-()  +  l  =  33, 

8x-l  =  ±V33.  __ 
.-.x  =  J(l±V33). 


16. 


V2x  +  a  +  V2x  —  a  =  6. 
2x  +  a  +  2V4X2  -  a2~+  2x  -  a  =  #, 


2V4x2-a2  =  *2-4x, 
lBx2  -  4a2  =  6*  -  8&*x  +  16x», 
862x  =  4a2  +  6*. 
4a2  +  &* 


86« 
17. 

V9x*  + 21x  +  l  -  V9^T6x+T  =  3x. 


9x2-f21x-fl-2V81x*  +  243xs  +  144xa  +  27x  +  l  +  9x2  +  6x+l=9gg 
2V81x*  +  243x»  +  144X2  +  27x  +  1  -  9x*  +  27x  +  2, 
324  x*  +  972x»  +  576x*  +  l08x  +'4  =  81  x*  +  729  x*  +  4  +  486  Xs 

243x*  +  486x»  - 189  x2  =  0,  +  36a;2  +  108*' 

27x2(9x2  +  18x-7)  =  0. 
.\x  =  0. 
Or,  9x2+18x=7, 

9x2  +  ()  +  9  =  16, 
3#  +  3  =  ±4, 
3x  =  l  or -7. 
.-.  »  =  Jor-2|. 


TKACHERS'    KDITI(JN.  389 

18. 

x*  —  4  a;'  +  x'  *  +  4aT*  =  —  J. 

(•S)-e4)~*- 

Since  re1  -  2  +  \  -  (a*  -  i  Y. 

(■Sy-'t-S)--* 

(^i,)-2.,, 

2a>*-2  =  3a*or  5a;*, 
2^-3^  =  2, 
16*1-0  +  9-25, 
4** -3- ±5, 

4a;*  =  8  or  -2, 
at  =  2  or  -  }, 
x*  =  8  or  -  |. 
.-.  a?  =  ±2V2or  ±J\/^2. 
Also,  2**-2  =  5a'» 

2a;* -5a;' =  2, 

16**-()  +  25-.41, 

4a-'-5  =  ±V41, 

a;»  =  J(5±Vil). 

.•.*-[!  (5  ±VE)]». 


390  ALGEBRA. 

19. 

(2*  +  3y)«-2(2x  +  3y)  =  8  (1) 

x*-y*  =  21  (2) 

Add  1  to  both  sides  of  (1), 

(2x  +  3y)»  -  2(2*  +  3y)  +  1  =  9. 

Extract  the  root.  (2*  +  3y)  - 1  =  ±  3, 

2x  +  3y  =  4or-2  (3) 

2  2 

^     16-24y+9y»  or  4+12y+9y» 
4  4 

Substitute  value  of  z*  in  (2), 

16-24y  +  9y»       .a21 
4  y 

5y*-24y  =  68, 

100y» -0  +  576  =  1936, 
lOy  -  24  =  ±  44, 

10y  =  68or-20. 
...  y  =  6$or-2. 

Substitute  second  value  of  ae1  in  (2), 

4  +  12y  +  9y»       a_gl 
4  y 

5y2  +  i2y  -  80, 
100^  +  0  +  144  =  1744, 

10y  +  12  =  ±4VT09, 

10y  =  -12±4Vl09. 
.•.y  =  -U±$VT09 
=  j(-3±>/l09). 
Substitute  values  of  y  in  (3),  a?  =  -  8J,  5,  }  (4  if  3  VT09). 


teachers'  edition.  391 

20. 

x  +  y  +  Vx  +  y  =  a, 
a  — y  +y/x  —  y  =  6. 
4(a;  +  y)  +  ()  +  l  =  4a  +  l, 
4(g-y)  +  ()  +  l-4ft  +  l, 
2(a;  +  y)*  + 1  =  ±V4a  +  l, 
2(a?-y)*  +  l=±V46  +  l, 
2(a?  +  y)*  =  - 1  ±V4o  +  lt 
2Qc-y)*  =  -l±V4&  +  l, 
4(a?  +  y)  -  4a  +  1  *  2V4a  +1+1, 
4(jt -y)  -  46  +  1  *  2V46TT  + 1, 

8aj  =  4a  +  46+4*2\/4a+1^2V46+l. 
.-.  s- J(a  +  6+l)^J(V4^+I*V46+l). 
8y-4o-46*2V5o  +  i*2Vl6  +  l. 

.-.  y  -J(«-ft)*H  V4a~+i  *  V4F+1). 

21. 

a>*-afy2  +  y4=-13  (1) 

a»-ay  +  y2  =  3  (2> 

Square  (2),      x*+3xy+y*-2x*y-2xy*  =  9  (3) 

Subtract  (3)  from  (1),  2arJy+2&ys-4afy2  =  4. 

2 
Divide  by  2  <ry,  a2  —  2 ay  +y2  =  —  (4) 

xy 

Subtract  (4)  from  (2),  xy  =  3 

.\  aty2-3ajy  =  -2, 
4s2y2-()  +  9  =  l, 
2a:y-3  =  ±lf 

ay  =  2  or  1  (5) 

Subtract  (5)  from  (2),      x*  -  2  xy  +  y2  -  1  or  2, 

a—  y  =  ±lor±V2  (6) 

Multiply  (5)  by  3,  and  add  to  (2), 

x*  +  2xy  +y2  =  9or6, 

x  +  y  =  ±3  or  ±\/§  (7) 

Add  (6)  and  (7),  2a?  =  ±  4,  ±  2,  or  ±  \/2  ±  y/6. 

.-.  a;  =  ±2,  ±1,  or  J(±\/2±V6). 
Subtract  (6)  from  (7),  2y  =  ±  2,  ±  4,  or  t  V2  ±  V& 

.-.  y  =  ±  1,  ±2,  or  J(tV2±V6).' 


392  ALGEBRA. 


22. 

x»+y*  +  x  +  y  =  48 

(1) 

•      2zy  =  24 

(2) 

Add  (1)  and  (2),        a* 

'+ 2xy +ya  +  a? +  y  =  72, 
(s  +  y)*  +  (s  +  y)=72. 

Complete  the  square, 

4(x+y)*  +  ()  +  l  =  289. 

Extract  the  root, 

2(x  +  y)  +  l  =  ±17, 

x  +  y  =  8  or  —  9 

(3) 

From  (2), 

12 
a?  =  — 

y 

Substitute  value  of  x  in  (3),              —  +  y  =  8  or  -  9, 

12+y*  =  8y  or-9y, 

y»-8y  =  -12, 

3^-0  +  16=- 4, 

y-4  =  ±2. 

.:y  =  6  or  2. 

Also, 

12  +  y*  =  -9y, 
y*  +  9y  =  -12. 

Complete  the  square, 

4y*  +  ()  +  81=33, 

2y  +  9«±V33, 

2y 9±V33. 

.-.y  =  J(-9±V33). 

Substitute  values  of  y 

n(3),                     *=2,  6,  }(~9tV33). 

23. 


.r2  +  xy  +  y*  =  a* 

a) 

a;  +>/aJy  +y  =  6 

(2) 

Divide  (1)  by  2, 

x-Vxy  +y  =  ^ 

•                                             0 

P) 

Subtract  (3)  from  (2), 

2V^-< 

Divide  by  2, 

Squaring, 

w 

TEACHERS      EDITION. 


393 


Add  (1)  and  (4),  x*  +  2xy  +  y2 

Extract  the  root,  x  +  y  —  ± 

From  (4), 

Add  (1)  and  (6),  x*  -  2  xy  +  y' 

Extract  the  root, 

(5)  is 

Add, 

Subtract  (7)  from  (5), 


a*  +  2a262  +  b* 


(5) 
(6) 


46* 
a*  +  62 
2b 

-3sv  =  -3(*4-2a262+"4) 
y  46* 

10q262-3a4-364 

'  462 

aj-y  =  ±lyi0a262-3a4-36*      (7) 

a*  +  62 

2a;  =  ±^-^±i-Vl0aa62-3a*--36*. 
26        26 

2y  =  ±  £±^  t  i-Vl0a262-3a4-364. 
*  26        26 


...  x  =  -^  [±(af  +  62)± Vl0a262-3a4-364]. 
46 

•"•  y  -  71  [*(«* +  i2)=FVl0o262-3a4-364]. 
46 


24. 

(*-tf-3(*-y)-10  (1) 

a?y-3a!y  =  54  (2) 

Complete  the  squareof  (1),  4(a?-y)2  - ( )  +  9  =  49, 

2(x-y)  +  3  =  ±7. 

.vx-y  =  5or-2  (3) 

Complete  the  square  of  (2),      4afy2  -  (  )  +  9  =  225, 

2  ay  -  3  =  ±  15. 

.-.  zy=9or-6  (4) 

From  (3),  y  =  x—  5  or  x  +  2. 

Hence  from  (4),    ■  a2  —  5a?  =  9  or  —6. 

Therefore  x2  —  5x  +  ^=-^-  or  h 

x-$  =  ±i  V61,  or±J,_ 
s  =  3,  2,  orJ(5±  V61). 

Putting  x  +  2  for  y  in  (4),  we  get  

3  =  -l±  VlO,  or-l±  V-5. 
Whence  y  =  -2,  —  3,  i(-5±  V61), 

1  ±  VlO,  1  ±  V^5. 


394  ALGEBRA. 


.26. 

yfx  —Vy  =  x^(x^  +  y*) 

(l) 

(*+y)j  =  2(x-y)» 

(2) 

From  (I), 

Vx—Vy  =  x  +  x*y* 

(3) 

Square  (3), 

x  —  2Vxy  +  y  =  a?  +  2xVxy 

4-^y. 

x  —  x1  —  xy+y  =  2xVxy  +2Vx~y 

(4) 

From  (2), 

x*  -f  2xy  +  y%  =  2x*  —  4xy  + 

2y», 

x*+y*  =  6xy 

(5) 

Subtract  2  xy  from  both  sides  of  (5), 

a*-2zy  +  y,  =  4ay. 

Extract  the  root, 

a;-y  =  ±2\/zy. 

Substitute  x  —  y  i 

or  2Vxy  in  (4), 

x—  a£  —  xy  +  y  =  x*  —  xy  +  x- 

-y. 

or  ar1  — y  =  0. 

.-.  y  =  x*. 

Substitute  x*  for 

y  in  (5),          a? +  3^  =  63* 

(«) 

a?  =  Oorl  +  a*  = 

■  6*. 

If 

a;  =  0,  y  =  0. 

From 

l  +  a?  =  6x, 
x»_6x  =  -l, 
x»-6a;  +  9  =  8, 

a-3  =  ±>/8, 
x  =  (3±2V2). 

Since 

y  =  (3  ±  2  V2)*. 
26. 

ginvf- 

(1) 

sy-(a;  +  y)  =  54 

(2) 

Square  (1), 

x  +  y             3* 

Simplify, 

9x*  +  **  +  2ay  +  y*  =  6  x»  +  6xy, 
4a?  —  4xy +  y*  =  0. 
2  x  -  y  =  0 

Extract  the  root, 

(3) 

\  y  =  2<c. 

Substitute  value  of  y  in  (2),  2x*  —  3a;  =  54, 
16^-()  +  9  =  441, 
4s-3  =  ±21, 

4a;  =  24  or -18. 
.-.  x  =  6  or  -4}. 
.\  y  =  12or-9. 


TEACHERS*   EDITION.  395 


27. 

. 

x  +  y  +Vxy  =  28 

(1) 

a*  +  y2  +  xy  =  336 

(2) 

Divide  (2)  by  (1), 

x—Vxy  +  y  =  12 

(3) 

Subtract  (3)  from  (1), 

2Vxy  =  16, 

a^  =  64 

(4) 

Add  (4)  and  (2), 

a?  +  2a;y +y2  =  400. 

Extract  the  root, 

x  +  y  =  ±  20 

(5) 

Multiply  (4)  by  3,  and  subtract  from  (2), 

x*  —  2xy  +  y2  =  144. 
Extract  the  root,  x  —  y  =  ±  12  (6) 

Add  (5)  and  (6),  2  a?  =  ±  32  or  ±  8. 

.\  x  =  ±16  or  ±4. 
Subtract  (6)  from  (5),  2y  =  ±8    or  ±  32. 

.\  y  =  ±4    or  ±16. 


-  -  Zax  =  V4a*  +  9oa;2  +  ?^t 
a  4 

4a;2  -  12a2a;  =4aa?V4a?  +  9a  -f  27a3, 

12a2a;  +  27a8  -  4aa?V4a;-+9a  =  4x2, 

3aa(4a?  +  9a)  -4aa;V4a;  +  9a  =  4  a?, 

36a2(4a?+9a)-48aa;V4a?  +  9a  +  16a;2  -  64a;2. 

Extract  the  root,  6a(4a;  +  9a)*  —  4a;  =  ±  8a;. 

Divide  by  2,  6a(4a;  +  9a)*  =  12a;  or  -  4a;. 

Square,  3a(4a;  +  9a)*  =*  6a;  or  -2ar 

Transpose,  •  36a2x  +  81a8  =  36  a?  or  4  a2 

36a?-36a2a;=81a8, 

and  4a?-36a2a;  =  81a8. 

36a;2-  36a2a;  +  9a4  -  9a4  +  81a8, 


6a;  -  3a2  =  ±  3aVa2  +  9a, 


6a;  =  3a2±3aVa2+9a. 
.-.  a;  =  |(a±N/a2+9a). 

From  4a;2-36a2a;  =  81a3, 

4a;2 -( )  +  81a4  -  81a2(a2  +  a), 
2a;-9a2  =  ±9a(a2  +  a)*. 

.%  a,*  =  -^(a±Va2  +  a). 
4 


396  ALGEBRA. 


(x  +  l+a^Xs-l+ar1)5"5*- 


? V' 

4x*  +  4a*  +  4  =  21*», 
4**-17x»  =  -4, 
64a?4- ()  4- 289  =  225. 
Extract  the  root,     8  Xs  -  17  -  ±  15, 

8a*  =  32  or  2, 

x*  =  4or}. 

,  ,\  a?  =  ±2  or  ±  J. 

30. 

2(x*-  l)-i-2(x*-4)-1  =  3(x*  -2)"1. 

2 2  3 

xl_l     a;4_4     a;*-2 

2a:  -  12s*  +  16  -  2  x  +  6 a*  -  4  =  3  a;  - 15 x*  +  12, 
-3a;  +  9a:*  =  0, 
3aj*(a*  +  3}  =  0, 

a?i  =,  _  3  or  0. 
,\  a>  =  9or0. 


Exercise  CIX. 

1.  log  6  =  log  (3x2)  3.  log  21  =  log  (7X3) 

-  log3  +  log  2.  -  log  7  +  log 3. 

log  3  =  0.4771  log  7  =  0.8451 

log   2  =  0.3010  log  3  =  0.4771 

.-.log  6  =  0.7781  .-.log 21  =  1.3222 

2.  log  15  =  log  (3x5)  4.   log  14  =  log  (7x2) 

=  log3+log5.  =log7+log2. 

log  3  =  0.4771  log   7  =  0.8451 

log  5  =  0.6990  log  2  =  0.3010 

.-.  log  15  =  1.1761  .%  log  14  =  1.1461 


teachers'  edition.  397 

5.    log 35  =  log (7x5)  11.  log42  =  log(7x2x3) 

—  log  7  +  log  5.  =  log  7  +  log  2  +  log  3. 

log   7  =  0.8451  log   7  =  0.8451 

log   5  =  06990  "  log   3  =  0.4771 

.•.log35-L6441  lo*   2  =  O3010 

.-.log  42  =1.6232 

,    log   9  =  log(3x3)  »  ^42°:S^f0-35X7) 
-log  3  + log  3.  8+log?+log7. 

log   3  =  0.4771  log    2'  =  0.6020 

log  3  =  0.4771  log    3  =0.4771 

.-.log   9  =  0.9542  }°g    I  -£££? 

6  log    7  =  0.8451 

.-.log  420  =  2.6232 

7-    ^   8:iSf+X4X2?Iogl      13.1ogl2  =  log(2x2x3) 
lot   2-03010  =log2  +  log2  +  log3. 

b|  2  =  a3olo  H  1  =  0.3010 

lol   2  =  03010  Jog   2  =  0.3010 

.Mog   8  =  0.9030  ...log  12  =  1^1 

14.  Iog60  =  log(2x  3X2X5) 

8.    log49  =  log(7x7)  =  log2»+log3  +  log5. 

=  log  7  + log  7.  log  2  =  0.3010 

log   7  =  08451  log   2  =  0.3010 

log   7  =  0.8451  }°g   3  =  04771 

e  log  5  =  0.6990 


.  log  49  =  1.6902 


,  log  60  =  1.7781 


15.  log75  =  log(5x5x3) 

9.    log 25  =  log (5x5)  =log5  +  log5+log3. 

=  log  5  +  log  5.  !       5  =  0.6990 

log   5  =  0.6990  log   5  =  0.6990 

log   5  =  0.6990  log   3  =  0.4771 

.\  log  25  =  1.3980  ...  log  75  =  1.8751 

16.  log7.5  =  log(3X  5X5X0.1) 

10.1og30  =  log(2x3x5)  =  ^+io#+logO  1 

=  log2  +  log3+log5.  j        3 -0.4m 

log  2  =  0.3010  log    5  =  0.6990 

log  3  =  0.4771  log    5  =  0.6990 

log   5  =  0.6990  log   .1^9.0000-10 

.-.  log 30  =  1.4771  ...  log  7.5  =  0.8751 


ay 

O                                          "           ALGEBRA. 

17. 

logO.  021 =log(7  X  3  X  0.001) 
=log7+log3 

+log0.001 
log        7=0.8451 
log        3=0.4771 
log0.001=7.0000-10 

22. 
23. 

24. 

) 

25. 
26. 

logl2.5=log(5*X0.1) 

=log6+log5+log5 
+logO,l. 
•log      5=0.6990 
log      6=0.6990 
log      5=0.6990 

.  Iog0.021=8.3222-10 

log0.36=log(7x  5X0.01) 
=log7+log5 

+logt).01 
log      7=0.8451 
log      6=0.6990 
log0.01=8.0000-10 

log  0.1— 9.0000— 10 

18. 

.log  12. 5=  1.0970 

log  1.25=log(5*X  0.01) 

=log6-Hog5-flog5 
-HogO.Ol. 
log      5=0.6990 
log      5=0.6990 
log      6=0.6990 
log0.01=8.0000-10 

.  Iog0.35=9.5441-10 

log0.0035=log(7  X  6  X  0.0001 
=log7+log6 

-HogO.0001. 
log '        7=0.8451 
log          5=0.6990 
logO.  0001  =6.0000-10 

19. 

.log  1.25=0.0970 

Iog37.5=log(58x3x0.1) 
=log5+log6+log6 
+logS+log0.1. 
log      6=0.6990 
log      5=0.6990 
log      5=0.6990 
log      3=0.4771 
log  0.1=9.0000-10 

. 

.log0.0035=7.5441-10 

log0.004=log(2  X  2  X  0.001) 
=log2+log2 

+log0.001. 
log        2=0.3010 
log        2=0.3010 
log0.001  =  7.0000-10 

20. 

.log37.5=1.5741 

log2.1=log  (3X7X0.1) 

=log3+log7+log0.1. 
log    3=0.4771 
•log    7=0.8451 
logO.  1=9.0000-10 

.log2. 1=0.3222 

21. 

.log0.004=7.6020-10 

Iog0.05=log(5x0.01) 

=log5+log0.01. 
log      5=0.6990 
logO.Ol  =8.0000-10 

logl6=log(2*) 

=log2+lpg2 

+log2+log2. 

log  2=0.3010 
log  2=0.3010 
log  2=0.3010 
log  2=0.3010 

.• 

.  Iog0.05=8.6990-10 

.log  16=1. 2040 

teachers'  edition. 


399 


27.   logO. 056=log(2X2X2X 7x0.001) 

=log2+log2+log2+log7+log0.001. 
log        2=0.3010 
log        2=0.3010 
log        2=0.3010 
log        7=0.8451 
log0.001=7.0000-10 

.-.log0.066=8. 7481-10 

28.  Iog0.63=log(3x 3x7x0.01)   30. 
=log3+log3 

+Iog7+log0.01. 
log      3=0.4771 
log      3=0.4771 
log      7=0.8461 
logO.Ol =8.0000-10 

Iogl05=log(6x3x7) 

=log5xlog3xlog7. 
log     6=0.6990 
log     3=0.4771 
log     7=0.8461 

.log  105=2.0212 

.-.logO.  63=9. 7993-10 

31. 

29.  logl.75=log(5X5x7x0.01) 
=log6+log5 

+Iog7+log0.01. 
log      5=0.6990 
log      5=0.6990 
log       7=0.8451 
log0.01=8.0000-10 

Iog0.0105=log(3x7x6 

X  0.0001) 
=log3+log7+log5 
-flogO.0001. 
log          3=0.4771 
log          7=0.8451 
log          5=0.6990 
logO.  0001  =6.0000-10 

.logl.75=0.2431 


.  Iog0.0105=8.0212-10 


32.  Iogl.06=log(7x3x5x0.01) 

=log7+log3+log6+log0.01. 

log  7=0.8451 
log  3=0.4771 
log  6=0.6990 
log0.01=8.0000-10 

.-.logl.05=0.0212 


Exercise  CX. 


1.  log28=3xlog2 
=3X0.3010 
=0.9030. 


log74=4xiog7 
=4X0.8451 
=3.3804. 


2.  log52=2Xlog6 
=2X0.6990 
=1.3980. 


4.  log38=8xlog3 
=8X0.4771 
=3.8168. 


400  ALGEBRA. 


5.  log7*  =  3xlog7  16.  W  7*  =  *oflog7 

=  3x0.8451  6       =  I  of  0.8451 

=  2.5353.  =5.2415. 


6.  log  5s  =  5  xlog5.  , 

=  5x0.6990  17.  log   5*  =  $  of  log  5 

=  3.4950.  =fof0.699C 

=  1.1650. 

7.  log2*  =  lofloe2 

=  i  of  0.3010  18.  log  3*  =  -X  of  log  3 

=  0.1003.  B        =  A  of  0.477 

l\  OCK\A 


8.  log  5*  =  J  of  log  5 


[771 
=  0.3904. 


03195  19-l08   ^  =  ioflog7 

°-3495-  =  J  of  0.8451 

=  2.9579. 

9.  log  3*  =  i  of  log  3 

1^96  »k8   3*  =  4oflo?3 

=  0.059b.  =  I  of  0.4771 

=  0.6361. 

10.  log  7*  =  J  of  log  7 

-  f  of  0.8451  - 

=  0.1690.  21.  log  5*=  $  of  log  5 

=  |of0/™ 

11.  log  5*  =  t  of  log  5 

=  *of0.P~ 


(990 
2.4465. 


,■  ux  v.6990  u 

6.1398.  22.  log  2 V=  J^.  of  log  2 

=  V  of  0.3010 

,0  ,     *A      1      m      *  =0.4730. 

12.  log  7A= -A- of  log  7 

= -A  of  0.8451 

-0.0768.  23.  log  5t  =  foflog5 

=  I  of  0.6990 

13.  log  2*  =  }  of  log  2  =0.5243. 

=  |  of  0.3010 

~  °2258-  24.  log  7*=  V  of  log  7 

=  V-  of  0.8451 

14.  log  5*  =  i  of  log  5  =1.3280. 

-J  of  0.6990 

"  °,466a  25.  log  21*  =  i  of log(7x 3). 

log   7  =0.8451 

15.  log  3?  =  4  of  log  3  log   3  =  0.4771 

=  4  of  0.4771  log  21   =1.3222 

=  0.2045.  i  of  1.3222  =  1.1569. 


teachers'  edition.  401 


Exercise  CXI. 

1.  log  $  =  log  2  +  colog  5.  8.    log  §  =  log  5  +  colog  2. 
log  2  =  0.3010  log  5  -  0.6990 

colog  5  =  9.3010  -  10  colog  2  =  9.6990  -  10 

. ..  log  i  =  9.6020  -  10  .  \  log  {  =  0.3980 

2.  log  $  =  log  2  +  colog  7.  9.    log  }  =  log  7  + colog  3. 
log  2  =  0.3010  log  7  =  0.8451 

colog  7  -  9.1549  -  10  coiog  3  -  9.5229  -  10 

.-.  log  *  =  9.4559  - 10  ...  i0g  j  =  0.3680 

3.  log  f  =  log  3  + colog  5.  ,*,.,*        i      o 
log 3 -0.4771  10-    log* -log 7  + colog 2. 

colog  5  =  9.3010 -10  log  7  =  0.8451 

5  colog  2  =  9.6990 -10 


.-.log  $  =  9.7781 -10 


6.    log  $  =  log  7  +  colog  5. 
log  7  =  0.8451 
colog  5  =  9.3010 -10 


log}  =  0.5441 


4.  log  $  =  log  3  -f  colog  7. 

log  3  =  0.4771  n.    bg  f  =  log  3  + colog  2. 

colog  7  -  9.1549  - 10  log  3  =  0  4771 

. ..  log  }  =  9.6320  - 10  colog  2  =  9.6990-10 

.-.log  J  =  0.1761 

5.  log  f  =  log  5  +  colog  7. 

log  5  =  0.6990 
colog7  =  9.1549-10  ^     log^=log7  +  cok*a* 

,.log*  =  9.8539-10  JogWMSl 

colog  0.5=  0.3010 


log  7  =  0.8451  .  • .  log  j^:  =  1. 1461 


.-.logJ  =  0.1461  006 

1$.     Iog-jp  =  log0.05+colog3. 

7.    log  i  -  log  5  +  colog  3.  Iog0.06  =  8.6990-10 

log  5  =  0.6990  colog      3  _  9  5229-IO 

colog  3 -9.5229-10  n  nft     

,      .      ZZZ2 .-.log^  =  8.2219- 10 

.\  log  f -0.2219  3 


402 


ALGEBRA. 


14.  log 


0.005 


Iog0.005+colog2.         20.  log 


logO.  005  =  7.6990  -10 
colog        2  =  9.6990  —  10 

.••log^  =  7.3980-10 


0.007 


:  logO.  007  -fcologO.02. 


0.02 

log0.007  =  7.8451 -10 
colog  0.02  =  1.6990 

••1°^7=e-5441-10 


15.  log  -J7—  =  log0.07  +  colog  5. 

•       log  0.07  =  8.8451-10 
colog        5  =  9.3010  —  10 

...log  5^1=8.1461-10 
o 


21.  Iog^|  =  log0.02+colog0.0f;7. 

log  0.02  =  8.3010-10 
colog0.007  =  2.1549 

•••l0W7  =  °-4569 


16.  log  —  =log5  +  colog0.07. 

log        5  =  0.6990 
colog  0.07  =  1.1549 


22.  Iog^^  =  log0.005+colog0.07. 

Iog0.005  =  7.6990- 10 
colog  0.07  =  1.1549 

.'.log^  =  8.8539-10 


17.  Iogj^y  =  log3  +  colog0.007. 

log        3  =  0.4771 
colog0.007  =  2.1549 
3 


.log 


0.007 


=  2.6320 


23.  log  ^  =  log0.03  +  colog7. 

log  0.03  =  8.4771-10 
colog        7  =  9.1549-10 


0.03 


•.log  ^=7.6320-10 


18.  log 


0.003 


logO.  003  +  colog7. 


Iog0.003  =  7.4771 -10 
colog        7  =  9.1549-10 

...log2^?  =  6.6320  -10 


**-  log^^  =  log0.0007+cologOi 

logO.0007  =  6.8451-10 

colog       0.2  =  0.6990 

,     0.0007     Z~77T<i     ™ 
.•.log-^y-  =  7.5441-10 


19-  Iog^|  =  log0.05+colog0.003.    25.  Iog^^=log0.022  +  colc 


log  0.05  =  8.6990-10 
colog  0.003  =  2.5229 


logO.  022  =  6.0O2O  -10 

colog      38  =  8.5687-10 

0  022     " 

...log^-  =  5.1707-10 


TEACHERS     EDITION. 


403 


33 
26.  log  —  =  log 33  +  cologO.022. 

log      38=1.4313 
colog0.022  =  3,3980 

•••logofe=4- 


27.  Iog-^  =  log78  +  colog0.022. 

log      78  =  2.5353 
colog0.022  =  3.3980 
78 


.log 


0.02*" 


:  5.9333 


28.  Iog|^8  =  log0.078  +  colog0.003». 

log  0.078  =  6.6353-10 
cologO.  0038  =7.5687 


0.078 
.log^ro  =  4.1040 


'O.OOS3 

29.  Iog^^  =  log0.0052+colog78. 

logO.  0062  =  6.3980  -10 
colog        78  =  7.4647-10 

...log2^  =  2.«7-10 


30.  log- 


*0.0062 
log        78=2.5363 
colog0.0052  =  4.6020 


=  log78  +  colog0.006a. 


Exercise  CXII. 


log     60  =  1.7782. 


-  6. 

log  3780  =  3.5775. 


log  101  =  2.0043. 

3. 

log  999  =  2.9996. 


log  54327  =  4. 7348  +  Jfo  of  0.0008 
=  4.7350. 


Iog90801  =  4.9581  +  Tfo  of  0.0005 
=  4.9581. 


Iog9901  =  3.9966  +  Jfc  of  0.0005  9t 

=  3  9957  log  10001  =  4.0000 +  T^  of  0.0043 

=  4.0000. 

5.  10. 

Iog5406  =  3.7324  +  ^  of  0.0008       log  10010  =  4.0000  +  ^ft  of  0.0043 
=  3.7329.  =4.0004. 


404  ALGEBRA. 


11.  22. 

log    70633=4.8488+^  of  0.0006       Antilogarithm  of  3.6330. 

=4.8490.  Number  corresponding  to  0.633Q 


12. 

log    12028=4.0792+^  of  0.0036 
=4.0802. 


is  4290+T5ff  of  10=4295. 
.  * .  number  required  is  4295. 


13. 


Antilogarithm  of  2.5310. 
,     ~  ~~.™    -~~,«    .~  Number  corresponding  to  0.5310 

log0.00987=7.9843-10.        -  fa  3390+A  of^3396^ 

]^  .  • .  number  required  is  339.6. 

logO.87701  =9.9430-10.  24. 

^  Antilogarithm  of  1.9484. 

*  ,™,     ~~™,    ,      **^m*       Number  corresponding  to  0.9484 
log  1.0001=0.0000+^  of  0.0043  ^  ^ 

=0.0000.  .  .     ,.   QOQ 

.  • .  number  required  is  88.8. 

16-  26. 

log  877.08=2.94^+TJ5of0.0005       AnffloglHWim  rf  4W17. 

Number  corresponding  to  0.7317 
17  is  6390+ 1  of  10=5391. 

log  73.896=  1.8681 +^  of  0.0005       • '  • numl)er  required  is  53910. 
=1.8686.  M 

jg  Antilogarithm  of  1.9730. 

log  7.0699=0.8488+^  of  0.0006       Number  corresponding  to  0.9730 
=0.8494.  is  9390+f  of  10=9398. 

.  • .  number  required  is  93.98. 
19. 

log  0.0897=8.9628-10.  **- 

Antilogarithm  of  9.8800-10. 

20.  Number  corresponding  to  0.8800 

log  99.778=1.0987+^  of  0.0004   is  7680+ J  of  10=7686. 

=  1.9990.  .  • .  number  required  is  0.7586. 

2L  28. 

Antilogarithm  of  4.2488.  Antilogarithm  of  0.2787. 

Number  corresponding  to  0.2488  Number  corresponding  to  0.2787 

is  1770+|  of  10=1773.  is  1890+fJ  of  10=1900. 

.  • .  number  required  is  17730.  .  • .  number  required  is  1.9. 


TEACHERS     EDITION. 


405 


29. 

Antilogarithm  of  9.0410—10. 
Number  corresponding  to  0.0410 
is  1090+fJ  of  10=1099. 
.  • .  number  required  is  0.1099. 

30. 

Antilogarithm  of  9.8420—10. 
Number  corresponding  to  0.8420 
is  6950. 
.  • .  number  required  is  0.6960. 

31. 

Antilogarithm  of  7.0216—10. 
Number  corresponding  to  0.0216 
is  1050+^-  of  10=1061. 
.  • .  number  required  is  0.001051. 

32. 

Antilogarithm  of  8.6580—10. 
Number  corresponding  to  0.6580 
is  4550. 
.•.  number  required  is  0.0455. 

33. 

948.76x0.043875. 
log    -948.76  =  2.9772 
log  0.043875  =8.6423 -10 

1.6195 
=  log  41.64. 

34. 

3.4097  X  0.0087634. 
log   3.4097  =  0.5328 
log  0.0087634  -  7.9427  - 10 

8.4755  - 10 
=  log  0.02989. 


35. 

830.75x0.0003769. 
log       830.75  =  2.9195 
log  0.0003769  =  6.5762  -  10 

9.4957-10 
-  log  0.3131. 

36. 

8.4395  x  0.98274. 
log   8.4395  =  0.9263 
log  0.98274  =  9.9925  -  10 

0.9188 
=  log  8.294. 

37. 

7564  x(-  0.003764). 
log  7564  =  3.8787 

log  (-0.003764)  =  7.5756*  -  10 
1.4543* 

-  log  -28.47. 

38. 

3.7648  x(- 0.083497). 

log  3.7648  =  0.5757 

log  (-  0.083497)  =  8.9217*  -10 

9.4974*- 10 

-  log  -0.3144. 

39. 

-5.840359  x(— 0.00178). 
log  (-5.840359)  =  0.7664* 
log  (-0.00178)  =  7.2504* -10 
8.0168  -  10 
=  log  0.0104 

40. 

-  8945.07  X  73.846. 

log  (-8945.07)  =  3.9515* 

log         73.846  =  1.8683 

5.8198* 
=  log  -660600. 


406 


ALGEBRA. 


41. 

-  log  70654  +  colog  54013. 


70654 

54013 

log     7Q§54  =  4.8491 

colog  54013  =  5.2675 

0.1166 
=  log  1.308. 


10 


43. 

M?i^  =  log  8.32165 

0.07891         e         -fcolog  0.07891. 

log     8.32165  =  0.9202 

colog  0.07891  =  1.1028 

2.0230 
=  log  105.4. 


58706 


=  log  58706  +  colog  93078. 


65039 


93078 
log     58706  -  4.7686 
colog  93078  =  5.0312  -  10 
9.7998  - 10 
-  log  0.6307. 


44. 

log  65039  + colog  90761. 

4.8132 


90761 

log        65039 
colog    90761  =  5.0421  -  10 
9.8553  - 10 
=  log  0.7167. 


7.652 


45. 


■  log  7.652  +  colog  (-  0.06875). 


-  0.06875 

log  7.652=   0.8838 

colog  (-  0.06875)  =  11.162? 
2.0465" 
-  log  -111.3, 


10 


0.07654 


83.947  X  0.8395 
log  0.07654  = 

colog  83.947  = 

colog  0.8395  = 


46. 

=  log  0.07654  +  colog  83.947  +  colog  0.8395. 


8.8839  - 10 

8.0760  - 10 

0.0759 

7.0358  - 10 

dog  0.001086. 


47. 


7564  x  0.07643 

8093  x  0.09817' 

log  7564  =  3.8787 

log     0.07643  =  6.8832-10 

colog      8093  =  6.0919  -  10 

colog  0.09817  =  1.0080 

9.8618  - 10 
-log  0.7277. 


48. 

89  X  753  x  0.0097 
36709  x  0.08497  ' 
log  89  =  1.9494 

log  753  =  2.8768 

log  0.0097=7.4352-10 
colog  36709  =  5.4352-10 
colog  0.08497  =  1.0708 

9.3190-10 
=  log  0.2084. 


teachers'  edition. 


407 


49. 

413  x  8.17  X  3182 

915  x 

728  x 

2.315 

log 

413 

=  2.6160 

log 

8.17 

=  0.9122 

log 
colog 

3182 

=  3.5027 

915 

=  7.0386  - 

10 

colog 

728 

=  7,1379  - 

10 

colog 

2.315 

=  9.6354  - 

0.8428 
=  log  6.963 

50. 

10 

212  x 

(-  6.12)  X(-  2008) 

365  X  (-  531)  x  2.576 
log  212  -  2.3263 

log  (-6.12)  =  0.7868* 
log  (-  2008)  =  3.3028* 
colog  365  =  7.4377  -10 

colog   (-531)  =  7.2749* -10 
colog       2.576  =  9.5891   -10 

0.7176* 
=  log -5.21 9. 

51. 

log  6.05  =0.7818 

3 

log  6.05s  =  2.3454 

-  log  221.5. 


log  1.051   =0.0216 

7 

log  1.0517  =  0.1512 

=  log  1.416. 

53. 

log  1.1768  =0.0707 

5 

log  1.17685  =  0.3535 

=  log  2.257. 


54. 

log  1.3178   =0.1198 

10 

log  1.317810=  1.1980 

=  log  15.78. 

55. 

log  0.78765  =9.8963-10 


log0.787656  =  9.3778  -10 
=  log  0.2387. 


56. 

log  0.691  =9.8395- 
9 


10 


log  0.6919  =  8.5555 -10 
=  log  0.03593. 

57. 

log  (ft)11  =  11  (log  73  +  colog  61) 
=  11(1.8633+8.2147-10) 
=  0.8580 
=  log  7.212. 

58. 

log  (M)7  =  7  (log  14+ colog  51) 

=  7(1.1461  +  8.2924-10) 
=  6.0695-10 
=  log  0.0001174. 


59. 


(io|)* -(w. 

-~)*  =  4(logc 

=  4(1.505f+  9.5229  -10) 


log(W 


1 32  +  colog  3) 


=  4.1120 
=  log  12940. 

60. 


(1 


log  (iff  =  8  (log  16  +  colog  9) 

=  8(1.2041  +  9.0458-10) 
=  1.9992 
=  log  99.82 


408  ALGEBRA. 


61.  log(Hi)6      =  6  (log  951+ colog  823) 
=  6(2.9782  +  7.0846-10) 
=  0.3768 
-  log  2.381. 


62.  (7*)""        =(H)0J8. 

log  (ff)0-88    -  0.38  (log  83  +  colog  1 1) 

-  0.38(1.9191  +  8.9586  -  10) 

-  0.3335 

=  log  2.155. 


63.   (3|B«        =(W)41T- 

log  (Vt1)4*17  =  41 7  (log  120  +  colog  31) 

-  4.17(2.0792  +  8.5086  -  10) 

-  2.4511 

=  log  282.6. 


64.  (W  -(H)* 

log  (H/8*2      =  3.2(log  13  +  colog  11) 

=  3.2(1.1139  +  8.9586-10) 
=  0.2320 
=  log  1.706. 


65.  (8})"  '-(W. 

log  (V)2'8    * =  2-3  (log  35  +  col°g  4) 

=  2.3(1.5441  +  9.3979  -  10) 

-  2.1666 

=  log  146.8. 


66.  (5||f875       =(Wf875- 

log(W)0'375  ■=  0.375(log  216  +  colog  37) 

=  0.375(2.3345  +  8.4318  - 10) 
=  0.2874 
=  log  1.938. 


teachers'  edition.  409 

67.  log  7  =  0.8451.  71.  log     906.80=2.9575. 

3)0.8451  4)2.9575 

log  7*  =  0.2817  log   906.80*  =  0.7394 
=  log  1.913.  -log  5.487. 

72.  log    8.1904  =  0.9133. 

68.  log  11  =  1.0414.  5)0.9133 

5)1.0414  log  8.1904*  =  0.1826 
log  11*  =  0.2083  =  log  1.523. 

=  log  1.616. 

73.  log  0.17643  =  9.2466  -  10 

5 

69.  log  783  =  2.8938.  46.2330  -  50 

31|^  6)56.2330-60 

log  783*  =  0.9646  0.17643*=  9.3722- 10 

=  l089-218-  =log0.2356. 

74.  log  2.5637    =0.4088 

70.  log    8379  =  3.9232.  3 

10)3.9232  11)1.2264 

log  8379*  =  0.3923  log  2.5637*  =  0.1115 

=  log  2.468  =  log  1.293. 

75.  log («i)*  =  l(log  431  +  colog  788). 


log     431  - 
colog  788  = 


431  -  2.6345 

7.1035  - 10 

9.7380  - 10 

10.         - 10 


2)  19.7380 -"20 

9.8690 -10  =  log  0.7397. 

76.  log  (*#**)*  =  *  (log  71  +  colog  43406). 

log  71  =  1.8513 

colog  43406  =  5.3624  -  10 
7.2137  - 10 

4 

28.8548  -  40 
30.  -  30 

7)58.8548-70 
""  8.4078  -  10  =  log  0.02558. 


410  ALGEBRA. 

77.  (9H)*  =  (W)*- 

log  (W)*  =  iflog  408  +  col°g  43)- 
log     408  =  2.6107 
colog   43  =  8.3665  -  10 
5)0.9772 

0.1954  =  log  1.568. 

8.  (llftf-W)1-  ,  ,     „„ 

log  (W)1  =  i  (log  802  +  colo8  71)' 
log     802  =  2.9042 
colog    71  =  8.1487  -  10 

1.0529 
$  (1.0529)  =  0.8423  =  log  6.955. 

?9     6/0.0075433'  x  78.343  x  8172.4*  X  0.00052 
*     64285*  X  154.27*  X  0.001  x  586.79* 

log0.0075433*=  5.7552  - 10 
log        78.343  =  1.8940 
log      8172.4*  =  1.3041 
log     0.00052  =  6.7160  -  10 
colog    64285*  =  8.3973  - 10 
colog    15427*  =  1.2468 -10 
colog      0.001  =  3.0000 
colog  586.79*  =  8.6158 -10 
5)36.9292-50 

7.3858 -10  =  log  0.002431. 

g0      •;         15.8328  x  5793.6*  X  0.78426 

^0000327*  X  768.94*  X  3015.3  X  0.007* 


log         15.832s  =  3.1 
log        5793.6*  =  1.2543 
log       0.78426  =  9.8445-10 
colog  0.000327*  =  1.1618 
colog     768.942  =  4.2282 -10 
colog      3015.3  -  6.5207  -  10 
colog      0.007*  =  1.0774 

27.7357  -  30 

20.         -  20 
5)47.7357-50 

9.5471 -10  =  log  0.3525. 


TEACHERS'    EDITION.  411 


81. 


5/7. 


82. 


7.1895  x4764.22x  0.00326s 
00048953  X  457s  X  5764.42" 

log  7.1895  =  0.8566 

log  4764.2s  =  7.3558 

log         0.003265=  7.5660 -20 
colog  0.00048953  =  3.3102 
colog  457s  =  2.0203 -10 

colog        5764.43  =  2.4786  -  10 
23.5875-40 
10.         - 10 
5)33.5875-50 

6.7175 -10  =  log  0.0005218. 
& '3.1416x4771.21x2.7183* 
>  30.103*  x  0.4343*  x  69.897* 


log      3.1416  =  0.4971 
log    4771.21  =  3.6786 
log    2.71834  =  0.2172 
colog  30.103*  =  4.0856  -  10 
colog  0.4343*=  0.1811 
colog  69.897*  =  2.6220  -  10 
11.2816-20 
30.         -  30 


5)41.2816-50 

8.2563 -10  =  log  0.01804. 


03     t  '0.032712  X  53.429  x  0.77542s 


"•a! 


32.769  x  0.000371* 

log       0.03271*  =    7.0292-10 
log  53.429=    1.7278 

log       0.77542s  =    9.6688-10 
colog      32.769=   8.4845-10 
colog  0.000371*  =  13.7224 
7)  10.6327 

1.5190  =  log  33.04. 
732.056'  x  0.0003572*  x  89793 
1.2798s  x  3.4574  x  0.00265185' 

log         732.0562=   5.7290 
log     0.0003572*=   6.2116-20 
log  89793=    4.9533 

colog     42.2798s  =   5.1217  -10" 
colog        3.4574=    9.4612-10 
colog  0.00265185  =  12.8825 
3)4  3593 

1.4531  =  log  28.39. 


412  ALGEBRA. 

3-     »/7932x  0.00657x0.80464 
\         0.03274x0.6428 


log  7932  =  3.8994 

log     0.00657  =  7.8176  -  10 
log     0.80464  =  9.9056  -  10 
colog  0.03274  =  1.4849  -  10 
colog   0.6428  =  0.1919 
3)3.2994 

1.0998  =  log  12.58. 


86. 


^ 


7.1206  xVO.13274  X  0.057389 


V0.43468  X  17.385  x  V0.0O96372 

log  7.1206  =  0.8525 

log         V0.13274  =  9.5615  - 10 
log  0.057389  =  8.7588  - 10 

colog      V0.43468  =  0.1809 
colog  17.385  =  8.7599-10 

colog  V0.0096372  =  1.0080 

3)29,1216-30 

9.7072  - 10  =  log  0.5096. 


_      f  3.075526*  x  5771.2*  X  0.0036984*  X  7.74  \  * 
t  72258  X  327.93*  X  86.97*  J 

log   3.075526s  =  0.9758 
log       5771.2*  - 1.8806 
log  0.0036984*  -  9.5136  - 10 
log  7.74  =  0.8887 

colog      72258  =  5.1412-10 
colog   327.93*  -  2.4526  - 10 
colog     86.97*  =  0.3030  - 10 
1.1555-20 
3 


3.4665-   60 
40  -   40 


5)43.4665-100 
8.6933-   20 
=  log  0.000000000004936. 


teachers'  edition.  413 


Exercise  CXIII. 

1.  Write  down  the  ratio  compounded  of  3 : 6  and  8 : 7.  Which 
of  these  ratios  is  increased  and  which  is  diminished  by  the  com- 
position? 

fxf-tf. 

As  f-fi 

and  f-«. 

3  :  5  is  increased. 
8  :  7  is  decreased. 

2.  Compound  the  duplicate  ratio  of  4 :  15  with  the  triplicate 
of  5 : 2. 

3.  Show  that  a  duplicate  ratio  is  greater  or  less  than  its  sim- 
ple ratio  according  as  it  is  a  ratio  of  greater  or  less  inequality. 

a?  :  b2  is  >  or  <  a  :  b. 


As 

6* 

is>or<~ 
6 

Asf 

a     a 
bXb 

is>or<£. 
6 

As 

a 
b 

is  >  or  <  1. 

4.  Arrange 

in  order  of  magnitude  the 

:  ratios 

3 

:4; 

23; 

:25; 

10 

s  11;  and  15 

:16. 

3 
23 

10 
15 

.  •.  the  order  of  magnitude  is  15  :  16,  23  : 

25,  10 : 

11, 

3: 

4. 

414  ALGEBRA. 

5.  Arrange  in  order  of  magnitude 

a  +  b  :  a  —  b  and  a2  +  6* :  a*  —  &*,  if  a>  b. 

a  +  6  :  a  -  6  is  >  or  <  a*  +  6* :  a2  -  6'. 

A,  «±|iB>or<4±|. 

a  —  b  a*  —  6* 

^       a«  +  2ai  +  yig>or<^±y. 

a*  —  6*  a2  —  6a 

As  a*  +  2  aft  +  b*  is  >  or  <  a*  +  6*. 

But         a*  +  2a&  +  &,is>a,  +  &*. 

.  • .  a  +  b :  a  -  b  is  >  a*  +  &* :  aJ  -  6*. 


6.  Ratio  compounded  of 

3:5;  10:21;  14:15, 

=  *xtfxtt 
-A 

-4:15. 

7.  Ratio  compounded  of 

7:9;  102:105;  15:17, 

=  2:3. 


8.  Ratio  compounded  of 


a*  +  ax  +  sc1  ,  a2  -  ax  4-  a* 


a8  —  cfo  +  flKc2—  a8  a  +  a; 

_  a4  +  a V  +  g* 
a*-s* 


9.  Ratio  compounded  of 


s2-9a?4-20  andga-13a;+42 
sc2  — 6a:  a2  — 5a? 

(s-5)(g-4)     (g-fl)(s-fl 
s(a>-6)  a>(a;-5) 

-(a* -11a; +  28)  jar2. 


teachers'  Edition.  415 

10.  Ratio  compounded  of 

a  +  b  :  a  -  b ;  a2  +  b2 :  (a  +  b)2 ;  (a2  -  b2)2  :  a*  -  6*. 
=  a  +  b      a2  +  b2      (a2  -  V)2 

a-b     (a  +  bf       a*-  b* 
=  a  +  b  a2  +  b2  (a-b)(a  +  b)(a-b)(a  +  b) 

a- 6     (a  +  6)(a  +  6)        (a2  +  b2) (a  +  6) (a -  6) 
=  1:1. 

11.  Two  numbers  are  in  the  ratio  2 : 3,  and  if  9  is  added  to 
each,  they  are  in  the  ratio  3 :  4.    Find  the  numbers.  • 

Let  2x  and  3  s  =  the  numbers. 
Then  2x  +  9:  3a?  +9: :  3:  4, 
8a; +  36 :9s +  27. 
.\  x  =  9. 
Hence,     2  a  =18, 
and  3  a;  =  27. 

12.  Show  that  the  ratio  a :  b  is  the  duplicate  of  the  ratio 
a  +  c :  b  +  c,  if  c*  ==  ab. 


a  =  fa  +  c\2 
mes,  when 

6      \6+i 


This  becomes,  when  c2  =  aft, 

■Va6V. 

\-Vab) 

a  _a2  +  2aVa6  +  aft 

6     62  +  26  Va6  +  a& 

a(o  +  2Va6  4-  6) 

b(b  +  2Vab  +  a) 

13.  Find  two  numbers  in  the  ratio  3 : 4,  of  which  the  sum  is 
to  the  sum  of  their  squares  in  the  ratio  of  7 :  50. 

Let  3  a;  =  first  number, 

and  4  a;  =  second  number. 

Then  3a;  +  4a;  or  7a;  =  sum. 
dx2 +  16a;2  or  25a;2  =  sum  of  squares. 
.-.  7a; :  25a;2  : :  7  :  50,     or  x  :  x2  : :  1 :  2,     or  1 :  x  :  :  1 :  2. 

.-.a;  =  2. 
The  required  numbers  are  6  and  8. 


416  ALGEBRA. 


14.  If  five  gold  coins  and  four  silver  ones  are  worth  as  much 
as  three  gold  coins  and  twelve*  silver  ones,  find  the  ratio  of  the 
value  of  a  gold  coin  to  that  of  a  silver  one. 


Let 

x  =  value  of  1  gold  coin, 

and 

y  —  value  of  1  silver  coin. 

Then 

5z  +  4y  =  3a:-f  12y, 

2a  =  8y. 

.-.  &  =  4y. 

That  is, 

x.y.A:  1. 

15.  If  eight  gold  and  nine  silver  coins  are  worth  as  much  as 
six  gold  and  nineteen  silver  coins,  find  the  ratio  of  the  value  of 
a  silver  coin  to  that  of  a  gold  one. 

Let  x  =  value  of  gold  coin, 

and  y  =  value  of  silver  coin. 

Then  8x  +  9y  =  6x  +  19y, 

or  2o;  =  10y. 

.-.  x=*5y. 

That  is,  y :  z  : :  1 : 5. 

16.  There  are  two  roads  from  A  to  B,  one  of  them  14  miles 
longer  than  the  other;  and  two  roads  from  B  to  C,  one  of  them 
8  miles  longer  than  the  other.  The  distance  from  A  to  B  is  to 
the  distance  from  B  to  C,  by  the  shorter  roads,  as  1  to  2 ;  by  the 
longer  roads,  as  2  to  3.    Find  the  distances. 

Let  x  =  shorter  road  from  A  to  B, 

and  x  +  14  =  longer  road  from  A  to  B. 

Then  y  =  shorter  road  from  B  to  C, 

and  y  +  8  =  longer  road  from  B  to  C. 

That  is,  x :  y : :  1 :  2, 

x  +  14  :  y  +  8  : :  2  :  3. 

.'.  2x  =  y  (1) 

And  3»+42  =  2y  +  16, 

or  3a;-2y--26. 

Substitute  2x  for  y,     x  =  26, 
x  +  14  -  40, 
y  =  52, 
y  +  8  =  60. 


teachers'  edition.  417 

17.  What  must  be  added  to  each  of  the  terms  of  the  ratio 
m :  n,  that  it  may  become  equal  to  the  ratio  p:  q? 

Let 


x-- 

=  number  to  be  added. 

.  m  +  x 

9. 

mq  +  qx- 
x(q-p)> 

.:  x-- 

=pn  +pxy 

=  pn  —  mq. 

pn  —  mq 

18.   A  rectangular  field  contains  5270  acres,  and  its  length  is 
to  its  breadth  in  the  ratio  of  31 :  17.    Find  Its  dimensions. 

Let  31  x  =  number  of  rods  in  length, 

and  17 a:  =  number  of  rods  in  width. 

Then      31a;  X  17a;  =  number  of  square  rods  in  area. 
But        160  X  5270  =  number  of  square  rods  in  area. 
.-.  31a;  X  17a;  =  160x5270. 
527  a:2  =  160x5270, 
a8  - 1600, 
a?  =  40, 
31  x  =1240  rods, 
17a;  =  680  rods. 


Exercise  CXIV. 


1.    Ita:b::c:d, 

2.   3a  +  b:b::Sc  +  d:d. 

a     c 
b~d 

Ifa:b::c:d,     £-1 
b     d 

Multiply  by  -, 
n 

Multiply  by  3,^  =  ^. 
o        d 

ma     me 

Add  1  to  each  side, 

nb      nd 
That  is, 

3«  +  1  =  3c+] 
o               a 

ma  :  nb  :  :  mc : 

nd. 

3a  +  o_3c  +  d 
b             d 
.-.  3a  +  6:  o:  :  3c  +  d:  d 

418  ALGEBRA. 


3.    Ifo:5::C:d,  5-    a  ■  <*  +  b:  '■  c  :  c  +  d- 

.  a      c 

then  -  =  -• 

b     a 


Add  2  to  each  side, 

6  a 

o4-26     c  +  2d 


If  a :  6  : :  C :  d,         T  = 

c 

6 

d 

By  inversion,       -  = 

d 

a 

c 

By  composition, 

6+  a 

d+  c 

a 

c 

By  inversion, = 

J                   a  +6 

c 

d  +  c 

.-.  a:  a  +  6: 

:  c:  C  +  d. 

b  d 

,a  +  2b:b::C  +  2d:d. 


6.   a:a  —  b::C:c  —  d. 

If    a:b::C:d,     -  =  — • 
b      a 

4.  Since  «:  6:  :«:<*,  By  inversion,       l-t 


a     c 


a      c 

b     d  By  division,    _-H-  =    ~    • 

a  c 

Cubing,        gj  =  Jy  B    inversion|  _«_  _     « 

°^      a  J  a-b      c—d 

.  ■.  os :  6s  :  :  c* :  ds.  •  %  a  :  a  —  6  :  :  c  :  c  —  d 


7.    If  a:  6::  e:  (2, 


a=  c 
b     d 
m  ma     mc 


Multaplyby-,  --^ 

By  composition  and  division, 

ma  +  nb  _mc  +  nd 
ma  —  n&     mc  —  nc? 
.  \  ma  +  w6  :  rwa  —  n6  :  :  mc  +  nd :  me  —  nrf. 


8.    If  a  :  6  :  :  c :  d, 


Also, 


£  =  £. 
b~  d 

a , 3=c      3 

6     2     d     2' 

2a-f36_  2c  +  3d 

26  2d 

.  2a  +  3&  =  2c  +  3rf 

6  d 

a__4  =  c  _4 

6      3      <*     3' 


(1) 


teachers'  edition.  419 


Dividing  (1)  by  (2), 


3a-46_3c-4d 

36  3d 

,3q-4&_3c-4d 

b  d 

2a  +  3b     2c  +  3d 


(2) 


3a-46     3c-4d 
.-.  2a+3&:3a-46::2c  +  3d:3c-4d 


9.    If  a  :  b  :  :  C  :  d, 
By  squaring, 


6     d 

b2     d2' 

•  ma'  =  nc>L 
m&2     nd2 

Let  i^  =  n 

nd2 

Hence,  ma2  =  mb2r,  and  nc2  =  ndV, 

ma2  +  nc2  =  (mb2  +  nd2)  r, 

m&2  +  nd2  62 

.*.  ma2  +  nc2 :  nib2  +  nd2 : :  a2 :  62. 


Then 


10. 

If  a:  b: 

:  c :  d,  by  alternation,  a :  c : 

:b:d. 

a  _ 
c 

b 

a2 

b2 
=  d2 

Also, 

c      c 

d     c 

a2 

ab_ 
=  cd 

62 
d2' 

,  ™&2 
mc2 

nab  _ 
ncd 

"pd2* 

.  ma2 

1  +  nab  +  pb2 

jp62 

62 

mc2 

+  ncd  +pd2 

pd2 

d2 

.•.  ma*  +  nab  +pb2 : 

mc2  +  ncd  +pd2  :  :  b 

2 :  d2. 

420  ALGEBRA. 

11.    If  a:b::b:c, 

by  composition,  a +  b  :  a:  :b  + c:  d; 

by  alternation,  a  +  b:b  +  c:  :a:b. 


12.    Ifa:6::6:C, 


a=6  ; 

b     c 

Multiply"  g-g  | 


or  a2  :  ft2  : :  ab  :  6c. 
By  alternation,  a2 :  aft  : :  62 :  be. 

By  composition,  a2 +  ab:db  :  :b* +  bc:bc. 

By  alternation,    a*  +  ab  :  6*  +  6c : :  ab  :  be. 
Cancelling  b  in  the  terms  of  last  ratio, 
a2  +  ab  :  b2  +  be : :  a :  c. 

13.    Ifa:6::6:c,  6»  =  ac. 

Multiply  by  (a  —  c),        ab2  -  b2c  =  a2c  —  ac2. 
Add  2a6c  to  both  sides, 

ab2  +  2a6c  —  b2c  =  a2c  +  2a6c  —  ac3. 
Transpose  —  b2c  and  —'ac2, 

all2  +  2a6c  +  ac2  =  a2c  +  2a6c  +  b2c, 
or  a(62  +  26c  +  c2)  =  c(a2  +  2a6  +  62), 
or  a(b  +  c)2  =  c(a  +  6)2. 

Divide  by  c(6  +  c)2,  "•  =  (^±_^!, 

c      (6  +  c)2 

or  a :  c : :  (a  4-  6)2  :  (6  -f  c)2. 


14.  When  a,  6,  and  c  are  proportionals,  and  a  the  greatest, 
show  that  a  +  c  >  2  b. 

a  :  6  :  :  6  :  C. 

Since  ?  =  -  and  a  >  6, 

6      c 

.-.  6>c. 

Also,  since  by  division   a-^—  =  — ^  and  6  >  c, 
6  c 

.-.  a  —  6>6  —  c. 

By  adding,  6  +  c  =  6  +  c, 

a  +  c>26. 


teachers'  edition.  421 


15.  If  - — y.  =  ? — -  =  - — - ,  and  x,  y,  z  are  unequal,  then  I  +  at 

I  m  n 

+  n  =  0. 

Let  — -i  «  r,    * =  r,     -=  r. 

cm  n 

Then  x  —  y  =  lr% 

y  —  z  =  ww\ 

z  —  a;  =  nr. 

*  —  V  +  y  —  z  +  z  —  x  =  (l  +  m  +  n)rt 

or  0  =  (Z  +  m  +  n)  r. 

.  \  Z  +  m  +  n  —  0. 

16.  Find  a  when         x  +  5:  2a?- 3  : :  5a? +  1 :  3*- 3. 
Equate  the  product  of  the  means  and  the  product  of  the 

extremes,         10a.a  _  i3a.  _  3  =  3^2.  +  12a.  _  15, 
7a;2 -25*  =  -12, 
•  196a* -()  4- 625  =  289, 

14s  -25  =  ±17, 
14a;  =  42  or  8. 
.  *.  x  -»  3  or  ^. 

17.  Find  x  when         a;  +  a  :  2a;  —  b  :  :  3a;  +  b  :  4a?  —  a. 

x  +  a  _  3x+  b 

2x  —  b     4a?— a 

Clear  of  fractions,  4  x*  +  Sax  +  a2  =  Gx*  —  bx  —  6s, 

2a?-a?(3a  +  &)  =  (&2-a2), 

16a*-  ()  +  (3a  +  bf  =  a2  +  Gab  +  9&2, 

4a?-(3a  +  6)  =  ±(a+3&), 

4a;  =  4a  +  46  or  2a  —  2b. 

.  r      a  — 6 

.  •.  jc  =  a  +  0  or 

2 

18.  Find  x  when 

\/x  +  V&  :  Vx—Vb:  :  a.b. 

bVx  +  bVb  =*  aVx  —  aVb, 
(a-b)Vx  =  (a  +  b)Vb, 
^  __  (a  -f  6)  V6 

a  — ft 
.  3._(a2  +  2a&  +  &*)& 
a2-2a&  +  62 


422  ALGEBRA. 

19.  Find  x  and  y  when  x :  27 : :  y :  9,  and  a; :  27 : :  2 :  x -y. 

x  :  27  : :  y  :  9. 
.-.  x  =  3y  (1) 

x:27::2:x-y. 
•\  X*  — xy  =  54. 
Substitute  3y  for  x,  9y2  -  3y2  -  54, 
6y*  =  54, 

y«  =  9. 
.-.y  =  ±3. 
Substitute  values  of  y  in  (1),  x  =  ±  9. 

20.  Find  &  and  y  when  x  +  y  +  1  :x  +  y+2::6:7,  and  when 
y  +  2x:y-2x::12x  +  6y-3:6y  —  12ac  — 1. 

x  +  y  +  l:x  +  y  +  2::6:7. 
By  division,  x  +  y+l:l::6:l. 

.\x  +  y  +  l  =  6, 

or  x  +  y  =  5  (1) 

y  +  2x :  y  -  2x : :  12x  +  6y-3  :  6y-12x— 1. 
By  composition  and  division, 

2y:4x::12y-4:24x-2, 
or  y:2x::6y-2:12x-l. 
••.  12xy  — y  =  12xy  — 4x. 

.'.4x-y  (2) 

From  (1)  and  (2),  x  =  1, 

and  y  =  4. 

21.  Find  x  when 

x2-4x  +  2:x2-2x-l.:x2-4x:x2-2x-2. 
By  alternation, 

x,-4x  +  2:x2-4x::x2-2x-l:xa-2x— 2. 
By  division,  2 :  x2  —  4  x : :  1 :  a?2  —  2x  —  2. 

.-.  2x2-4x-4  =  x*-4x. 
.-.Xs -4. 
.-.   x  =  ±2. 

22.  A  railway  passenger  observes  that  a  train  passes  him, 
moving  in  the  opposite  direction,  in  2  seconds ;  but  moving  in 
the  same  direction  with  him,  it  passes  him  in  30  seconds.  Com- 
pare the  rates  of  the  two  trains. 


teachers'  edition.  423 

Let  x  =  rate  of  the  faster  train, 

and  y  =  rate  of  the  slower  train. 

Then  x  +  y :  x  -  y : :  30 :  2. 

By  composition  and  division, 

2a;  :2y::  32:  28. 

.\s:y::8:7. 

23.  A  and  B  trade  with  different  sums.  A  gains  $  200  and  B 
loses  $  50,  and  now  A's  stock :  B's : :  2 :  J.  But,  if  A  had  gained 
#100  and  B  lost  #85,  their  stocks  would  have  been  as  15: 3  J. 
Find  the  original  stock  of  each. 


Let 
id 

x  =  original  stock  of  A. 
y  =  original  stock  of  B, 

• 

Then 

a;  +  200:y-50::2:  J. 

Simplify, 

x  +  200  =  4y  -  200, 

a;-4y  =  -400 

a) 

Also, 

a?  +  100:y-85::15:3J. 

Simplify, 

13x  +  1300  =  60y- 5100, 

13  a  -60y--  6400 

(2) 

Multiply  (1)  by  15,   15  x  -  60y  -  -  6000 

(3) 

Subtract  (2) 

from  (3),            2x  =  400. 
.-.a;  =  200. 
200-4y  =  -400. 
.-.  y  - 150. 

24.  A  quantity  of  milk  is  increased  by  watering  in  the  ratio 
4  •.  5,  and  then  3  gallons  are  sold ;  the  remainder  is  mixed  with 
3  quarts  of  water,  and  is  increased  in  the  ratio  6 :  7.  How  many 
gallons  of  milk  were  there  at  first? 

Let  x  =  number  of  quarts  of  milk  at  first, 

and  y  =  number  of  quarts  of  water  put  in  at  first. 

Then   x  +  y  =  number  of  quarts  of  mixture  after  watering. 

.*.  x:  x  +  y  :  :  4:  5, 

x         4 
or  =  -, 

x  +  y     5 

5a?  =  4a; +  4y, 


424 


ALGEBRA. 


x  +  y  —  12  =  number  of  quarts  in  remainder  before  watering. 
*  +  y  —   9  =  number  of  quarts  in  remainder  after  watering. 


.  *  +  y- 

-12 

6 

=  7 

x  +  y- 

-9 

7x  +  7y- 

-84  = 

=  6x  +  6y- 

-54. 

X  — 

4y  = 

-   0 

(1) 

x  + 

5y  = 

=  30 
=  30 

=  6. 

(2) 

Substitute  value  of  y  in  (1), 

x- 

-24  = 

=  0. 

. 

\  x  = 

=  24  quarts 

or  6 

gallons. 

25.  In  a  mile  race  between  a  bicycle  and  a  tricycle  their  rates 
were  as  5  :  4.  The  tricycle  had  half  a  minute  start,  but  was  beaten 
by  176  yards.     Find  the  rate  of  each. 

Let         x  =  number  of  yards  bicycle  goes  per  minute, 
and  y  =  number  of  yards  tricycle  goes  per  minute. 

x  :  y  : :  5  :  4, 
4&  =  5y. 

••■-  4 

=  number  of  minutes  tricycle  was  going  after 

V        2         bicycle  started, 

=  number  of  minutes  bicycle  was  going. 


1584 

y 

1 
2  = 

-1760, 

X 

84  s- 

xy  _ 
'  2  = 

-  1760y, 

3168  a:  -ary  =  3520y. 
Substitute  ^  for  x,  5y2  =  1760y. 

...  y  =  352, 
and  x  -  440. 


teachers'  edition.  425 


26.  The  time  which  an  express-train  takes  to  travel  180  miles 
is  to  that  taken  by  an  ordinary  train  as  9 :  14.  The  ordinary 
train  loses  as  much  time  from  stopping  as  it  would  take  to  travel 
30  miles ;  the  express-train  loses  only  half  as  much  time  as  the 
other  by  stopping,  and  travels  15  miles  an  hour  faster.  What 
are  their  respective  rates? 

Let  y  =  number  of  miles  ordinary  train  goes  per  hour, 

and  y  +  15  =  number  of  miles  express-train  goes  per  hour. 

Then      =  number  of  hours  required  for  ordinary  train. 

Also, + —  =•  number  of  hours  required  for  express-train. 

y  +  15     y 

180  +  30      180        15     1yl    0 
.\ : =-=  +  —  : :  14  :  9. 

y       y  +  15     y 

1890  ^  2520   210t 
y    ""  y  +  15   y  * 
1680^  2520 

y      y  +  15' 

1680y  +  25200  =  2520y, 
840y  -  25200. 
.-.y  =  30, 
and  y  +  15  =  45. 

27.  A  line  is  divided  into  two  parts  in  the  ratio  2:3,  and  into 
two  parts  in  the  ratio  3:4;  the  distance  between  the  points  of 
seetion  is  2.    Find  the  length  of  the  line. 


Let 

x  =  one  part, 

id 

y  =  the  other  part. 

.-.  x :  y  : :  2 :  3. 

3x=2y, 

Sx-2y  =  0 

(1) 

Also, 

aj  +  2:y-2::3:4, 

4x-3y  =  -14 

(2) 

Multiply  (1)  by  3, 

9z-6y  =  0 

Multiply  (2)  by  2, 

8z-6y  =  -28 

Subtract, 

x           =28 

Substitute  value  of  x  in 

i(l),             2y  =  84, 

y-42. 

.%  x  +  y  -  70. 

426  ALGEBRA. 


28.  A  railway  consists  of  two  sections ;  the  annual  expendi- 
ture on  one  is  increased  this  year  5%,  and  on  the  other  4%,  pro- 
ducing on  the  whole  an  increase  of  4T8ff%.  Compare  the  amount 
expended  on  the  two  sections  last  year,  and  also  this  year. 

Let  x  +  y=*  amount  expended  last  year. 

x  =  amount  expended  on  first  section  last 
year, 
and  y  —  amount  expended  on  second  section 

last  year. 

Then    ^-^ V-  =  amount  expended  on  whole  this  year. 

1000  .  v  J 

But  x  +  — —  =  amount  expended  on  one  part  this 

year. 

and  y  +  — %-  =  amount  expended  on  other  part  this 

year. 


Then        105s      104  y  _  1043  s  +  1043  y 

100        100  1000 

Simplify,  7s  =  3y. 

ar_3 

„   .  105  s     315     „ ,     „ . 

and  WIT  728       A  =  7A- 


29.  When  a,  6,  c,  d  are  proportional  and  unequal,  show  that 
no  number  x  can  be  found  such  that  a  +  x,  6  +  x,  c  +  x,  d  +  x  shall 
be  proportionals. 

If  a  :  b  : :  c  :  d,  ad=bc; 

and  if  a+x:b  +  x::c  +  x:  d  +  x, 

ad+dx  +  ax  +  x*  =  bc  +  cx  +  bx  +  x*. 
Transpose,  and  cancel  x*, 

ax  —  bx  —  cx  +  dx  =  bc  —  ad. 
But  ad  =  be. 

•\  x(a  —  b  —  c  +  d)  =  0. 
.-.  a?  =  0. 


teachers'  edition.  427 


Exercise  CXV. 

1.  If  A  oc  B,  and  A  =  4  when  B  =  6,  find  A  when  B-  12. 
Here  A  =  wi5, 

.-.  m  =  f 
And  if  f  and  12  be  substituted  for  m  and  5, 
4  =  $Xl2. 

2.  If  -4  oc  B,  and  when  B  =  J,  -4=  J,  find  4  when  £=  J. 

Here  A  =  mB. 


m 

.4 
B 

.*.  m  = 

=  i 

Substitute  §  for 

m  and  j  for 

5, 

4  = 

=  }x 

I 

.-.4  = 

-f 

3.  If  -4  varies  jointly  as  £  and  C,  and  3,  4,  5  are  simultaneous 
values  of  A,  B,  C,  find  A  when  B  =  C  =  10. 

Here  4  =  ro.BC. 

Substitute  3  for  A,  4  and  5  for  B  and  (7, 

Then  A  =  A  X  10  X  10. 

.\4  =  15. 

4.  If  A  <x  — ,  and  when  <4  =  10,  B  =  2,  find  the  value  of  B  when 
.4  =  4.  B 

Here  .4  =  — » 

m-AB. 
.•.  m  =  20. 
Substitute  values  of  m  and  -4, 

a     20 
4  =  -, 

4£=20. 
.-.J?  =  5. 


428  ALGEBRA. 


5.  If^oc— ,  and  when  A  =  6,  #  =  4,  and  0=  3,  find  the  value 

of  A  when  B=5  and  C=  7. 

Here  A  =  ^ 

0 
mB  =  J.C, 
4m  =  18. 
.-.  m  =  4}.' 
Substitute  value  of  2?,  C,  and  m, 

.:A  =  Z&. 

6.  If  the  square  of  X  varies  as  the  cube  of  F,  and  X  =  3  when 
F=  4,  find  the  equation  between  Xand  F. 

Here  X*  =  mY*t 

.-.  m  =  &. 
Substitute  value  of  m,         X*  =  &  7*, 
64X*  =  9F». 

7.  If  the  square  of  X  varies  inversely  as  the  cube  of  7,  and 
X=  2  when  F=  3,  find  the  equation  between  X  and  Y. 

Here  X2  =  -§*' 

m-X'F8. 
.\  m  =  108. 

Substitute  value  of  m,         X»  =  ^- 

ys 

8.  If  Z  varies  as  X  directly  and  F  inversely,  and  if  when  Z  =  2, 
X=  3,  and  F=  4,  find  the  value  of  Z  when  X=  15  and  Y=  8. 

Here  S-2^. 

™     ^F 

.•.m  =  f  =  2J. 
Substitute  values  of  m,  X,  and  F, 

5_  2^x15 
8 


teachers'  edition.  429 


9.  If  A  <x  B  +  c  where  c  is  constant,  and  if  A  =  2  when  2?  =  1, 
and  if  A=  6  when  5  =  2,  find  J.  when  B=&. 

As  A  =  mi?  +  c. 

Substitute  first  values  of  A  and  5, 

2  =  m  +  c  ■  (1) 

Substitute  second  values  of  J.  and  B, 

5  =  2m  +  c  (2) 

Subtract  (1)  from  (2),         m  =  3. 
Whence,  from  (1),  c  =  —  1. 

But  A  =  m5  +  c. 

Substitute  for  to,  JB,  and  c  their  values  3,  3,  and  —1, 
A  =  8. 

10.  The  velocity  acquired  by  a  stone  falling  from  rest  varies 
as  the  time  of  falling;  and  the  distance  fallen  varies  as  the 
square  of  the  time.  If  it  is  found  that  in  3  seconds  a  stone  has 
fallen  145  feet,  and  acquired  a  velocity  of  96}  feet  per  second, 
find  the  velocity  and  distance  at  the  end  of  5  seconds. 

Let  v  =  velocity, 

T=  time, 
d  =  distance. 
Then  v  <x  Tt 

and  dccT2. 

Let  v  =  mT. 

Substitute  96J  for  v  and  3  for  Tt 

96*  =  3  m. 


.-.  m  =  A$ft 

When  T=  5, 

t,  =  ^X5  =  161f 

Let 

d  =  mT\ 

.-.  145  =  32m. 

m  =  if± 

When  T=  5, 

d  =144x5*  =402}. 

11.  If  a  heavier  weight  draw  up  a  lighter  one  by  means  of  a 
string  passing  over  a  fixed  wheel,  the  space  described  in  a  given 
time  will  vary  directly  as  the  difference  between  the  weights,  and 
inversely  as  their  sum.  If  9  ounces  draw  7  ounces  through  8 
feet  in  2  seconds,  how  high  will  12  ounces  draw  9  ounces  in  the 
same  time? 


430  ALGEBRA. 


Let  x  =  heavy  weight, 

y  —  light  weight, 

2  =  space. 

x  —  y 
zee 2. 

x'+y 

z_(x-y)m 
x  +  y 

x-y 

Substitute  values,  ,  m-(7  +  9)8- 

'  9-7 

.-.  m  =  64. 

6i     (12+9)« 

64  ~    12-9- 

3 

72  =  64. 

.-.s-9f 

12.  The  space  will  vary  also  as  the  square  of  the  time.  Find 
the  space  in  Example  11,  if  the  time  in  the  latter  case  is  3 
seconds. 

We  have  from  last  example,  9}  feet  for  2  seconds. 
Since  space  varies  as  square  of  time,  we  have 

9}  :  x  : :  2*  :  3*. 

.•.4a  =  9x^f 
x  =  9xV 

-20f 

20}  feet.  Am. 

13.  Equal  volumes  of  iron  and  copper  are  found  to  weigh  77 
and  89  ounces  respectively.  Find  the  weight  of  10}  feet  of  round 
copper  rod  when  9  inches  of  iron  rod  of  the  same  diameter  weigb 
31^  ounces. 


teachers'  edition.  431 

Let      x  —  required  weight. 
9  inches  =  f  of  a  foot. 

If  £  of  a  foot  weigh  31.9  ounces,  \  of  a  foot  would  weigh  10.03J 
ounces,  and  10J  feet  would  weigh  446.60  ounces. 

And,  as  equal  volumes  of  iron  and  copper  weigh  77  and  89 
ounces  respectively, 

77  :  89  :  :  446| :  a. 
.*.  x  =  516£  ounces. 

14.  The  square  of  the  time  of  a  planet's  revolution  varies  as 
the  cube  of  its  distance  from  the  sun.  The  distances  of  the 
Earth  and  Mercury  from  the  sun  being  91  and  35  millions  of 
miles,  find  in  days  the  time  of  Mercury's  revolution. 

Let  x  =  time  of  Mercury's  revolution. 

91s  :  35s  :  :  P  :  x\ 
13s  :  58  :  :  1 :  x*. 
Whence,  a*  =  0.056895. 

.*.  x  =  0.238,  time  in  years, 
=  87.1,  time  in  days. 

15.  A  spherical  iron  shell  1  foot  in  diameter  weighs  J^  of 
what  it  would  weigh  if  solid.  Find  the  thickness  of  the  metal, 
it  being  known  that  the  volume  of  a  sphere  varies  as  the  cube  of 
its  diameter. 

Let    D  =  diameter  of  shell, 

d  =  diameter  of  sphere  required  to  fill  the  shell, 
and  1  represent  the  weight  of  iron  sphere  having  diameter  =  D. 
Then  1  —  -ffa  will  represent  the  weight  of  iron  sphere  having 
diameter  =  d. 

Now  the  weights  vary  as  the  cubes  of  their  diameters, 

.-.  2P:#::1:1-Jft. 
That  is,  IP  :  d*  :  :  1  :  Jff  ; 

or,  by  extracting  the  cube  root  of  each  term, 
D:d::l:#, 
oxd=%D. 
Since  the  thickness  of  the  shell     =  J(D  —  d), 
the  thickness  of  the  shell  =  J(l  —  f)  =  ^. 

Hence,  the  thickness  of  the  shell  is  ^  of  a  foot,  =  1  inch. 


432  ALGEBRA. 


16.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diameter. 
Compare  the  volume  of  a  sphere  G  inches  in  diameter  with  the 
sum  of  the  volumes  of  three  spheres  whose  diameters  are  3,  4, 5 
inches  respectively. 

Let  x  —  volume  of  first  sphere, 

and  y  =  sum  of  volume  of  other  three. 

Then  x:y::(6)»:(3)8  +  (4)8  +  (5)8, 

x  :  y  : :  216  :  216. 
Therefore,  the  ratio  is  a  ratio  of  equality. 

17.  Two  circular  gold  plates,  each  an  inch  thick,  the  diame- 
ters of  which  are  6  inches  and  8  inches  respectively,  are  melted 
and  formed  into  a  singular  circular  plate  1  inch  thick.  Find  its 
diameter,  having  given  that  the  area  of  a  circle  varies  as  the 
square  of  its  diameter. 

Let    Oi  =  area  of  gold  plate  6  inches  in  diameter, 
a,  =  area  of  gold  plate  8  inches  in  diameter, 
Oj  =  area  of  gold  plate  formed  from  the  other  two  plates, 
and         x  =  diameter  required. 

Then  c^  +  a,  :  a, : :  6*  +  8* :  x*. 

Since  the  first  ratio  is  a  ratio  of  equality,  the  second  is  also. 
Therefore,        x*  =  6*  +  8*  =  100. 
.-.  x  =  10. 

18.  The  volume  of  a  pyramid  varies  jointly  as  the  area  of  its 
base  and  its  altitude.  A  pyramid,  the  base  of  which  is  9  feet 
square,  and  the  height  of  which  is  10  feet,  is  found  to  contain  10 
cubic  yards.  What  must  be  the  height  of  a  pyramid  upon  a  base 
3  feet  square,  in  order  that  it  may  contain  2  cubic  yards? 


Let 

and 
Then 

v  =  volume, 

b  =  area  of  base, 

a  =  altitude. 

v  oc  ba, 

v  =  mba                                    (1) 

When 
and 

V 

v  =  10  cubic  yards  =  270  cubic  feet, 
b  =  9  x  9  =  81  square  feet, 
a  =10  feet. 

Then 

—  S-H»-*- 

From  (1), 
When 

a  =  — 
mb 
m  =  J, 

v  =  2  cubic  yards  =  54  cubic,feet, 
6  =  3x3  =  9  square  feet. 

Then 

°=^=18- 

/ 


TEACHERS     EDITION. 


467 


Let 
Then 

Hence, 


x  and  y  —  first  and  last  terms. 


2  ay 
x+  y 


and 


=  middle  term. 

,       ,    2xy  _^ 

*+»+jrn;-11 


Square  (1),  and  subtract  (2)  from  the  result, 

6xy=  72,  and  ay  =  12 
Substitute  12  for  xy  in  (1),  and  clear  of  fractions, 

(x  +  y)2-ll(x  +  y)  =  -24. 
Complete  the  square  and  extract  the  root, 


0) 
(2) 

(3) 


(1) 


21.  A  number  consists  of  three  digits  in  geometrical  progres- 
sion. The  sum  of  the  digits  is  13 ;  and  if  792  be  added  to  the 
number,  the  digits  in  the  units'  and  hundreds'  places  will  be  in- 
terchanged.    Find  the  number. 

Let  x  =  first  digit, 

rx  =  second  digit, 
and  r*x  =  third  digit. 

x  +  rx  +  r8®  ^  13 
100a;  +  lOra;  +  r*x  +  792  =  100  r^x  +  lOrx  +  x, 
-99r*a;  +  99a;*=-792. 

r*x-x=    8  (2) 

x  +  rx  +  r*x  —  13 

Subtract,  2x  +  rx=   5 

....    * 

2+r 
Substitute  value  of  x  in  (2), 

5r* 5_  =  g 

2+r      2+r 

5r2-5  =  16  +  8r, 
5r2-8r  =  21, 
100r2-()  +  64  =  484, 
10r-8  =  ±22, 
10r  =  30. 
.-.  r  =  3. 
From  (1),  x  +  3a;  +  9a?  -=  13. 

.-.  s-1. 
Hence,  the  number  is  139. 


432  ALGEBRA. 


16.  The  volume  of  a  sphere  varies  as  the  cube  of  its  diameter. 
Compare  the  volume  of  a  sphere  C  inches  in  diameter  with  the 
sum  of  the  volumes  of  three  spheres  whose  diameters  are  3, 4, 5 
inches  respectively. 

Let  x  =  volume  of  first  sphere, 

and  y  =  sum  of  volume  of  other  three. 

Then  o;:y::(6)8:(3)8  +  (4)8  +  (5)»l 

x  :  y  : :  216  :  216. 
Therefore,  the  ratio  is  a  ratio  of  equality. 

17.  Two  circular  gold  plates,  each  an  inch  thick,  the  diame- 
ters of  which  are  6  inches  and  8  inches  respectively,  are  melted 
and  formed  into  a  singular  circular  plate  1  inch  thick.    Find 
riiam*±*r.  h&yinj?  idran.  that  the  area  of  a  circle  varies  as 


»8. 
l-r     1-i 


(2)  a  =  i, 

r-f  <n 


"l-r      1_§        * 

(3)  a  =  t, 
r J. 

_a J_  W 

"l-r~l-(-« 

(4)  a-1, 
r  — f  (9) 

•'T^"i-(-» 

(5)  a-i, 
r-J.  (10) 

.      a  * 


a  = 

=  0.1, 

r  = 

=  0.1. 

a 
'"'  l-r" 

-  01  -1 

1-0.1     ! 

a  = 

-0.86, 

r  = 

=  0.01. 

a 
•'•l-r  = 

0.86 
1-0.01 

a  = 

aTTV» 

r  = 

•it- 

a 

-4s-* 

l-r 

A 

A  +  A  = 

-H- 

a  = 

=  iooO' 

r  = 

=  TW- 

1  — r 

=  1000  _  JL 

AV  **• 

A  +  A  = 

-H- 

/. 


467 


Let  x  and  y  =  first  and  last  terms. 

Then  — ; —  =  middle  term. 

x  +  y 

Hence,  x  +  y  +  ^  =  ll  (1) 

and  X2+y2+A^-  =  ^  (2) 

Square  (1),  and  subtract  (2)  from  the  result, 

6  xy  =  72,  and  xy  =  12  (3) 

Substitute  12  for  xy  in  (1),  and  clear  of  fractions, 

(x  +  y)*  -  11  (x  +  y)  =  -  24. 
Complete  the  square  and  extract  the  root, 

x  +  y=S  (4) 

Square  (4),  x2  +  2xy  +  y%  =  64 

From  (3),  4xy  =48 

x2-2xy  +  y8  =  16 

x-y=±4  (5) 

From  (4)  and  (5),  x  =  6,  and  y  =  2. 

Hence,  the  numbers  are  6,  3,  2. 

11.  When  a,  6,  c  are  in  harmonical  progression,  show  that 
a:c::a  —  b:  b—c. 

If  a,  6,  c  are  a  harmonical  series, 

1  _  1  _  1  _  1. 
6      a      c      6 
Multiply  by  a6c,  ac  —  bc^ab  —  ac, 

or  c  (a  —  b)  =  a  (6  —  c), 
or  a:  c::  a  — 6:  6  — C. 


Exercise  CXIX. 

1.  How  many  different  permu-  (Rule  V)  the  number  of  different 

tations  can  be  made  of  the  let-  permutations  of  the  letters  is 

ters  in  the  word  ecclesiastical,  i^ 

taken  all  together  ?  j|j|j|j|j5j|  =  454,063,600. 

The  word  contains  t  once,  a,  e, 

i;  I,  and  s  each  twice,  and  c  three  2.  Of  all  the  numbers  that  can 

times;  in  all  14  letters.     Hence  be  formed  with  four  of  the  digits 


468 


ALGEBRA. 


5,  6,  7,  8,  9,  how  many  will  begin 
with  66? 

The  last  two  digits  of  the  num- 
bers may  be  selected  in  any  way 
from  7,  8,  9.    This  is  possible  in 

.—  =  6  ways. 

Hence  6  numbers  can  be  formed 
of  the  required  kind. 

3.  If  the  number  of  permuta- 
tions of  n  things,  taken  4  together, 
is  equal  to  12  times  the  permuta- 
tions of  n  things,  taken  2  together, 
find  n. 

By  the  given  conditions 

In  In 

—     =  12     *-     , 
|n-4  [n-2 

or    n(n  —  1)  (n  —  2)  (n  —  3) 

=  12n(n-l). 
Hence  (n  —  2)  (n  —  3)  =  12 
n2-5n  +  6=12 
n2-5n  —  6  =  0 
(n+l)(n-6)  =  0 

n  =  —  1  or  6. 
Therefore  n  =  6. 

4.  With  3  consonants  and  2 
vowels,  how  many  words  of  3 
letters  can  be  found,  beginning 
and  ending  with  a  consonant, 
and  having  a  vowel  for  a  middle 
letter? 

The  2  consonants  can  be  chosen 
from  3  in  6  ways,  and  the  vowel 
can  be  chosen  from  2  in  2  ways. 


19  in 


Hence  the  number  of  words  of 
the  required  kind  that  can  be 
found  is  6  X  2  =  12.  Ans. 

5.  Out  of  20  men,  in  how  many 
different  ways  can  4  be  chosen 
to  be  on  guard  ?  In  how  many 
of  these  would  one  particular 
man  be  taken,  and  from  how 
many  would  he  be  left  out  ? 

(1)  Four  men  can  be  selected 

120 
from  20  in  rrrr^  =  4845  ways. 

(2)  If  one  particular  man  is  to 

be  included,  the  remaining  3  can 

be  selected  from  the  remaining 

110 
^—969  ways. 

(3)  If  a  particular  man  is  to  be 
excluded,  the  4  can  be  selected 
from  the  remaining  19  in 

[19 
^  =  3876  ways. 

3876  +  969  =  4845. 
Hence   the    total   number  of 
selections  is  4845;    including  a 
particular  man,  969;  excluding 
a  particular  man,  3876. 

6.  Of  12  books  of  the  same  size, 
a  shelf  will  hold  5.  How  many 
different  arrangements  on  the 
shelf  may  be  made  ? 

The  number  of  arrangements  is 

112. 


II 


=  95,040. 


TEACHERS'    EDITION. 


469 


7.  Of  8  men  forming  a  boats' 
crew,  'one  is  selected  as  stroke. 
How  many  arrangements  of  the 
rest  are  possible?  When  the  4 
who  row  on  each  side  are  decided 
on,  how  many  arrangements  are 
still  possible  ? 

(1)  The  7  remaining  men  can 
be  arranged  in  the  7  remaining 
seats  in  [7  =  5040  ways. 

(2)  The  3  men,  besides  the 
stroke,  who  row  on  the  stroke 
side  can  be  seated  in  [3  =  6  ways. 
The  4  men  on  the  other  side  can 
be  seated  in  [4  =  24  ways.    Hence 


the  total  number  of  arrangements 
under  the  given  conditions  is 
6  X  24  =  144. 

8.  How  many  signals  may  be 
made  with  6  flags  of  different 
colors,  which  can  be  hoisted  either 
singly  or  any  number  at  a  time  ? 

A  set  of  r  flags  can  be  selected 
.         I? 


fcli: 


;  ways,  and  each  set  can 


be  hoisted  in  [r  different  orders. 
Hence  the  number  of  different 

signals  with  r  flags  is  tttzt  ' 

The  number  of  signals,  therefore, 


with  1  flag  is  6 

=       6 

"    2  flags  is  6  X  5 

=     30 

44    3       44      6X6X4 

=    120 

44    4       44      6X6X4X3 

=    360 

44    5      44      6X5X4X3X2 

=    720 

44    6      44      6X6X4X3X2X1 

=    720 

In  all, 

1956 

9.  How  many  signals  may  be  made  with  8  flags  of  different  colors, 
which  can  be  hoisted  either  singly  or  any  number  at  a  time  one 
above  another  ? 


The  number  of  signals 

with  1  flag  is  8 

2  flags  is  8  X  7 


8X7X6  = 

8x7x6x6  = 

8x7x6x5x4  = 
8X7X6X5X4X3  = 
8X7X6X5X4X3X2  = 
8x7x6x6x4x3x2x1= 
In  all, 


8 

56 

336 

1,680 

6,720 

20,160 

40,320 

40,320 

109,600 


470  ALGEBRA. 

10.  How  many  different  signals  one  light  is  used  must  be  counted 
can  be  made  with  10  flags,  of  only  once;  all  other  cases  four 
which  3  are  white,  2  red,  and  the  times.  The  number  of  signals  in 
rest  blue,  always  hoisted  all  to-  a  single  line 

gether  and  one  above  another  ?       ^th  l  light  is    4  =4 

The  number  of  different  signals      "     2  lights  is  4  X  3  =12 

is  equal  to  the  number  of  arrange-  "  3  "  4x3x2  =24 
ments  of  10  things,  of  which  2  "  4  "  4x3x2X1  =  24 
are  alike,  3  are  alike,  and  5  are  In  all,  60  +  4. 

alike.    This  number  is  (Rule  V)        Hence    the    total    number  of 
tig     _  signals  is  4  X  60  +  4  =  244. 

I?  I?  15  14.    From   10  soldiers   and  8 

11.  How  many  signals  can  be  sailors»  now  many  different  par- 
made  with  seven  flags,  of  which  ties  of  3  soldiers  and  3  sailors 
2  are  red,  1  white,  3  blue,  and  can  *»  formed  ? 

1  yellow,   always  displayed    all       The  3  soldiers  can  be  selected 

together  and  one  above  another?         110 

t>    t>  i    xr  *v  v     .  in  foi*  =  120  wavs>  and  toe  3 

By  Rule  V  the  number  is  [3[7 

II        .OA  sailors  in  rz77=  56  ways.    Hence 

f2J3  =  420.  [8  [6 

the  party  of  3  soldiers  and  3  sailors 

12.  In  how  many  ways  may   can  be  selected  in 

the  8  men  serving  a  field-gun  be  120  x  66  =  6720  ways, 
arranged  so  that  the  same  man 

may  always  lay  the  gun  ?  15-  How  manv  sign*18  <*&  be 

„«-_..                       ,  made  with  3  blue  and  2  white 

The  7  remaining  men  may  be  _            _.  ,             ,       ,.    ,      .  . 

,  .    ,-       °J/X  flags,   which    can    be   displayed 

arranged  in  7  =  5040  ways.  Jr        .     ,                         v      * 

•—  either  singly  or  any  number  at 

13.  Find  the  number  of  signals  a  time  one  above  another  ? 
which  can  be  made  with  4  lights  a  single  flag  maybe  either  blue 
of  different  colors  when  displayed  or  white.  2  ways. 
any  number  at  a  time,  arranged  Two  flags  may  be  both  blue  or 
above  one  another,  side  by  side,  both  white,  or  1  blue  and  1  white, 
or  diagonally.                                   In  the  lafit  caj3e  they  may  be  ar- 

The  number  of  arrangements  ranged  in  two  ways, 

in  vertical,  horizontal,  or  either  In  all,  4  ways, 

diagonal    line    is    evidently   the  Three  flags  may  be  all  blue,  or 

same.     But  the  case  where  only  2  blue  and  1  white  or  2  white 


teachers'  edition.  471 

and  1    blue.     In    the   last  two  18.  From  12  soldiers  and  8  sail- 
cases  they  may  be  arranged  in  3  ore,  how  many  different  parties 
ways.                      In  all,  7  ways,  of  3  soldiers  and  2  sailors  can  be 
Four  flags  may  be  3  blue  and  found? 

1  white,  4  ways ;  or  2  blue  and  The  3  soldiers  can  be  selected 

2  White'  1212  "  6    WayS    (RU16  from  12  in  }=-  =  220  ways ;  the 
V).           ^^      In  all,  10  ways.  [3  [9                 Jf 

Five  flags  must  be  3  blue  and  2  sailors  from  8  in  r3r  =  28 


JL-- 


1216 


2  white,   rzrrz  =  10  ways  (Rule  ways.    The  party  can  therefore 

V).            l-L     in  all,  10  ways,  be  formed  in 

There   are,    therefore,  in  all,  220  X  28  =  6160  ways. 
2  +  4  +  7  +  10  + 10  =  33  possible 

signals.  19*  Find  the  number  of  com- 

„.   „    ^  binations  of  100  things,  97  to- 

16.  In  how  many  ways  can  a  . 

party  of  6  take  their  places  at  a 

round  table?  B^    Rule  vm   *■»   *"P*** 

,  ,  number  is 

One  person  may  take  any  one  .  1Q0 

of  the  6  seats;   the  other  6  can  L=^  =  10°  x  "  x  98 

then  be  seated  in  [6  =  120  differ-  l?Z  12             3  x  2 

ent  orders.     Hence  the  answer  is  =  161,700. 

6  X  120  =  720.  2k  wuh  2Q  congonants  ajld  5 

17.  Out  of  12  Democrats  and  vowels,  how  many  different  words 
16  Republicans,  how  many  differ-  can  be  formed  consisting  of  3 
ent  committees  can  be  formed,  different  consonants  and  two  dif- 
each  consisting  of  3  Democrats  ferent  vowels,  any  arrangement 
and  4  Republicans  ?  of   letters    being    considered    a 

The  3  Democrats  can  be  selected  word  ? 

1 12  The  3  consonants  can  be  selected 

from  12  in  rrr^  =  220  ways ;  the  |2Q 

L-L                       [16  from  20  in  r^==  =  1140  ways; 

4  Republicans  from  16  in  liilll            15 

*—  ^—  the  2  vowels  from  5  in  .7=5  =  10 

=  1820  ways.     The  committee,  l±  l£ 

consisting  of  any  3  Democrats  ways.     The  number  of  combina- 

and  any  4  Republicans,  can  there-  tions  of  3  different  consonants 

fore  be  formed  in  and  two  different  vowels  is  there- 

220  X  1820  =  400,400  ways.  for©      10  X  1140  =  11,400. 


472 


ALGEBRA. 


The  five  letters  of  each  com- 
bination can  then  be  arranged  in 
[6  =  120  ways.  Hence  the  total 
number  of  words  of  the  required 
kind  is 

120  X  11,400  =  1,368,000. 

21.  Of  30  things,  how  many 
must  be  taken  together  in  order 
that,  having  that  number  for 
selection,  there  may  be  the  great- 
est possible  variety  of  choice  ? 

By  §  417,  the  number  taken 
must  be        30 

f=15- 

22.  There  are  m  things  of  one 
kind  and  n  of  another.  How 
many  different  sets  can  be  made 
containing  r  of  the  first  and  s  of 
the  second  ? 

The  r  of  the  first  kind  can  be 


selected  from  m  in  .- 


[m 


any  seat ;  the  other  9  can  then  be 
seated  in  [9  =  362,880  different 
orders.  For  each  order  of  the 
10  persons  and  the  reverse  order, 
each  person  has  the  same  neigh- 
bors, and  this  is  true  only  in  this 
case.  Hence  the  number  of  ar- 
rangements required  is 

?»=  181,440. 

24.  The  number  of  combina- 
tions of  71  things  taken  r  together, 
is  3  times  the  number  taken  r  —  1 
together,  and  half  the  number 
taken  r  +  1  together.  Find  n 
and  r. 

By  the  given  conditions 

In       o        I* 


lr[n^ 


=  3 


=  * 


k|   .  ways; 

the  s  of  the  second  kind  from  n 

\n 

in  — : The  total  number 

|g  |n  —  3 

of  sets  of  r  of  the  first  kind  and 
s  of  the  second  is  therefore 
\m\n 

\r  \s  \m  —  rln  —  s 


r-l[n-r+l 

l» 
r+lln-r-1 


ln-r  +  1  \r 

'  ,  =S-^=-i  (l) 

\n—  r  \r—  1 

In-r  [r+1 

=  2*=^.  (2) 


lr 


23.  In  how  many  ways  may  10 
persons  be  seated  at  a  round  table 
so  that  in  no  two  of  the  arrange- 
ments may  every  one  have  the 
same  neighbors  ? 

Only  the  order  of  seating  being 
of  account,  one  person  may  take 


\n-r-l 
From  (1), 
n  —  r+  1  =  3r, 
n+l  =  4r. 
From  (2), 

n-r  =  2(r+l), 
n  =  3r  +  2. 
.-.  3r+2  =  4r-l, 
r  =  3, 
n  =  3r  +  2 
=  11. 
Therefore, 

w=  11,  r  =  3. 


TEACHERS^  EDITION.  473 

25.  In  how  many  ways  may  12  elapse  before  the  same  20  men 

things  be  divided  into  3  sets  of  4  ?  go  on  guard  the  second  time. 

Here  the  sets  are  indifferent,  The  number  of  guard  details 

and  the  answer  is  which  include  a  particular  man  is 

112  189 

—     -  =  6776.  L—     - 


26.  How  many  words  of  6  let-  28.  Supposing  that  a  man  can 
ters  may  be  formed  of  3  vowels  Place  himself  in  3  distinct  atti- 
and  3  consonants,  the  vowels  tudes,  how  many  signals  can  be 
always  having  the  even  places  ?  made  bv  4  men  P^ced  side  by  side  ? 

The  3  consonants  can  be  ar-  S^^ach  man  can  place  him- 

ranged  in  the  3  odd  places  in  self  in  3  attitudes,  the  total  num- 

18  =  6  ways,  and  the  3  vowels  can  ber  of  distinct  attitudes  of  the 

be  arranged  in  the  even  places  in  &0**  of  4  men  M  3*.  =  81'  ,       , 

6  ways.     The  total  number  of  But  one  is  the  attitude  of  rest. 

,.«        .          ,      *  .,             .     ,  Therefore  34  —  1  =  80  signals, 

different  words  of  the  required  & 

kind  is  therefore  6  X  6  =  36.  29.    How  many  different   ar- 

rangements can  be  made  of  11 

27.  Fromacompanyof  90men,  cricketerSj  SUpp0Sing  the  same  2 
20   are  detached    for  mounting  always  to  bowi  ? 

guard  each  day.  How  long  will  The  Q  remaining  cricketers  can 
it  be  before  the  same  20  men  are  ^  BrnsDSBd  m  m  =  362,88o  ways, 
on  guard  together,  supposing  the  And  gince  the  two  who  bowl  may 
men  to  be  changed  a*  much  a*  change  placegj  the  total  number 
possible  ;^  and  how  many  times  of  arrangemente  is 

2X362,880=725,760. 
30.  Five  flags  of  different  colors 
can  be  hoisted  either  singly  or 

90  in  ,4=-  ways;   and  this  is  ^  number  at  a  time  one  above 
1*5;  170  another.      How  many  different 

the  number    of   days  that  will  signals  can  be  made  with  them  ? 

The  number  of  signals 

with  1  flag  is  5  =5 

u    2  flags  is  6  X  4  =20 

"    3      "      5X4X3  =60 

"4       "      5X4X3X2         =120 

44     6       "      5X4X3X2X1  =  120 

In  all,      325 


will   each  man   have   been 
guard? 
20  men  can  be  selected  from 


474  ALGEBRA. 

3L  How  many  signals  can  be  with  the  lights  one  above  another, 
made  with  5  lights  of  different  col-  Every  possible  arrangement  mast 
ors,  which  can  be  displayed  either  therefore  be  counted  twice ;  but 
singly,  or  any  number  at  a  time  if  only  one  light  is  used,  this  can 
side  by  side,  or  one  above  another  ?   be  counted  only  once. 

The  number  of  signals  with  the  The  number  of  arrangements 
lights  side  by  side  is  equal  to  that  in  a  single  line 

with  1  light  is  5  =5 

"    2  lights  is  5  X  4  =20 

"3       "       5X4X3  =60 

"    4       "       5X4X3X2        =120 

"    6        "       6  X  4  X  3  X  2  X  1  =  120 

In  all,  320  +  5. 

The  total  number  of  possible  signals  is  therefore 

2X320+6=645. 

32.  The  number  of  permuta-  have  425  times  as  many  hands  as 
tions  of  n  things,  3  at  a  time,  is  there  are  cards  in  the  pack.  How 
6  times  the  number  of  combina-  many  cards  are  there  ? 

tions,  four  at  a  time.    Find  n.  Let    n  =  the  number  of  cards ; 

By  the  conditions  of  the  prob-  l»  , 

,  ^  then  rz\ s  =  the    number   of 

lem»                                                             3  n-~3  _       ,  .  , , 

in  \n  •  •— ' hands  one  can  hold. 

HenC6[3^3  =  425n 

w(n-l)(n-2)  =  425nl3 

(n-l)(n-2)=2550 

w2-3n- 2548  =  0 
(n-52)(n  +  49)  =  0 

33.  At  a  game  of  cards,  3  being  •'•  n  =  62- 

dealt  to  each  person,  any  one  can       There  are  52  cards  in  the  pack. 

1.  (l  +  2x)*  EXEBCISE     CXX. 

=  (1)6+5  (1)4(2  x)  + 10  (1)8(2  x)2+ 10  (1)2(2  x)»+ 5  (1)  (2  x)4+  (2s)* 
=  1  +  lOx  +  40  x2  +  80x»  +  80  x*  +  32  a* 

2.  (x  -  3)8  =  x*  -  8  x7(3)  +  28  x°(3)2  -  56  x*(3)8  +  70  x*(3)* 

-  56x8(3)6  +  28x2(3)«  -  8x(3)7  +  (3)8 
=  x*  -  24x7  +  252x6  -  1512X5  +  5670x* 

-  13608  x8  +  20412  x2  -  17496  x  +  6561. 


h 

-3 

>|n- 

-4 

• 

6 

[n-8 
|n-4 

4  = 

71—3 

n  = 

7. 

teachers'  edition.  475 

3.  (2x-  3y)* 

=  (2s)4  -  4(2a0»(3y)  +  6(2xf(3yy  -  4(2s)(3y)»  +  (3y)* 
=  16s*  -  96afy  +  216^-216^  +  81  y*. 

4.  (2-xf 

=  (2)'-3(2)*(s)  +  3(2)(z)>-(*)a 
=  8- 12s  +  6s2 -a3. 

-(!)•  -  5(1).  0*)  *  10(1C (^'_  10(1). (& J 

^<"(3f)'-(^T 

1      15y     45y»      135.V3      405y*      243y* 
4  8  32  256        1024* 

* HI 

+M(l)--»(fy+9(f)'-l(|j 

i      q     ,  a„*     28a*     14x*      14ar>     28s6     4s'      ^  ^ 


9  9  27        243      243     729     19683 

7.  The  fourth  term  of  (2  x  —  5  y)1*. 

Substitute  in  formula  values  of  n  and  r, 

w(n~1) fr-r  +  2)  an_r+i  ^.i 

1X2 (r-1) 

12X11X10, 


1x2x3 
=  -  14080000  afy8. 


(2s)»(5y)» 


8.  The  seventh  term 

_  10  X  9  x  8  X 
1x2x3x4 
35  aV 

"  1944  ' 


r(D'(f)' 


ZTl\. 


.(.'     V!  Ill*    «»1        ' 

•    j»-  a    j   if   i   ur:r 


I  V  .// 


'  1  ,'  ,y    * 


TEACHERS*   EDITION.  477 

15.  The  rth  term  from  the  end  of  (2  a  +  x)* 

-  n(n  ~  *) (n  ~  r  +  2)(2aHaf^i 

1X2 (r-1)       V     T 

16.  The  (r  +  4)th  term  of  (a  +  x)» 

=  n(n  -  1) (n-r-  2)a_^  ^ 

1X2 (r  +  3) 

17.  The  middle  term  of  (a +  z)*» 

=  2n(2n-l) (2n-(»+l)  +  2)fl^Watft 

1X2 (n  +  1  -  1) 

,2n(2n-l) (n+1)^ 

1X2 n 

Multiply  both  terms  by  [n, 

(w  +  1)  th  term  =  ===-  aV. 
(In)* 


18.  Expand  (2a  +  x)w,  and  find  the  sum  of  the  terms  if  a  =  1, 
x=-2. 

(2a)u  +  12(2a)na;  +  66(2a)1(V  +  220(2a)»x8  +  495(2a)8s* 
+  792(2a)7ari  +  924(2a)«a*  +  792(2afxt  +  495(2a)*x8 
+  220(2a)sa*  +  66(2a)2a^°  +  12(2a)x11  +  «". 

Substitute  1  for  a  and  —2  for  a?, 
(2-2)u=0. 


Exercise  CXXI. 
1.    (1  +  as)*  to  four  terms 

1      _L  _1_ 

2~8     16~ 


2.   (1  +  sc)3  to  four  terms 

-1  +  fas-^  +  ^s"- 


478  ALGEBRA. 


3.    (a  +  x)*  to  four  terms 


=    |       3z        3z*         5g8 
~a       4ai     32a*     128a* 

,/,3x     3x2       5x«  \ 

-aV+4^~3^+128^~ )' 


4.  (1  —  x)-4  to  four  terms 

By  substituting  1  for  a  and  — x  for  x  in  the  formula  (a+z)B 
,      ,  n(w— 1)      ,  0  ,  n(n— l)(n— 2)(n— 3)     _, 
=  a»4-na^1x  +  -L^-^a»-2x2  +  -* |j* ** + 

=  1  +  4x  +  10x2  +  20x8  4- 

5.  (a2  —  x2)*  to  four  terms 

=  (a2)1  ~  t  (a2)1-1  +  ^-p^a2)*^ 


.     5a8x2  .  15ax*      5x«  , 


_Mi-iKi^(a2)Hx»+ 

15 


6.    (x2  4-  xy)~i  to  four  terms 

=  (x'2)-3  -  }(x2)-H  (xy)  +  ""W""*-1)  (x2)-!-2  (xy)2 


+  -f(     I     *)(     f     2)  x2-*-b  (xy)»  + 


=  ,-s_|x-,,  +  i^_§^+. 


7.   (2x  — 3y)-i  to  four  terms 

=  (2x)-i  -  (-  i)(2g)-t(8y)  +  (~  *M~  ^  (2x)-t  (3y)2 

-(~^^~y~J)(2x)^(3y)'+ 

=  2*^{l  +  fx-iy  +  -^ar»^  +  #ftar*y»  + } 


teachers'  edition.  479 

8.  Vl  — 5a;  to  four  terms 

=  l-x--2JB2-6ar,- 

9.  ——z==zl  to  four  terms 
V(4a8-3aa;)3 

=  (4a2-3oa;)-3 

1  X  « 

_  (-t)(-»(~t)  (4a2)-«(3  ^ji 
1x2x3      v      '    v       ' 

-  J_  {  l  +  ^£  +  135a;2  .    945  x8      ) 

=  8as(         8  a     128  a2      1024  a8     J 

10.  a/ .  to  four  terms 

>(1  — 3y)5 

=  l+§J/  +  5^  +  ?35^     

2  8  48 

11.  (1  +  jc  +  a;*)*  to  four  terms 

«[1 +(*  +  «*)]» 

- 1  + 1(0  +  ^)  +  ii^l)(*+ ^  +  *»zMzl9  (*  +  <*> 

12.  (1  —  x  +  s2)*  to  four  terms 

-[1 -(*-*•)]• 

-  1  -  fa;  +  f  a"  +  (far2  -  £  s»)  -  (-  ^ar*) 


£g0  ALGEBRA* 

,  .„»  u!fhi  of  (a  +  x)» 

Hi  -1) (i -r-H  2)^-^1^-1 

"1X2 (r-1) 


d)(-i)(-i)(-i) m^ 


lr-1 


n-(l)(l)(D(l) (^').^ 


[r-J. 
Multiply  both  terms  by  ¥ _1 

.  /     !y  1X3X5 (2r-5)  fl3  2 V* 

14.  The  rtffc  term  of  (a  —  ac)  -» 

=  (-3)(-4) (-r-1)  fl_r.  Jaf  ^ 

1x2x3 r-1 

_  3X4 (r-i)(r)<r+i) > 

K     1}  1X2X3X4 (r-1) 

15.  \/66  to  five  decimal  places 

-{64(1  + *»* 

=  8(1+ A)4 

-8{l+(»(*)+*£^(lW,l 

-8(1+t1t-t^^) 
-8(1+  0.00781  -  0.00003) 
=  8.06224 

16.  \fijfo  to  five  decimal  places 

={1(1  +  A)}i(*-D 

-1 +0.01111  —  0,00012 
-1.01099. 


teachers'  edition.  481 


17.   v'l29  to  six  decimal  places 

=  {128(1  +TiT)}» 
=  {2'(1+Tk)}* 

.v&ia-2Ci+T«* 

=  2(1  +0.001116—0.000004) 
=  2.002224. 


18.  (1— 2x  +  3a?)i  to  four  terms 
=  {l-(2x-3^)}-* 
=  1  -  (- 1)  (2c  -  Zx>)  +  fc<).(--*'(2*  -  3a»)» 


.(-l)(-*)(-V)(2,_3aiy 

1x2x3       v  ' 


1  + 


f4£  _  6aA      /28z*  _  84a3     63_^\      /224^  _ \ 

V  5        5  )      {  25         25         25  J      V  125       '"  J 


_1      4a;     2s*      196r»     

5        25        125 

19.  Ci+^i!  to  coefficient  of  a* 

(l  +  3s)3 

=  (l  +  2a:)2(l+3a;)-3 

=  (1  +  4z  +  4s2)  (1  -  9<e  +  54s2  -  270x*  +  1215a4). 

The  terms  containing  re4  will  be 
1215s*  -  4s(270s8)  +  4a2(54s2) 
=  351  a*. 

20.  (1  +  as)*  expanded 

u».+iS^,,+K»-i)(»-»v 

,HJ-l)tt-2)ft-8) 

+  0 

-1+J* L-s.+      1X3     -...    1X3X5      . 


2x2*         2x3x2"         2x3x4x2« 
When  i  =  l,  this  becomes  (1  +  1)' 
_1+I 1       ,      1x3  1x3x5      , 


2     2x2s     2x3x2»     2x3x4x2« 


482  ALGEBRA. 


Exercise   CXXII. 

1.  If  I  throw  a  single  die,  what  Out  of  7  chances,  5  are  favor- 
is  the  chance  that  it  will  turn  up  able  and  2  unfavorable.    Hence 
(i.)  An  ace  ?  the  chance  of  success  is  f,  and  of 
(ii.)  An  ace  or  a  two  ?  failure  f. 

(iii.)  Neither  an  ace  nor  a  two  ?  _ 

5.  The  chance  of  an  event  is 
Since  a  die  has  6  faces,  f     Find  the  odds  for  or  against 

(i.)  The  chance  of  an  ace  is  J.    the  eyeat 
(ii.)  The  chance  of  an  ace  is  1, 
and  the  chance  of  a  two  is  £.       °*  ot  9  chances<  2  "»  aTOr- 

Hence  the  chance  that  either  an  able  for  *"  event  and  7  nnfayor- 
.  «iii    able.     Hence    the   odds  against 

ace  or  a  two  turns  up  is  4+  I  *u  c'     XAC"°°    °  c    wuo  °*> 

_  i  the  event  are  7  to  2. 

(iii.)  There  being  4  faces  be-  6,    What  is  the  chance  of  a 

sides  the  ace  and  the  two,  the  year,  not  a  leap  year,  having  53 

chance  that  neither  an  ace  nor  a  Sundays  ? 

two  turns  up  is  J  =  f  jf  a  year  0f  365  days  has  53 

%  The  chance  of  a  plan  sue-  Sundays,  it  must  begin  on  Sun- 

ceeding  is  \.    What  is  the  chance  day.    But,  as  the  year  may  begin 

that  it  fails?                                  .  on   any  day  of    the  week,  the 

Since  there  is  one  chance  in  4  chancethat  it  will  begin  on  Sun- 
that    it    succeeds,    there    are    3  day  ta  *• 

chances  in  4  that  it  fails.     Hence  7.  Two  numbers  are  chosen  at 

the  chance  of  failure  is  £ .  random.     Find  the  chance  that 

«    ,,  ,       , ,  ~  .        their  sum  is  even. 

3.  If  the  odds  are  10  to  1  against 

an  event,  what  is  the  probability       The  numbers  may  * 

of  its  happening?  (i.)  both  even, 

(ii.)  both  odd, 

(iii.)  the  first  odd,  the  second 

even, 

(iv.)  the  first  even,  the  second 
probability  of  its  happening  is  1  _, , . 


Out  of  11  chances  there  are  10 
that  the  event  will  not  happen 
and  1   that  it  will.     Hence  the 


in  1 1  or  T*T.  and  an  these  f QUr  cages  are  eqUaii} 

4.  If  the  odds  are  5  to  2   in  likely  to  occur.     But  the  sum  is 

favor  of  the  success  of  an  experi-  even  only  in  the  first  two  cases, 

ment,   what   are   the   respective  and  the  chance  that  one  of  these 

chances  of  success  or  failure  ?  will  happen  is  J  =  £. 


teachers'  edition.  483 

8.  If  4  cards  are  drawn  from  a  45.  But  the  number  of  adjacent 
pack,  what  is  the  chance  that  places  is  now  10,  since  the  first 
they  will  all  be  hearts  ?  and    tenth    are    now    adjacent. 

Four  cards  can  be  selected  from   Hence  the  chance  that  the  two 
1 52  assigned  persons  will  stand  to- 

the  52  in  the  pack  in  rrrjz  ways,    gether  is  JJ  =  J. 

Four  hearts  can  be  selected  from       n'  Show  that  if  n  persons  sit 

1 13  down  at  a  round  table,  the  odds 

the  13  hearts  in  r^r?  ways.  against  2  particular  persons  sit- 

"—  ■—  ting  next  each  other  are  n  —  3 

Since  any  set  of  four  cards  is  to  2 

aslikelytobedrawnasanyother,  0ne  of  the  two  persons  being 
the  chance  that  they  will  all  be  geated?  there  are  2  seats  next  to 
hearts  is  laim  and  n  __  3  not  next  to  him 

iAJ  _^   [jjg    _  iijljg  The  second  person  being  equally 

|4[9  *  [4[48       [9  [62  likely  to  take  any  seat,  the  odds 

_  13  X 12  X 1 1  x  10  that  he  does  not  sit  next  the  first 

~~"  62X51X60X49  one  are  n  —  3  to  2. 

=  -—  •  12.  If  two  letters  are  selected 

at  random  out  of  the  alphabet, 

9.  If  10  persons  stand  in  a  line,  what  is  the  cnance  that  both  will 
what  is  the  chance  that  2  assigned  be  vowels  ? 

persons  will  stand  together  ?  with  the  26  letters  of  the  alpha- 

The  total  number  of  pairs  of  [26 

1 10  bet,  .    .      =  326  pairs  can   be 

places  among  the  10  is  7575-  =  45.  L  L_ 

L£l°  formed  and  from  the  5  vowels 

Of  these,  9  are  adjacent ;  viz. ,  the  [5 

first  and  second,  second  and  third,  |2J3  =  10  pairs*  Hence  the  chance 

etc.,  up  to  the  ninth  and  tenth.  ^  ft  random   pair  should   be 

Hence  the  chance  that  the  two  TTrtTwrolo  •     1 0  —  2 

,                      ,,.           ,  voweis  is  -tot  —  -gx. 
assigned  persons  will  stand  to- 
gether is  ?93  =  J.  13-  Five  men»  A>  B>  c>  D,  E» 

speak  at  a  meeting,   and   it   is 

10.  If  10  persons  form  a  ring,  known  that  A  speaks  before  B. 
what  is  the  chance  that  2  assigned  What  is  the  chance  that  A  speaks 
persons  will  stand  together  ?  immediately  before  B  ? 

As  in  Ex.  9,  the  total  number  The  five  men  can  be  assigned 
of  pairs  of  places  among  the  10  is  to    speak  in  [5  =  120   different 


484  ALGEBRA. 

orders.     In  one-half  of  these  A  prize  of  $100.     What  is  his  ex- 
will  precede  B.     Also  there  are  pectation  ? 
4  ways  in  which  B  can  immedi-  From  the  60  tickets  in  the  bag, 
ately  follow  A ;  viz.,  A  may  speak  [50 
first  and  B  second,  or  A  second  2  can  be  drawn  m  f^jig  =  1225 

and  B  third,  or  A  third  and  B   wavs-  

fourth,  or  A  fourth  and  B  last.  Two  tickets  entitling  to  prizes 

The  3  remaining  speakers  may  be  <**  be  drawn  ^^  ^  ten  m 

arranged  in  each  case  in  6  differ-    Ii2  __  Af>mn„a  n„A  +1>„  «™,rtxft 
°,  .-^tz  =  45  ways,  and  the  expecta- 

ent  orders.  [2  [8 

Hence  the  chance  that  A  speaks  tion    from   drawing  2  prizes  is 

immediately  before  B  is  T*5?  x  §200  =  $\6^. 

4X6  _  One  ticket  entitling  to  a  prize 

60           '  and  one  blank  can  be  drawn  in 

14.  A,  B,  C  have  equal  claims  40  X  10  =  400    ways,   and   the 
for  a  prize.     A  says  to  B,  "  You  expectation    from   the   event  is 
and  I  will  draw  lots,   and  the  t\°A  *  $100  =  fifja. 
winner  shall  draw  lots  with  C  for  Hence  the  total  expectation  is 
the  prize."    Is  this  fair?  $3fl0^160Q=  $40. 

The  chance  that  A  will  win  is.  One  of  two  events  must 

from  B  is  £.     If  he  wins  from  B,  happen.     If  the  chance  of  one  is 

the  chance  that  he  will  win  from  $.  0f  the  chance  of  the  other,  find 

C  is  also  i.     To  get  the  prize  he  the  odds  on  the  first, 

must  win  from  B  and  then  from  Since  one  of  the  two  events 

C,  and  his  chance  of  doing  this  must  happen,  the  sum  of  their 

is  i  x  £  =  ±.  probabilities  is  1 ;    and  as  the 

Similarly,  B's  chance  of  win-  chance  of  the  first  is  £  of  the 
ning  is  £ ;  but  C's  chance,  since  chance  of  the  second,  their  re- 
he  is  sure  of  a  trial  and  has  to  gpective  chances  are  f  and  f. 
win  only  once,  is  £.  Hence  the  odds  on  the  first  are 

Originally  A,  B,  C  had  each  a  2  to  3. 

chance    of   £.     A's    proposal   is       ^   „„  n  .   „  „ 

•  .    x     a       j  t»        j       17.  There  are  3  events,  A,  B,  C, 

equally  unfair  to  A  and  B,  and  luwcaicuo  cu»,n,    ,   , 

..       f  .    .    „  one  of  which  must  happen.    The 

more  than  fair  to  C.  , ,  n  .         , « A   r 

odds  are  3  to  8  on  A,  and  2  to  5 

15.  A  person  is  allowed  to  draw  on  B.     Find  the  odds  on  C. 

2  tickets  from  a  bag  containing  The  chance  that  A  happens  is 
40  blank  tickets,  and  10  tickets  fy;  that  B  happens,  f.  Hence 
each  entitling   the  holder  to  a   (Rule  V),  since  A,  B,  or  C  must 


teachers'  edition. 


485 


happen,  the  chance  that  C  hap- 
pens is  1  —  (f  +  fV)  =  f f ;  and 
the  odds  on  C  are  34  to  43. 

18.  In  a  bag  are  7  white  and  5 
red  balls.  Find  the  chance  that  if 
one  is  drawn  it  will  be  (i.)  white 
or  (ii.)  red  ;  or,  if  two  are  drawn, 
that  they  will  be  (i.)  both  white, 
(ii.)  both  red,  or  (iii.)  one  white 
and  the  other  red. 

Since  there  are  12  balls  in  all, 
the  chance  of  drawing  a  white  ball 
is  ^  j  of  drawing  a  red  ball,  T*y. 

From  the  12  balls,   2  can  be 

112 
drawn  in  r^rr^  =  66  ways.    Two 

white  balls  can  be  drawn  from 

II 
the  7  in  — —  =  21   ways ;    two 

^  L6 

red  balls  from  the  5  in  r^-r  =  10 
[2  [3 

ways ;  one  white  ball  and  one  red 
ball  in  7  X  5  =  35  ways.  Hence 
the  chance  of  drawing  2  white 
balls  is  |4  =  A ;  2  red  balls, 
i?  =  -fg ;  one  white  ball  and  1 
red,  ff. 

19.  If  3  cards  are  drawn  from 
a  pack,  what  is  the  chance  that 
they  will  be  king,  queen,  and 
knave  of  the  same  suit  ? 

Three  cards  can  be  drawn  from 

1 52 
the  52  in  the  pack  in  -=t-  =  22,100 
[3|49 

ways.  King,  queen,  and  knave 
of  the  same  suit  can  be  drawn  in 
4  ways. 


Hence  the  chance  of  drawing 
king,  queen,  and  knave  of  the 
same  suit  is  „$„  =  5^y. 

20.  A  general  orders  2  men  by 
lot  out  of  100  mutineers  to  be 
shot,  the  real  leaders  of  the 
mutiny  being  ten  in  number. 
Find  the  chance  (i.)  that  1  only, 
(ii.)  that  two,  of  the  leaders  will 
be  shot. 

Two  men  can  be  selected  from 

1100 
100  in  -)=  =  4960  ways.     One 

leader  and  one  follower  can  be 
selected  in  10  X  90  =  900  ways ; 

Ii? 
two  leaders  in  rr-z  =  45  ways. 

If  l£ 

Hence  the  chance  that  only  one 

leader  will  be  shot  is  ?9g0T°0  =  ^ ; 

that  two  leaders  will  be  shot, 

?iiff  —  riff- 

21.  Show  that  the  odds  are  8 
to  1  against  throwing  9  in  a  single 
throw  with  2  dice. 

The  total  number  of  throws 
with  2  dice  is  6  X  6  =  36.  The 
number  of  ways  of  throwing  9 
is  4;  viz.,  3  and  6,  6  and  3,  4 
and  6,  5  and  4.  Hence  the  odds 
against  throwing  9  is  32  to  4,  or 
8tol. 

22.  Show  that  in  a  throw  with 
3  dice  the  chance  of  either  a  trip- 
let or  a  doublet  is  f . 

The  total  number  of  throws 
with  3  dice  is  6x6x6  =  216. 
The  number  of  ways  of  throwing 


486  ALGEBRA. 

a  triplet  is  6,  since  it  may  be  will  be  alternately  white  and  black 
three    l's,    three    2's,    etc.      A  i8|xjxfxjx?x|x|x} 
doublet  may  happen  with  any  2  =  T£7. 
of  the  three  dice,   and  accord- 
ingly the    number  of    ways    of  24.    A  bag   contains  2  white 
throwing  a  doublet  is  equal  to  balls»   3  black  balls,   and  5  red 
the  number  of  doublets,  6,  multi-  Dalls-    K  4  balls  are  drawn,  find 
plied  by  the  numbers  of  pairs  of  tne  chance    that   there  will  be 
dice,  3,  and  multiplied  further  by  among  them  : 
the  number  of  ways  the  odd  die  (*-)  Botil  the  white  balls, 
may  fall.     This  last  Dumber  is  5,  ("•)  Two  m&V  of   l^e   Wack 
since  the  case  of  a  triplet  has  balls, 
already  been  considered.  0"-)  Two  at  least  of  the  red 

Hence  a  triplet  can  be  thrown  balls, 

in  6  ways,  and  a  doublet,  but  pour  Dang  ^n  ^  drawn  from 

not  a  triplet,  in  6X3X5  =  90  mq 

ways.   Either  a  doublet  or  a  trip-  the  10  in  the  bag  in  r^rz  =  210 

let  can  be  thrown  in  6  +  00  =  96  ways.                            *-L 

ways.     The  chance  of  throwing  (L)  M  the  4  balls  are  to  include  . 

a  doublet  or  triplet  is  therefore  the  2  white  oneg?  the  other  2  can 

*rV  —  %'  be  drawn  from  the  remaining  8 

23.  In  a  bag  are  5  white  and  4  in  J=_  =  28  ways.    Hence  the 

black  balls.     If  drawn  out  one  L?15 

by  one,  what  is  the  chance  that  chance  that  the  2  white  balls  are 

the  first  will  be  white,  the  second  drawn  among  the  4  is  -ff^  =  fa 

black,  and  so  on,  alternately  ?  (iL)  Two  black  Mls  can  be 

The  chance  that  the  first  ball  drawn  from  the  3  black  balls  in 

will  be  white  is  f .  13 

The    first    ball    having   been  [2Q  =  3ways-   « exactly 2 black 

drawn,  if  it  is  white,  the  chance  „                ,.,,,.,    j 

that  the  second  one  is  black  is  baUs  are  to  be  included  in  the  4 

4  _  1  drawn,  the  other  2  can  be  selected 

1  T^e  chance  that  the  third  ball  f rom  the  7  white  and  red  balls 

is  white  is  then  }  ;  that  the  fourth  in  _L_  _  21  wayg     Hence  ^ 

is  white,  £  =  J  ;  that  the  fifth  is  I?  IJ> 

black,  | ;  that  the  sixth  is  white,  chance  of  drawing  exactly  2  black 

|  =  i  ;  and  so  on.  _    ...   3  X  21 

balls  is =  -Ac 

Hence  the  chance  that  the  balls  Utt  °  °    210        **" 


TEACHERS7    EDITION. 


487 


(iii.)  If  no  red  ball  is  drawn, 
the  4  balls  can  be  selected  from 
the  5  white  and  black  balls  in 

I* 

— —  =  5  ways.    Hence  the  chance 

[4[1 

that  no  red  ball  is  drawn  is  jfo 
=  ^.  If  exactly  one  red  ball  is 
to  be  drawn,  it  can  be  selected 
from  the  5  red  balls  in  5  ways. 
The  3  remaining   balls  can  be 


selected  from  the  5  white  and 
black  balls  in  —  =  10  ways. 

[3  [2 

Hence  the  chance  that  exactly  1 

red  ball  is  drawn  is  =  ^T. 

The  chance  of  drawing  either  no 
red  ball  or  exactly  1  is  &  +  -fa 
=  \\ ;  and  the  chance  of  draw- 
ing more  than  one  red  ball  is 


Exercise   CXXIII. 


1.  The  chance  that  A  can  solve 
a  certain  problem  is  £,  and  the 
chance  that  B  can  solve  it  is  $. 
What  is  the  chance  that  the 
problem  will  be  solved,  if  both 
try? 

The  chance  that  A  will  fail  is 
£ ,  and  the  chance  that  B  will  fail 
is  £.  Hence  the  chance  that  both 
will  fail  is  £  X  \  =  £ ;  and  this  is 
the  chance  that  the  problem  will 
not  be  solved.  The  chance  that 
it  will  be  solved  is,  therefore, 
l-i  =  *. 

2.  What  is  the  chance  of  throw- 
ing at  least  one  ace  in  2  throws 
with  one  die  ? 

The  chance  of  not  throwing  an 
ace  is  J  each  time.  Hence  the 
chance  of  not  throwing  an  ace 
both  times  is  £  X  £  =  f  f  ;  and 
the  chance  of  throwing  at  least 
one  ace  is  1  - 1|  =  \\. 


3.  If  n  coins  are  tossed  up, 
what  is  the  chance  that  one,  and 
only  one,  will  turn  up  head  ? 

The  chance  that  any  given  one 
will  be  head  is  £,  and  the  chance 
that  the  rest  will  be  tails  is  (i)n_1. 
Hence  the  chance  that  a  given 
one  will  be  head  and  the  others 
tails  is  i  X  (i)«-i  =  ($)».  And 
the  chance  that  some  one  will  be 
head  and  the  rest  tails  is  n(£)B. 

4.  What  is  the  chance  of  throw- 
ing double  sixes  at  least  once  in 
3  throws  with  2  dice  ? 

The  chance  of  throwing  double 
sixes  any  given  time  is  J  x  J  =  ^ 
and  the  chance  of  not  throwing 
them  any  given  time  is  1  —  -fa 
=  }§.  Hence  the  chance  of  not 
throwing  them  in  any  of  the  3 
throws  is  (f  J)8;  and  the  chance 
that  they  will  be  thrown  at  least 
once  is  1-  (||)8  =;&&V 


488 


ALGEBRA. 


5.  A  copper  is  tossed  3  times. 
Find  the  odds  that  it  will  fail : 

(i.)  Head  and  2  tails  without 
regard  to  order. 

(ii.)  Head,  tail,  head. 

The  coin  may  fall  head  three 
times,  which  can  happen  in  only 
1  way ;  or  head  twice  and  tail 
once,  which  can  happen  in  3 
ways;  or  head  once  and  tail 
twice,  3  ways;  or  tails  three 
times,  1  way.  All  these  8  ways 
are  equally  likely  to  happen. 
Hence  the  chance  that  it  will  fall 
head  and  two  tails  is  f . 

The  chance  that  it  will  fall 
head,  tail,  head  isix^x|  =  i. 

6.  If  a  copper  is  tossed  4  times, 
find  the  odds  that  it  will  fall  2 
heads  and  2  tails  sooner  than  4 
heads. 

The  total  number  of  ways  it 
can  fall  is  24  =  16.  It  can  fall 
head  every  time  in  only  1  way. 
It  can  fall  head  twice  and  tail 
twice  in  as  many  ways  as  2  selec- 
tions can  be  made  from  4;  viz., 

Ii 
— rr  =  6.     Hence  the  odds  are  6 

L±L£ 

to  1  that  it  will  fall  head  twice 
and  tail  twice  rather  than  head 
four  times. 

7.  If  from  a  lottery  of  30  tick- 
ets,   marked    1,   2,   3, four 

tickets  are  drawn,  what  is  the 
chance  that  1  and  2  will  be  among 
them? 


Four  tickets  can  be  drawn  from 

130 
30  in  r~^  ways.     If  1  and  2 

are  to  be  among  them,  the  other 

two  can  be  drawn  from  the  re- 

[28 
maining  28  in  Yj^  ways.   Hence 

the  chance  that  1  and  2  will  be 
drawn  is 

[2  [26  '  [4  [26      [30  [2      "*• 

8.  If  2  coppers  are  tossed  3 
times,  find  the  odds  that  they 
will  fall  2  heads  and  4  tails. 

There  are  3  ways  of  getting  2 
heads  and  4  tails.  First,  the  first 
copper  may  fall  head  twice,  all 
the  other  cases  giving  tails ;  sec- 
ond, each  copper  may  fall  head 
exactly  once  ;  and  third,  the  sec- 
ond copper  may  fall  head  twice, 
all  other  cases  giving  tails. 

The  chance  that  the  first  cop- 
per will  fall  head  twice  and  tail 
once  is  3x}x|x^  =  |,  since 
there  are  three  ways  of  getting 
two  heads  and  one  tail.  The 
chance  that  the  second  copper 
will  fall  tail  every  time  is  i  X  £ 
x  £  =  £.  Hence  the  chance  that 
the  first  copper  gives  two  heads 
and  a  tail,  and  the  second  copper 
3  tails,  is  f  x  £  =  ^ ;  and  the 
chance  that  one  of  the  two  cop- 
pers gives  two  heads  and  a  tail, 
and    the    other   three    tails,    is 

2xA  =  TV 


TEAUHBKS'    EDITION.  489 

The  chance  that  each  copper  10  are  6,  4, 1  and  5,  3,  2.     5, 4, 1 
gives  1  head  and  2  tails  is  f  x  £  can  be  drawn  in  6  ways.    The 
=  &.  chance  that  5  will  be  drawn  the 
Hence  the  chance  of  tossing  first  time  is  ^.     5  not  being  re- 
two  heads  and  four  tails  is           •  placed,  the  chance  of  drawing  4 
.3  +  9  —  is  the  second  time  is  J.    And  the 
chance  of  drawing  1  the  third 

9.  There  are  10  tickets,  five  of  time  is  J.  Hence  the  chance  of 
which  are  numbered  1, 2,  3,  4,  5,  drawing  5,  4,  1  in  this  order  is 
and  the  other  five  are  blank.  ^  X  J  X  J  =  7^j.  Therefore  the 
Find  the  chance  that  the  sum  of  chance  of  drawing  5,  4,  1  in  any 
the  numbers  on  the  tickets  drawn  order  is  6  X  7^  =  T£7.  And 
in  3  trials  will  be  10,  one  ticket  similarly,  the  chance  of  drawing 
being  drawn  and  then  replaced  5,  3,  2  is  T^.  Hence  the  chance 
at  each  trial.  of  drawing  a  total  of  10  is 

The  numbers  drawn  must  be  ,     ,     1    __  . 

one  of  the  five  sets :  T**     T3nr "  A* 

5,  5,  0 ;   5,  4,  1 ;   5,  3,  2 ;  11.  A  bag  contains  4  white  and 

4,  4,  2 ;    4,  3,  3.  6  red  balls.     A,  B,  and  C  draw 

_ .  ,         ^    _             ,     ,  each  a  ball,  in  order,  replacing. 

Of  ifrese  the  first  can  be  drawn  Fmd  the  chance  ^  th      haye 

as  5,  5,  0  or  5,  0,  6  or  0,  5,  5,  drawn 
and  the  0  may  be  any  one  of  the 

five   blanks,    giving   3X6=16  O-)  Each  a  white  ball, 

ways.    The  fourth  and  fifth  sete  (ii.)  A  and  B  white,  C  red. 

can  be  drawn  in  3  ways  each.  <hi'>  Two  white  «*  1  red' 

The  other  two  sets  can  be  drawn  (i.)  Tne  chance  that  one  draws 

in  6  ways  each,  making  in  all  a  white  ball  being  T%  =  §,  the 

15  +  6  +  12  =  33  ways  of  draw-  chance  that  they  all  draw  white 

ing  a  sum  of  10.  Dalls  fa  (2)8  =  ^ 

The  total  number  of  possible 

drawings  is  10  X 10  X 10  =  1000.  (**•)  Tne  chance  that  A  and  B 

Hence  the  chance  of  drawing  a  draw  white,  and  C  red,  is 

sum  of  10  is  T$fo.  (2)2  X}  =  ^. 

10.  Find  the  chance  in  Ex.  9  (iii.)  Multiplying  the  last  result 
if  the  tickets  are  not  replaced.  ^y  3^  we  ^Yq  ^  ag  tne  chance 

If  the  tickets  are  not  replaced,  that  some  one  of  the  three  draws 

the  only  combinations  which  give  red  and  the  other  white. 


490 


ALGEBKA. 


12.  Find  the  answer  to  Ex.  11 
if  the  balls  are  not  replaced. 

The  chance  that  A,  B,  and  C 
successively  draw  white  balls  is 
now  ^  x  »  x  I  =  3V- 

The  chance  that  A  and  B  draw 
white,  and  C  red  is  T%x  fx  5=^. 

The  chance  that  A  draws  white, 
B  red,  and  C  white  is  ,4ff  x  «  x  } 
=  ^ ;  and  the  chance  that  A 
draws  red,  and  B  and  C  white 
is  A  x  i  x  I  =  *0.  Hence  the 
chance  that  two  white  balls  and 
one  red  are  drawn  is  ^. 

13.  A  draws  4  times  from  a 
bag  containing  2  white  and  8 
black  balls,  replacing.  Find  the 
chance  that  he  has  drawn 

(i.)  Two  white,  two  black, 
(ii.)  Not  less  than  two  white, 
(iii.)  Not  more  than  two  white, 
(iv.)  One  white,  three  black, 
(i. )  Two  white  balls  and  2  black 

can  be  drawn  in  — -r  =  6  differ- 

ent  orders.  The  chance  of  draw- 
ing a  white  ball  is  at  every  trial 
T2<j  —  h  an(*  tne  chance  of  draw- 
ing a  black  one  is  f .  Hence  the 
chance  of  drawing  2  white  balls 
and  2  black  is 

«  X  (i)2  x  d)2  =  *%. 
(ii.)  The  chance  of  drawing  a 
white  ball  4  times  is  (£)*;  of 
drawing  a  white  ball  3  times  and 
a  black  ball  once,  4  X  ( £)*  +  f ; 
of  drawing  2  white  balls  and  2 


black  ones,  /z65,  as  just  shown. 
Hence  the  chance  of  drawing  not 
less  than  two  white  balls  is 

(S)4  +  4x(!)»xf  +  Vft  =  iif- 

(iii.)  The  chance  of  drawing  2 
white  balls  and  2  black  is  ^*5; 
of  drawing  1  white  and  3  black, 
4X  JX  ( j)8 ;  of  drawing  4  black, 
(J)4.  Hence  the  chance  of  draw- 
ing not  more  than  2  white  balls  is 

^  +  4XiX    (4)8+  (4)4  =   MJ. 

(iv.)  The  chance  of  drawing  1 
white  and  3  black  is 

14.  Find  the  odds  against  throw- 
ing one  of  the  two  numbers  7  or 
11  in  a  single  throw  with  2  dice. 

To  give  7  or  11,  the  number 
thrown  must  be  one  of  the  sets 

6,  1;    6,  2;    4,  3;    6,  5; 
and  each  of  these  can  be  thrown 
in  two  ways;  for  example,  6,  1 
as  6,  1  or  as  1,  6. 

The  total  number  of  throws 
with  2  dice  is  36.  Hence  the 
chance  of  throwing  7  or  11  is 

4X2 

oa    =  },  and  the  odds  against 
oo 

it  are  7  to  2. 

15.  If  a  copper  is  tossed  5  times, 
what  is  the  chance  that  it  will 
fall  head  either  2  times  or  else  3 
times? 

The  number  of  ways  it  can  fall 
either  head  twice  and  tail  three 
times,  or  tail  twice   and  head 


teachers'  edition. 


491 


three   times  is 


I*   _ 


10.      The 


12  I* 

chance  that  it  will  fall  head  twice 
and  tail  three  times  in  given 
order  is  (i)6.  Hence  the  chance 
that  it  will  fall  head  twice  and 
tail  three  times  in  any  order  is 
10  X  (£)6  =  -& ;  and  the  chance 
that  it  will  fall  head  exactly 
twice  or  exactly  3  times  is 

16.  Find  the  same  chance  if 
the  copper  is  tossed  6  times. 

The  number  of  ways  the  cop- 
per can  fall  head  exactly  twice  is 

I? 
now  .—77  =  15,  and  the  number 

[2  [4 

of  ways  it  can  fall  head  exactly  3 
•timesisi|  =  20. 

The  chance  that  it  will  fall 
head  exactly  twice,  in  one  order 
or  another,  is  15  x  (£)6,  and  the 
chance  that  it  will  fall  head 
exactly  three  times,  in  any  order, 
is  20  x  (*)«. 

Hence  the  chance  that  it  will 
fall  head  exactly  twice  or  exactly 
three  times  is 

15x(i)«  +  20X(i)«  =  if. 

17.  In  one  bag  are  10  balls  and 
in  another  6;  and  in  each  bag 
the  balls  are  marked  1,  2,  3,  etc. 
What  is  the  chance  that  on  draw- 
ing one  ball  from  each  bag  the 
two  balls  will  have  the  same 
number  ? 


The  number  of  pairs  of  balls 
that  can  be  drawn  is  6  x  10  =  60, 
and  there  are  6  pairs  which  have 
the  same  number.  Hence  the 
chance  of  drawing  such  a  pair  is 

A  =  TV 

18.  A  bag  contains  n  balls.  A 
person  takes  out  one  ball  and 
then  replaces  it.  He  does  this  n 
times.  What  is  the  chance  that 
he  has  had  in  his  hand  every  ball 
in  the  bag  ? 

The  number  of  sets  that  can  be 
drawn  in  n  times  is  nw.  The 
number  of  orders  in  which  the  n 
balls  can  be  arranged  without  rep- 
etitions is  [n.  Hence  the  chance 
that  there  are  no  repetitions  in 

\n 
the  drawing  is  —  • 

19.  If  on  an  average  9  ships 
out  of  10  return  safe  to  port, 
what  is  the  chance  that  out  of  5 
ships  expected,  at  least  3  will 
return  ? 

The  chance  that  all  the  5  ships 
will  return  is  (-&)*;  that  4  will 
return  and  one  be  lost  is  5x  (r9ff)4 
X  T^ ;  that  3  will  return  and  2  be 
lost  is  10  X  (T%)»  X  (tV)2.  Hence 
the  chance  that  at  least  3  will 
return  is 

x(A)'x(tW2 

=  (&)>(&*+ Ms +m 

—  12893 


492  ALGEBRA. 


20.  What  is  the  chance  of  the  other  times  is  (J)3  x  ($)J. 
throwing  doable  sixes  at  least  Hence  the  chance  that  it  will  be 
once  in  3  throws  with  a  pair  of  thrown  exactly  3  times  out  of 
dice  ?  the  5  is  10  x  (£)»  x  (j)«  =  flft. 

The  chance  that  they  will  be  (|L)  ^  chance  ^  m  m 
thrown  any  particular  time  is  ^  not  ^  t^^  at  au  jg  (f)5. 
J  x  J  =  A,  and  the  chance  that  ^  ^  ^  ^  ^  thrown  onJy 
they  will  not  be  thrown  any  once  ^  6  x  (|)4  x  ,  =  (s)6.  ^ 
particular  time  is  1  -  *=  ft  ^  ^  wU1  be  thrown  exactly 
Hence  the  chance  that  they  will  twice  k  10  x  (|)i  x  „)2  Hence 
not  be  thrown  any  time  is  <ff )»,  ^  chance  ^  ^  ^  wiu  be 
and  the  chance  that  they  will  be  ^^  ^^  timeB  or  m0M  fa 
thrown  is  l-(^)*  =  AWs- 

ML  What  is  the  chance  of  ^x^+^^AV 
throwing  15  in  one  throw  with  (iii.)  The  chance  that  an  ace 
3  dice  ?  will  be  thrown  exactly  4  times  is 

The  sum  of  15  can  be  thrown  6  x  (i)4  x  h  and  *****  **  wiU  ^ 

as         6,  6,  3        3  ways,  thrown  6  times  is  (J)5.    Hence 

a    k'  a        a  mn„J  *he  chance  that  it  will  be  thrown 

o,  6,  4        o  ways, 

5,  5,  5        1  way,  more  than  3  times  te  6  X  ®*  X  * 

10  ways.  +  <*)*  =  tHj.   and  *he  chance 

that  an  ace  will  be  thrown  not 


more  than  3  times  is  f  f  J|. 


The  3  dice  can  fall  in  6  X  6  x  6 
=  216  ways.     Hence  the  chance 

of  throwing  15  is  ^  =  tJt-  23.  In  a  bag  are  3  white,  6  red, 

22.  In  5  throws  with  a  single  and  7  black  balls,  and  a  person 

die  what  is  the  chance  of  throw-  draws    three    times,    replacing, 

ing  an  ace  Find    the    chance    that   he  has 

(i.)  Three  times  exactly  ?  drawn : 

(ii.)  Not  less  than  three  times?  (i.)  A  ball  of  each  color, 

(iii.)  Not  more  than  three  times?  (ii.)  Two  white,  one  red. 

(i.)  An  ace  can  be  thrown  3  ("*•)  Three  red. 

1 5  (iv.)  Two  red,  one  black, 
times  out  of  5  in  rrrr  =  10  ways. 

12 1±  (i.)  In  this  case  the  balls  can 

The  chance  that  it  will  be  thrown  be  drawn  in  6  different  orders, 

any  particular  time  is  £,  and  the  The  chance  of  drawing  one  ball 

chance  that  it  will  be  thrown  3  of  each  color  in  a  particular  order 

particular  times  and  not  thrown  is  f»5  X  ^z  x  T^  =  ^|T,   and  the 


493 


chance  of  drawing  a  ball  of  each  6    is    5  X  (})*  x  j.      Hence   his 

color  in  any  order  is  6  x  ^$,  =  ^f .  chance  of  losing  3  games  or  less, 

(ii.)    Here    the    balls    can    be  tnat  k»  nis  chance  of  winning  2 

drawn  in  only  3  different  orders,  games  or  more,  is 
and  the  chance  is  1  -  [(f)5  +  6  X  (»)*  X  f  ] 

(iii.)  The  chance  of  drawing  3 
red  balls  is  (^)8  =  jj. 


26.  The  skill  of  A  is  double 

that  of  B.    Find  the  odds  against 

(iv.)  The  chance  of  drawing  2  A,g  winni      4               Mfm  B 

red  and  1  black  is  .„  0  0 

wins  2. 

In  order  that  A  may  wm  4 

24.  A  and  B  play  at  chess,  and  games  before  B  wins  2  it  is  neces- 
A  wins  on  an  average  2  games  gajy  and  sufficient  that  A  should 
out  of  3.  Find  the  chance  of  A's  wm  at  least  4  games  out  of  the 
winning  exactly  4  games  out  of  first  5. 

the  first  6,  drawn  games  being  The  chance  that  A  will  lose 

disregarded.  any  particular  game  is  i,  and  the 

A  may  win  4  games  out  of  the  chance  that  he  will  lose  some  one 

15        „r               TT.  of  the  first  five  games  and  win 

first  6  in  j-tt^  =  15  ways.     His  ..           .    .     c  ._ °  ._  /tX4       -0 

[4|2                J  the    rest   is    5  X  £  X  (f)4=  /ft. 

chance  of  winning  any  particular  The  chance  that  he  will  win  all 

game  is  £ ,  and  of  winning  any  4  the  games  is  (f  )5  =  /ft.     Hence 

particular  games  and  losing  the  the  chance  that  he  will  win  4 

other 2  is  (f)*  Xtt)2=  Jft.    Hence  .     .  .,     «    .  .  .   80  +  32 

i..     ^            *     .     .               a    *  games  out  of  the  first  5  is —rTT— 

his  chance  of  winning  some  4  of  24d 

the  6  games  and  losing  the  other  2  =  JJf ,  and  the  odds  against  his 

is            16  X  /ft  =  /ft.  doing  it  is  131  :  112. 

25.  A  and  B  engage  in  a  game  ^     If  B,a  gkm  ^  a  certain 

in  which  A's  skill  is  to  B's  as  game  fa  equal  t0  tnree_fifths  of 

2:3.    Find  the  chance    of  A's  A^  find  A,g  chance  of  w|nning 

winning  at  least  2  games  out  of  fi  gameg  out  of  g 

the   first    5,  drawn    games    not  . .     ,             ...           .     , 

,  .              .    ,  A's  chance  of  winning  a  single 

bemg  counted.  .    R      XT.      _             -     7 

game  is  $ .     His  chance  of  win- 
As  chance  of  losing  any  par-  ning  exactly  fi  gameg  QUt  of  g  fa 

ticular  game  is  f .     His  chance  of 

losing  all  the  games  is  (f)*,  and  JL  x  (<>\5X  (§\8  =  55  x  5*x38. 

of  losing  just  4  games  out  of  the  [5[3     \8/      \8/                  88 


494 


ALGEBRA. 


His  chance  of  winning  exactly 

6  games  is 

His  chance  of  winning  exactly 

7  games  is 

o     /6\7     3      0      67x3 

8X(8)X8  =  8X-8^- 

His  chance  of  winning  all  the 

games  is      /5\8_  — 

\s)  ~~  88* 

Hence  his  chance  of  winning 

at  least  5  games  out  of  8  is 

6^X3»  6^X^ 

66X— — +  28X  — — 


65, 


88       '   %s 

=  j£  (66  X  27  +  28  X  45  +  24 

X  25 +126) 
56 
=  g8X3497. 

28.  A  bag  contains  4  red  balls 
and  2  others,  each  of  which  is 
equally  likely  to  be  red  or  white. 
Three  times  in  succession  a  ball 
is  drawn  and  replaced.  Find  the 
chance  that  all  the  drawn  balls 
are  red. 

The  chance  that  the  2  un- 
identified balls  are  both  red  or 
both  white  is  in  each  case  ^x{ 
=  i ;  that  one  is  red  and  the 
other  white  is  £. 

If  the  balls  are  all  red,  the 
drawn  balls  are  of  course  all  red. 
The  chance  of  this  is  £. 

If  two  of  the  balls  were  white, 
the  chance  that  the  drawn  balls 


are  all  red  would  be  (t)8  =  ?V 
Hence  the  chance  that  2  balls 
are  white  and  the  drawn  balls  all 
red  is  J  X  ^  =  fa 

If  only  1  ball  were  white,  the 
chance  that  the  drawn  balls  are  all 
red  would  be  ($ )8  =  iff.  Hence 
the  chance  that  there  is  only  1 
white  ball  and  that  the  drawn 
balls  are  aU  red  is  i  X  Jff  =  y|. 

Therefore  the  chance  that  the 
drawn  balls  are  all  red,  whatever 
the  2  unidentified  balls  may  be,  is 

29.  A  man  has  left  his  umbrella 
in  one  of  three  shops  which  he 
visited  in  succession.  He  is  in 
the  habit  of  leaving  it,  on  an 
average,  once  in  every  four  times 
that  he  goes  to  a  shop.  Find  the 
chance  that  he  left  it  in  the  first, 
second,  and  third  shops  respec- 
tively. 

The  chance  that  he  would  leave 
it  in  the  first  shop  is  £. 

The  chance  that  he  would  not 
leave  it  in  the  first  shop,  but 
would  leave  it  in  the  second,  is 

***  =  *• 

The  chance  that  he  would  not 
leave  it  in  the  first  or  second 
shop,  but  would  leave  it  in  the 
third,  is  (f)2xi=TrV 

Hence  the  probability  that  he 
would  leave  it  in  one  of  the  three 

shops  is  i  +  A+A=H- 

But  as  the  umbrella  was  cer- 
tainly left  in  one  of  the  shops,  all 


teachers'  edition.  495 

the  preceding  chances  must  be  he  draws  a  white  ball  he  is  to 

increased  in  the  ratio  of  64 :  37  receive  $1,    and  every  time  he 

(Rule  IX.).  draws  a  black  ball  he  is  to  pay  60 

Hence  the  chance  that  it  was  cents.     What  is  his  expectation  ? 

left  in  the  first  shop  is  f  7  x  J  =  J7 ;  A's  expectation  is  the  same  for 

in  the  second  shop,  §*  x  A  =  i*  >  eacn  drawing,  viz.,  T%  x  $1  —  A 

in  the  third  shop,  1 1  x  &  =  sV  x  $$  =  —  $  A     His  expectation 

Hence,  16 :  12  :  9.  from    6    drawings    is    therefore 

30.  A  bets  B  $10  to  $1  that  he  "*  $h    The  chances  are  that  he 
will  throw  head  at  least  once  in  wiU  lose  26  cenfcs- 

3  trials.  What  is  B's  expecta-  32.  From  a  bag  containing  2 
tion  ?  What  would  have  been  a  eagles,  3  dollars,  and  3  quarter- 
fair  bet  ?  dollars  A  is  to  draw  one  coin  and 

The  chance  that  A  will   not  then  B  three  coins;  and  A,  B, 

throw  head  at  all  in  3  trials  is  and  C  are  to  divide  equally  the 

(i)8  =  i ;  and  the  chance  that  he  value  of  the  remainder.     What 

will  throw  head  at  least  once  is  are  their  expectations  ? 

1  —  £  =  £.      B's    expectation    is  In  the  final  division  A,  B,  and 

therefore  £  of  gaining  $10,  and  C  receive  equal  shares. 

i  of  losing  $1 ;   that  is,  his  ex-  Consider     their     expectations 

pectation  is  if-  —  J  =  $f.  from  this  division. 

A  fair  bet  would  have  been  in  There  are  $23.75  in  the  bag. 

the  ratio  of  the  chances  of  win-  A  and  B  are  to  draw  out  4 

ning  and  losing,  that  is,  7  to  1.  coins  at  first. 

o-i     a  j          c  *.-        /      i     •     \  4  coins  can  be  selected  from  8 

31.  A  draws  5  times  (replacing) 

from  a  bag  containing  3  white  jn   I—   =  ^q  wavs# 

and  7  black  balls.     Every  time  [4[* 


A  and  B  may  draw :             wutyT 

Amount 
remaining 

Expectation  for  the 
division. 

(1)  2  eagles,  2  dollars                   A 

$1.75 

7\x    $1.75  =  $AV 

(2)  2  eagles,  1  dollar,  1  quarter  A 

$2.50 

Ax    $2.50  =  $^ 

(3)  2  eagles,  2  quarters                A 

$3.25 

AX    $3.25  =  $&% 

(4)   1  eagle,  3  dollars                    A 

$10.75 

A  X  $10.75  =  $TV* 

(5)   1  eagle,  2  dollars,  1  quarter   7# 

$11.50 

7$X  $11.50  =  $W 

(6)  1  eagle,  1  dollar,  2  quarters   7§ 

$12.25 

ft  X  $12.25  =  |ft 

(7)   1  eagle,  3  quarters                  A 

$13.00 

A  X  $13.00  =  $Jf 

(8)  3  dollars,  1  quarter                 A 

$20.50 

a  x  $20.50 = mi 

(9)  2  dollars,  2  quarters                A 

$21.25 

^X  $21.26=  *W 

(10)   1  dollar,  3  quarters                 A 

$22.00 

AX  $22.00  =$H 

Total  expectation  for  the  division  $3AV_  or  ^H- 


496  ALGEBRA. 


Hence  the  expectation  of  each  from  the  division  is  £  of  $11 f£, 
or  $3.862V 

Hence  C's  entire  expectation  is  $3.86^. 

Again,  A  may  draw  an  eagle.  The  probability  of  this  is  },  or  £. 
The  corresponding  expectation  is  $2.50. 

Or,  he  may  draw  a  dollar.  The  probability  of  this  is  |.  The  cor- 
responding expectation  of  this  is  37£  cents. 

Or,  he  may  draw  a  quarter.  The  probability  of  this  is  f .  The 
corresponding  expectation  is  9f  cents. 

Hence  A's  expectation  from  his  draw  is  $2.96$. 

And  A's  total  expectation  is  $6.83^. 

A  and  C's  expectations  together  are  $10.69^. 

Hence  B's  expectation  is  $23.76  —$10.69^*  =  $13.05}|f 

33.  A,  B,  and  C,  staking  each  and  C  all  fail  the  first  time,  and 

$5,  draw  from  a  bag  in  which  A  the  second  time,   and  that  B 

are  4  white  and  6  black  balls,  draws  a  white  ball  the  second 

each  drawing  in  order,  and  the  time  is  (|)4  x}=  (f  )*  x  £ ;  and 

whole  sum  is  to  be  received  by  so  on. 

him  who  first  draws  a  white  ball.  Hence  B's  chance  of  drawing 

What  are  their  expectations :  a  white  ball  first  is 

(i.)  Replacing  the  balls  ?  A  +  A  x  (!)8  +  A  x  (!)6 + 

(ii.)  Not  replacing  the  balls  ?  =  A  "=■  L1  -  (I)8]  =  tt- 

(i.)  The  chance  that  A  draws  c's  chance  of  winning  is  ther*- 

a  white  ball  the  first  time  is  £ ;  f ore  1  —  \ i  -  it  =  &• 

that  A,  B,  and  C  fail  the  first  Hence  the  expectations  of  A, 

time  and  that  A  draws  a  white  B>  and  C  respectively  are 

ball  the  second  time  is  (J)8  x|;  }(  x  $15  =  $7f  f , 

that  they  all  fail  twice  and  that  Jf  X  $15  =  $4f  ? , 

A  draws  a  white  ball  the  third  and        &  X  $15  =  $2ff 

time  is  ($)«  X  $ ;  and  so  on.  The  chances  are  that  A  will 

Hence  A's  chance  of  drawing  gam  $2f$,  that  B  will  lose  $\l 

a  white  ball  first  is  and  that  C  will  lose  $2Jf 

_  *  +  *  (!)8  +  I  (*)6  + (ii.)  The  chance  that  A  draws 

-  f  "*-  C1  -  (!)8]  -  tt-    (§  396-)  a  white  ball  the  first  time  is,  as 

The  chance  that  A  fails  the  first  before,  j.     The  chance  that  A, 

time  and  that  B  draws  a  white  B,   C  all  draw  black  balls  the 

ball  is  |  x  -|  =  ^ ;    that  A,   B,  first  time,  and  that  A  draws  a 


TEACHERS7    EDITION. 


497 


white  ball  the  second  time,  is 
f  x  |  x  |  x  $  =  fa.  The  chance 
that  A,  B,   C  fail  each  time  is 

jxjxf  xjxf  x  i  =  Th.  ^d 
Amusi  then  draw  a  white  ball. 
Hence  A's  chance  of  drawing  a 
white  ball  first  is 

*+A  +  ii*=t 

The  chance  that  A  fails  the 
first  time  and  that  B  draws  a 
white  ball  is  $  X  J  =  f5.  The 
chance  that  A,  B,  and  C  fail  the 
first  time,  and  A  the  second  time, 


and  that  B  draws  a  white  ball 
the  second  time  is 

|xf  x|xfx  $=fa. 

Hence  B's  chance  of  drawing 
a  white  ball  first  is  ^5  4-  fa  =  £j, 
and  C's  chance  is  1  —  J  —  ii  ==  t J- 

Hence  the  expectations  of  A, 
B,  and  C  respectively  are 
ix$15  =  $7J, 
"X$15=$4*, 
and        i$  x  $15  =  $2^. 

The  chances  are  that  A  will 
gain  $2£,  that  B  will  lose  $$,  and 
that  C  will  lose  $2^ . 


Exercise   CXXIV. 


Note.  — Four-place  logarithms  do  not  give  very  accurate  results 
in  some  of  the  following  problems. 

1.  In  how  many  years  will  $100 
amount  to  $1050  at  5  per  cent 
compound  interest  ? 
P=100,  12=1.05,  -4  =  1050. 
100  X  (1.05)»=1050, 
(1.06)»=10.5, 
nlog  1.05  =  log  10.5, 


(ii.)  B=AR». 

.-.  n  log  B  =  log  B  —  log  A, 
_  log  B  —  log  A 
n~~         log* 

The  time  is  — ^—. — ^*     years. 
log-R 


_  log  10.5 
n     log  1.05 
_  1.0212 
"  0.0212 
=  48,  nearly. 
The  time  is  48  years,  nearly. 

2.  In  how  many  years  will  $-4. 
amount  to  $2?  (i.)  at  simple  in- 
terest, (ii.)  at  compound  interest, 
r  and  R  being  used  in  their  usual 
sense?  • 

(i.)  —£-  years. 


3.  Find  the  difference  (to  five 
places  of  decimals)  between  the 
amount  of  $1  in  2  years,  at  6  per 
cent  compound  interest,  accord- 
ing as  the  interest  is  due  yearly 
or  monthly. 
If  the  interest  is  due  yearly, 
A  =  (1.06)2  X  1 
=  1.1236. 
If  the  interest  is  due  monthly, 
A  -  (1.005)»  x  1 
=  1.12923. 
Hence  the  difference  is 
$0.00563. 


498 


ALGEBRA. 


4  At  5  per  cent,  find  the 
amount  of  an  annuity  A  which 
has  been  left  unpaid  for  4  years. 

By  §  452,  the  amount  due  is 

S(ifr-l)_^[(1.05)«-l] 
r  0.05 

=  A  (0.2155) 

0.05 
=  4.31.4,  nearly. 

5.  Find  the  present  value  of  an 
annuity  of  $100  for  5  years, 
reckoning  interest  at  4  per  cent. 


1 


_  100       1.045 
~  0.04  X      1.04& 
=  444. 
The  present  value  is  $444. 

6.  A  perpetual  annuity  of  §1000 
is  to  be  purchased,  to  begin  in  10 
years.  If  interest  is  reckoned  at 
3£  per  cent,  what  should  be  paid 
for  it  ? 

8 


P  = 


12*  (22-1) 
1000 


P  = 


S 


12-1 


12»-1 

2J» 


1.035W  x  0.035 
=  20,270,  nearly. 
The  amount  paid  should  be 
$20,270. 


7,  A  debt  of  $1850  is  discharged  by  two  payments  of  $1000  each, 
at  the  end  of  one  and  two  years.    Find  the  rate  of  interest  paid. 

Amount  of  $1850  for  2  years  =  1850  E2, 
Amount  of  $1000  for  1  year  =100012, 
Balance  due  =  1000. 
.-.  1850  122  -  1000  B  =  1000, 
37222-2012  =  20, 

__  10  +  V840 
*~         37 
=  1.0535. 
Hence  the  rate  of  interest  is  5.35  per  cent. 

8.  Reckoning  interest  at  4  per  __  80 
cent,  what  annual  premium  should 
be  paid  for  30  years  in  order  to 
secure  $2000  to  be  paid  at  the 
end  of  that  time,  the  premium 
being  due  at  the  beginning  of 
each  year  ? 

Ar 


P  = 


12(22«-1) 

2000  X  0.04 
1.04  (1.04*>  -  1) 


1.04  X  2.2426 
=  34.402. 
The  annual  premium  should  be 
$34.40. 

9.  An  annual  premium  of  $150 
is  paid  to  a  life  insurance  com- 
pany for»  insuring  $5000.  If 
money  is  worth  4  per  cent,  for 
how  many  years  must  the  pre- 


TEACHERS7   EDITION. 


499 


mium  be  paid  in  order  that  the 
company  may  sustain  no  loss  ? 


2J»=1  + 
(1.04)»=1  + 


r 

At 

T' 

6000X0.04 


160 


=  1  +  *^ 
T150 


n  log  1.04  =  log  7  —  log  3, 
=  log  7  -log  3 
71  log  1.04 

_  0.36798 
0.01703 
=  22  nearly. 

The  premium  must  be  paid  for 
22  years. 


10.  What  may  be  paid  for  bonds  due  in  10  years,  and  bearing 
semi-annual  coupons  of  4  per  cent  each,  in  order  to  realize  3  per 
cent  semi-annually,  if  money  is  worth  3  per  cent  semi-annually  ? 

ByS464,         PQ+^a+^W  +  V-1! 

P  =  9  +  r(l  +  g)»  —  r 
8  q  (1  +  x)» 

_  0.03  +  0.04  X  1.032Q  -  0.04 
0.03  X  (1.03)«> 
0.03  +  0.04  X  1.8063  -  0.04 
0.03  X  1.8063 
=  0.062262 
0.064189 
=  1.16,  nearly. 
The  price  paid  should  be  116,  nearly. 


11.  When  money  is  worth  2  per  cent  semi-annually,  if  bonds 
having  12  years  to  run  and  bearing  semi-annual  coupons  of  3i  per 
cent  each  are  bought  at  114|-,  what  per  cent  is  realized  on  the 
investment  ? 

1+x=  /ag  +  arq  +  qy— a-y 
i 


-( 


0.02  +  0.035  (1.022*  -  i)  v  A 
1.14126  0.02  / 


600  ALGEBRA. 


-( 


1     0.02  +  0.035  X  0.6084  \  A 
1.14125         0.02       / 


_  /  2.0647  \  A 

a.U125/ 
=  1.025. 
The  per  cent  realized  is  2£  semi-annually ;  that  is,  5  per  cent. 

12.  If  $126  is  paid  for  bonds  due  in  12  years  and  yielding  3j  per 
cent  semi-annually,  what  per  cent  is  realized  on  the  investment, 
provided  money  is  worth  2  per  cent  semi-annually  ? 


1+x=/^+5L(L+1)^15rxI 


/_1_  0.02  +  0.035  (1.022*  -  i)  v  A 
Vl.26  0.02  / 

=  /_1_  0.02  +  0.035  X  0.6084  \  A 
Vl.26  0.02  / 


/2.0647\A 
V   126  ) 
=  1.0207. 
The  per  cent  realized  is  4.2,  nearly. 


13.  A  person  borrows  $600.25.  How  much  must  he  pay  annually 
that  the  whole  debt  may  be  discharged  in  35  years,  allowing  simple 
interest  at  4  per  cent  ? 

Let  P  =  amount  of  debt. 

S  =  annual  payment. 

Then  PR,  =  §I^ZD. 

r 

P=  600.25, 

r=0.04, 
B  =  1.04, 

w=35, 

e      600.25  X  0.04  X  1.04*6 
o  —  — 


1.0485  -  1 

-  2.935"  32*3' 
He  must  pay  $32.30  per  year. 


teachers'  edition.  501 

14.  A  perpetual  annuity  of  $100  a  year  is  sold  for  $2600.  At 
what  rate  is  the  interest  reckoned  ? 

$100  is  the  interest  on  $2500  for  1  year.  Hence  the  rate  is  4 
per  cent. 

15.  A  perpetual  annuity  of  $320,  to  begin  10  years  hence,  is  to  be 
purchased.  If  interest  is  reckoned  at  3  J  per  cent,  what  should  be 
paid  for  it  ? 

Let  P  denote  the  amount  paid.  Then  at  the  end  of  10  years  P  is 
worth  P  x  1.03210  =  P  x  1.37026.  The  interest  on  this  amount  at 
3  J  per  cent  should  be  $320.     Hence 

P  X  1.37026  X  0.032  =  320 

10000 
1.37026 
=  7298. 

The  price  paid  should  be  $7298. 

16.  A  sum  of  $10,000  is  loaned  at  4  per  cent.  At  the  end  of  the 
first  year  a  payment  of  $400  is  made,  and  at  the  end  of  each  follow- 
ing year  a  payment  is  made  greater  by  30  per  cent  than  the  pre- 
ceding payment.    Find  in  how  many  years  the  debt  will  be  paid. 

The  amount  of  the  original  loan  at  the  end  of  n  years  is 

10,000  x  1.04". 
The  amount  of  the  several  payments  at  the  end  of  the  n  years  is 

400  X  1.04"-1  +  400  X  1.30  X  1.04»"2 

•f  400  x  1.302  X  1.04»-s  + 

rl.30> 


--""[(ST-'] 

1.30 
1.04 


0.26 
When  the  debt  is  paid, 

-"•-[(jS)'-'] 

10,000  X  1.04»  = jj^g < 


502  ALGEBRA. 

10000  X  0.26       /1.30> 


400 


ll.04j  +  400 

=  7.5, 


n  log  —  =  log  7.6, 


log  7.5 


log  1.30  -  log  1.04 
_  0.87506 

0.11394  -  0.01703 
=  9,  nearly. 
The  debt  will  be  paid  in  9  years. 

17.  A  man  with  a  capital  of  $100,000  spends  every  year  $9000. 
If  the  current  rate  of  interest  is  5  per  cent,  in  how  many  years  will 
he  be  ruined  ? 
The  amount  of  the  original  capital  at  the  end  of  n  years  is 

100.000  x  1.06". 
The  sums  spent  yearly  amount  at  the  end  of  the  n  years  to 

9000  X  1.06^-1  +  9000  X  1.05—2  +  9000  x  1.06*-«  + 

9000  X  LOfrg-ip-Ofr-*—  1). 

1.05-1-! 

When  the  money  is  all  spent, 

ioo.oooxi.os-^0000^106:^-1), 

—  0.06 

ioo,ooo  =  <)000<1-106"<)' 

0.00 
l-1.06-»  =  !» 

1.06-*  =  |> 

_  log  9  —  log  4 
n~      log  1.06 
=  0.3521 
0.0212 
=  17t  nearly. 
He  will  be  ruined  in  17  years. 


teachers'  edition. 


503 


18.  Find  the  amount  of  $365 
at  compound  interest  for  20  years 
at  5  per  cent. 

The  amount  is 

$365  X  1.05»  =  |969. 

19.  In  how  many  years  will 
§20  amount  to  $150  at  4  per  cent 
compound  interest  ? 

20  X  1.04*  =  150, 

log  150  —  log  20 
W~~         log  1.04 
_  2.17609  -1.30103 

0.01703 
=  51,  nearly. 
The  required  time  is  51  years. 

20.  At  what  rate  per  cent,  com- 
pound interest,  will  $2500  amount 
to  $3450  in  7  years  ? 

2500  i*7=  3450, 

7/3450 
/2600 


R  = 


=  V1.38 
=  1.047. 
The  rate  is  4^  per  cent. 

21.    If    the    population    of    a 
State  increases  in  10  years  from 
2,009,000  to  2,487,000,   find  the 
yearly  rate  of  increase. 
2,009,000  Jjw  =  2,487,000, 
>/2487 
/2009 
=  1.0216. 
The  annual  rate  of  increase  is 
2\  per  cent,  nearly. 


10/2 


22.  The  population  of  a  State 
now  is  1,918,600,  and  the  yearly 
rate  of  increase  is  2.38  per  cent. 
Determine  its  population  10  years 
hence. 

The  population  10  years  hence 
will  be 

1,918,600  X  1.0238W  =  2,428,000. 

23.  A  banker  borrows  a  sum 
of  money  at  3£  per  cent,  interest 
payable  annually,  and  loans  the 
same  at  5  per  cent,  interest  pay- 
able quarterly.  If  his  annual 
gain  is  $441,  determine  the  sum 
borrowed. 

If  the  sum  is  A,  it  will  amount 
in  one  year,  at  3£  per  cent 
annually,  to 

A  x  1.035. 

At  5  per  cent,  payable  quar- 
terly, it  will  amount  to 

A  x  1.0125*  =  AX  1.051. 


Hence 

A  (1.051  - 

1.035) = 

441, 

A  X  0.016  = 

441, 

A  = 

441 
0.016 

27,563. 

The  sum  borrowed  was  $27,563. 


504  ALGEBRA. 


Exercise   CXXV. 

1.   Find  continued  fractions  for  \tf;  ty;    V6;    VlT;  4V6; 
and  And  the  fifth  convergent  to  each. 

r\        !??  =       1       =         1 

157      l  +  £      1+      * 


123  21 

3  +  34 


1  +  3-v  1  +  1 


3  +  ^r  3-f1 


1  +  2  1+1 


21  1  +  i- 

r13 


1  +  1 


3  +  1 


1  +  1 


1+1 


1  +  1 


1  +  1 


1  +  1 


Fifth  convergent  =  \\. 


... ,        169      .  ,  18 
(«•)         4f=3  +  47 

=  3  +  -* 


-1 


2  +  1 


1  +  1 


1  +  1 


1  +  1 


-I 


Fifth  convergent  =  3^. 


teachers'  edition.  505 


(iii.)  Let 
then 

X 

1 

X_  V5-2 

=  Vs  +  2. 

Let 
then 

V6 

+  2  =  4  +  i> 

y 

1 

""V5-2 

=  x. 

Hence 

./7_9  ,       1 

vo  —  M            - 

4+     i 

4  +  4  +  etc. 

-.+*■ 

Fifth  convergent  =  2^*- 


(iv.)  Let 
then 

Let 
then 

Let 
then 


VH=3  +  i> 


X  = 

1 

Vii-3 

Vn  +  3 

2 

2 

+  3_ 

8  +  1. 

y 

y- 

2 

Vn-3 

Vn  +  3. 

Vn 

+  3  = 

.+*. 

z  = 

l 

Vll-3 

X. 

606  ALGEBRA. 


Hence  Vll  =  3  + 


3-h1- 


6  +  3  +  etc. 


-3  +  1      1- 


Fifth  convergent  =  3j^97. 
(v.)  Let  4V6=9  +  1 


4^6-9 

.4V6  +  9, 

15 


4V6  +  9,.,  ,  1 
**  — 15--1  +  y* 

15 


4V6-6 
2V6  +  3 
—2—' 


2V6  +  3_0  ,  1 
Let  ———g  +  j. 


.-.  z  = 


2V6-3 

4V6  +  6, 

15 


4V5  +  6,,   ,   1 

**        — is- -1+s* 

15 


4V6-9 
=  4V6  +  9 

=  18  +  i» 
x 

.    m     ~  .  i      !      1       1 
,.4V6  =  9  +  I+3  +  T+ig; 


Fifth  convergent  is  %°- 


teachers'  edition. 


507 


2.  Find  the  continued  fraction  for  $T;  J$J;  UHl  *!$£'>  and 
find  the  third  convergent  to  each. 


(i.) 


4T_ 
257' 


5  +  - 


2065. 
4626" 


(iii.) 


2  +  - 


2  +  - 


4  +  ^ 


7  +  ; 


Third  convergent  =  J j. 


6  + 


1 


8  + 


^  =  2  + 
204         ^ 


(u.) 
1 


Third  convergent  = 
2991 


10 


568 


=  6  + 


(iv.) 

1 


3  +  - 


4  + 


1 


1  + 


1 


6  +  ; 


3  + 


•+s 


Third  convergent  =  |JJ.  Third  convergent  =  6^. 

3.  Find  continued  fractions  for  V21;  V22;  V33;  V56. 

_  V21  +  3 


(i.) 


Let  V21  =  4  +  -» 

x 


then 

Let 
then 

Let 
then 


V21-4 


3 
3  u 


V21  +  4  ^en 


V21+4      ,   ,1 

K =   1  +  -' 

6  y 


V21-3 
V21  +  3 


y  = 


V21-1 
V21  +  1 


Let     ^±3=1  +  1, 
4  v 


then 


V21  +  1      ,   ,  1 

=  1  +  -» 

4  z 


V2T-I 
V21  +  1 


V21-3 


Let  M±i  =  1+i, 

5  w 


508  *  ALGEBRA. 

»=^f-7  Let     ^  =  2  +  i 


w     V21-4  ""        ¥~~"^V 

=  V21  +  4.  t.                                     3 

,  then                   v  =  — }= 

Let     V5l  +  4  =  8  +  l.  ^-2 

1  _  V22  +  2 

•  then      '             t  =  —= 6 

^-4  T4      V22  +  2      ,  .    1 

Hence  6                  w 

•  ft 

-v/oT  —  ii1      1     *      1      *      *      then  10  =  —7= 

V21-4+l  +  l  +  2+l  +  T+8"  ^22-4 

=  V22  +  4. 

(".)  _                    ! 

,—       j   ,  1  Let     V22  +  4  =  8  +  -> 

Let            V22  =  4  +  - »  * 

x  1 

1  then                   t  =  —= 

then                  x  =  —7= V22  —  4 

V22-4  =x 

=  V22  +  4  Hence 

r-                        6  /— r^.llllli 

Lrt      V=±i=1  +  3f  ^  =  4  +  l  +  2  +  4  +  2  +  l  +  8' 
6                   y 


6 


then                  y  =  —p= (iii.) 

V22-2  v     ' 

V22  +  2  Let             V33  =  5  +  -> 

=r  •  X 

3 

T          V22±_2__,l  then                   x  =  —rJ 

L<*   — 3 —  =  2  +  ;'  V33-5 

3  V33  +  5 

then                   z  =  — f= = « 

V22-4  8 

_V^2  +  4  Let     M±*=1  +  l, 

""2  8                     y 

T         V22  +  4_  A  ,  1  then                   y  =  -pJ 

Let    — 2 —     4+V  V33-3 

then                    «=_L^,  =  V33  +  3. 

V22  —  4  3 

V22  4-4  _    .      V33  +  3       .   ,   1 

=  — A Let      — A"  =  2  +  ^' 


509 


*  =  #i  -t    ^±I-.+i. 

8  V65  -  6 

^    ^3±3  =  1  +  I,  -V66  +  5 

8  u  6 

then  «=-7=K  Let    ^|±-5  =  2  +  i, 


V33-5  "»    — T 


z 


=  V33  +  6. 


then 


Let     V33  +  6=10  +  i.  ^~5 

«  _  V66  +  5 

then  o  =  -?^ ,—  6 

V38-6  Vtt_+5  =  1, 

=  x.  6  u 

Hence  0 

i      1      1       1  ^en  u  —  "7= 

=  V65  +  7. 

(iv.)  Let     V65+7  =  14  +  i* 

1  W 

Let  V66=7  +  i»  t-  1 

x  tnen  v  —  — = 

!  V65-7 

then  x  = 


VS5  -  7  Hence 

6  V0°      '^2  +  2  +  2+14 

4.  Obtain  convergents,  with  only  twojigures  in  the  denominator, 
that  approach  nearest  to  the  value  of  VlO;  Vl5;  Vl7;  Vl8;  V20. 


(i.) 

Let 

Vl0  =  3  +  i> 
X 

1 

ten 

~~  VlO-3 

=  Vio  +  3. 

Let 

Vl0  +  3  =  6  +  i» 

then  y  = 


VlO-3 


Hence       Vl0  =  3  +  i» 
o 

Convergents  =  },  ^,  J#, 

The  required  convergent  of  VlO 


510  ALGEBRA. 


(U)  Let      Vl8  +  4  =  8  +  *> 

Let  Vl6=3  +  i' 


z 


X 


then 


then 


1  Vl8-4 

Vl5-3  =  x. 

^15  + 3  Hence       Vl8  =  4+7 


6 


4  4-8 


w77,o  ,  Quotieats      =4,8,4,8, 

6        =  1  +  - '  Convergent*  =  fc  Jft  W» 

6  The  required  convergent  of  Vl8 


^  ^Vl5-3  l-W 

=  Vl6  +  3.  (iv<) 

Let     Vl6  +  3=6  +  i>  j 


2 
1 

then 


Let  V20  =  4  +  - 

x 


Vl5-3 

=  X. 


then 


V20-4 


Hence        Vl6  =  3  +  i      1.  _  V20  +  4, 

1  +  6  —        4 

Quotient*  =  1, 6,  1, 6, -v/on  +  4             1 

Convergents  =  f ,    f,    V.    ¥»  Let     -^j^  =  2  +  -> 

w.  -w> / 

The  required  convergent  of  Vl6  then                   y  =    r— 

is  W- 


(iii.) 


=  V20  +  4. 


Let 


Vl8  =  4  +  -»  Let      V20  +  4  =  8  +  i» 

x  z 


then  x  =  — p= then 


Vl8-4  V20-4 

_  Vl8  +  4  =*» 

2  / —  i     i 

_  Hence        V20  =  4  +  -,  5» 
_  ,     V18  +  4       .  ,  1  2  +  8 

Let     r =  4  +  - » 


then  2/  = 


2  y  Quotients      =  2,  8,  2,  8, 

_2  Convergents  =  f ,  f ,  ff ,  W> 

V 18  —  4  The  required  convergent  of  V20 

=  Vl8  +  4.  is  W-. 


teachers'  edition.  511 

5.  Find  the  proper  fraction  6.  Find  the  next  convergent 
which,  if  converted  into  a  con-  when  the  two  preceding  conver- 
tinued  fraction,  will  have  quo-  gents  are  ^  and  Jf ,  and  the  next 
tients  1,  7,  5,  2.  quotient  is  5. 

Quotients  =  1,  7,  5,  2.  Quotients  = ,5, 

Converges  =  h  h  *,  H)  »     jg^T"  =  '    **    * 

The  fraction  is  $$.  The  next  convergent  is  ¥9/*- 

7.  If  the  pound  troy  is  the  weight  of  22.8157  inches  of  water, 
and  the  pound  avoirdupois  of  27.7274  inches,  find  a  fraction  with 
denominator  <  100  which  shall  differ  from  their  ratio  by  <  0.0001. 

22.8157  =      1 
27.7274  1 


4  +  1 


1  +  1 


1  +  1 


1  +  1 


4  +  1 


1  +  1  +  etc. 
Quotients     =  1,  4,  1,  1,  1,  4,  1,  1, 

Convergent^  f,  {,  f,  J,  A,  ih  «»  it.  ltt> 

The  nearest  convergent  with  denominator  <  100  is  therefore  Jf . 
The  difference  between  it  and  the  actual  value  of  the  ratio  is 

^o^TTts0^0-0001- 

8.  The  ratio  of  the  diagonal  to  the  side  of  a  square  being  V2, 
find  a  fraction  with  a  denominator  <  100  which  shall  differ  from 
their  ratio  by  <  0.0001. 

Let  V2  =  l+i> 


then  x  = 


x 

1 


V2-1 
=  V2  +  1. 

Let  V2  +  l  =  2  +  i> 

y 


512  ALGEBRA. 

then  v  =  — := r-  1 

V2-1  Hence     V2  =  l+±- 

=  x.  2 

Quotients      =  2,  2,  2,  2,  2,  2, 

Convergent^  J,  f,  J,  H,  fi>  rh  !B. 

The  nearest  convergent  with  denominator  <  100  is  therefore  fg. 

The  difference  between  it  and  V2  is  <  .„  w  100  or  <  0.0001. 

70  X   lot* 

9.  The  ratio  of  the  circumference  of  a  circle  to  its  diameter  being 
3.14150265,  find  the  first  three  convergent^,  and  determine  to  how 
many  decimal  places  each  may  be  depended  upon  as  agreeing  with 
the  true  value. 

3.14150265  =  3  +  — Kr 
7  +  - 


15  +  1 


1  +  -*- 
T  288  +  etc. 

Quotients      =  7,  15,  1,  288, 

Convergent^  f,  ^,  flfc,  fff.  "WW. 

The  first  convergent,  ^ ,  differs  from  the  true  value  by  <  7 
or  <  0.01;  the  second,  f  Jf,  by  <  1Q6  *  ng  or  <  0.0001 ;  the  third, 

tft.  ^  <113x132657  °r  <  0-000001- 

10.  Two  scales  whose  zero-points  coincide  have  the  distance 
between  consecutive  divisions  of  the  one  to  those  of  the  other  as 
1 : 1.06677.    Find  what  division  points  most  nearly  coincide. 

1  + 


15  +  1 


4  +  1 


1  +  1 


8  +  1 


11  +  1 


2  +  1 


1  +  1 


-I 


teachers'  edition.  513 

Quotients      =  1,  15,  4,  1,  8,  11,  2, 

Convergent^  J,  },  ft,  ft,  Jf,  ftf,  ft|J, 

Hence  the  15th,  61st,  76th, divisions  of  the  first  scale  nearly 

coincide  with  the  16th,  65th,  81st,  ....  divisions  of  the  other,  the 
coincidence  becoming  closer  as  the  series  proceeds. 

11.  Find  the  surd  values  of 

o+i  !.  i  1  !.  1+i  i  i. 

M  +  6'    3+1  +  6'       ^2+3+4 

i       i  27x  +  4x2=7  +  x, 

(L)  x~l  +  6  4x2+.26x=7, 

1  -  13  +  Vl97 


1  + 


6  +  x 


4 
ill 


__6  +  x  (iii.\          x  = 

x2  +  6x  =  6,  =      1 

x2  +  6x+9=15,  2  +  i 

x  +  3=±Vl6,  __                                               3  +  _„ 


x=-3±Vl6.  4  +  x 

Hence  the  required  value  is  = 1 

3-3±Vl5  =  Vl5.  2  +  jJf^ 


1      1      i  =13  +  3x, 

3+1  +  6  30+7x 

x  30x+7x2=13  +  3x, 

:         i~  7x2  +  27x=13, 
3  +  ^ 


1  Q»     L     Vl093 

T6  +  x  T¥         14 

.         1  The  required  value  is 

"o,6  +  x  VJ093 

3+7  +  x  ^tt  +  ^M" 

7  +  x  VJ093-13 

'27  +  4x'  *          14 
The  surd  values  are 

A/Tr    -13  +  VJ97         ,    V1093-13 
V15; ;  and   - 


514  ALGEBRA. 


12.  Show  that  the  ratio  of  the  diagonal  of  a  cube  to  its  edge  may 
be  nearly  expressed  by  97  :  56.  Find  the  limit  of  the  error  made  in 
taking  this  ratio  for  the  true  value. 

The  true  ratio  is  V3. 

By§403,  V§=1  +  I+|" 

Quotients      =  1,  2,  1,  2,  1,  2,  1,  2, 

Convergents=  f,  |,  J,  J,  A,  H»  If.  tt.  iii- 

The  7th  convergent  is  If}  or  §J,  and  it  differs  from  V3  by 
<56xl63°r<^nr' 

13.  Find  a  series  of  fractions  converging  to  the  ratio  of  5  hours 
48  minutes  51  seconds  to  24  hours, 

6  hrs.  48  min.  51  sec.  =  20,931  sec. 
24  hrs.  =  86,400  sec. 

20931  =      1 

86400      A  ,  1 
4  + 


7  +  i 


1  +  i 


4  +  i 


1  +  i 


1  +  1 


1  +  i 


1  +  i 


3+i 


2  +  i 


*  +  l 


Quotients      =  4,  7,  1,  4,  1,  1,  1,  1,  3,  2,  1,  2. 

Convergent*  =  f,  \,  &,  &,  #T,  ^,  f&,  iff,  ftf,  #&,  Hff, 


TEACHEKS'    EDITION.  515 

14.  Find  a  series  of  fractions  converging  to  the  ratio  of  a  cubic 
yard  to  a  cubic  meter,  if  1  cubic  yard  =  0.76453  of  a  cubic  meter. 

0.76453  =      X 

'     1+— T 

3+—T 

4+— T 
19  +  — j- 

1+— 

Quotients      =  1,  3,  4,  19,  2,  3,  1,  1,  5,  1,  2. 

Convergents=J,  {,   *,   ^  fWf  m   im,  m  m   }m 

Exercise    CXXVI. 
1.  Expand  - — z—  to  four  terms  in  ascending  powers  of  x. 

2  —  oX 

Let         — ^-r-  =  A  +  Bx+Cx*  +  Dx*  + , 

2  — 3x 

tken                     1  =  24  +  (2B-SA)x  +  (2C-35)x2 
+  (2D-3C)x3+ 

.•.24  =  1,  4  =  *, 

25-3.4  =  0,  B=& 

2C-3£  =  0,  C=f, 

2D-3C=0,  D=ih 

Hence,  3^  =  i+ **+ f  *2  +  Bx*  + 


1  +  x 
2.  Expand  to  four  terms  in  ascending  powers  of  x. 

Let        -y^"  =  A  +  Bx+Cx*  +Dx*  + 

2  +  3x 


516  ALGEBRA. 

then  l  +  x=2^4  +  (2B  +  ZA)z  +  (2C  +  3B)x2 

+  (2D  +  3C)xH 

.-.2.4  =  1,  A  =  ±, 

2B+SA  =  1,  J5=-i, 

2C  +  3£  =  0,  C=J, 

2Z>  +  3C=0  D=-^, 


Hence'       ^p^=i-ia;  +  i^-A^  + 


3.  Expand  '  __       to  four  terms  in  ascending  powers  of  z. 

I**         Sa  ~~  I X  =  ^  +  Bx  +  Cz*  +  Dx«  + 

4  — 3x 


then  3-2x  =  4.4  +  (45-3.4)x 

+  (4C-3£)x2  +  (4D-3C)x»  + 

.-.4.4  =  3,  -4  =  f, 

4J5-3^1  =  -2,        B=T\, 
4C-3£=0,  C=&> 

4D-3C=0,  D=jh> 


Hence,         f^=  !  + A*+ &*2  + rf**8  + 


1 $ 

4.  Expand  ~ — ^— ^  to  four  terms  in  ascending  powers  of  x. 

Let        ,  1"~*    ,  =  4  +  Bz  +  Cx2  +  Dx*  + 

1  —  X  +  X2 


then          l-x  =  .A  +  (£-4)x  +  (C-]3  +  4)x2 
+  (D-C+£)x8+ 

.-.-4  =  1,  -4  =  1, 

J5T  — 4  =  -lf  5=0, 

C-.B  +  ^  =  0,  C=-l, 

D-C+5  =  0,  D=-lf 


Hence -£- ;=1  —  x2  —  x»  + 

1  —  x  +  x2 


teachers'  edition.  517 


5.  Expand  —        2  to  four  terms  in  ascending  powers  of  x. 

***        ; — o  1-lq  2  =  A  +  ^x+  Cx2  +  Bx  + 

1  —  2x+  3x2 

then  l  =  A  +  (B-2A)x  +  (C-2B+ZA)x* 

+  (D-2C+32*)x8  + 

.\A  =  1,  A=l, 

B-2A  =  0y  B  =  2, 

C-2B  +  3A  =  0,  C  =  l, 

D-2C  +  3B  =  0,  D=-4, 


Hence        . — —   ,  0  , =  1  +  2x  +  x2  —  4x8- 

1  —  2  x  +  3  x2 

5  —  2x 
6,  Expand      ,  q — 3-^  to  four  terms  in  ascending  powers  of  x. 


L^t       ,   f  o  2X  o  =  ^  +  Bx  +  0x2  +  i)X8  + 

1  +  3x  — x2 

then  6-2x  =  ^+(B  +  34)x+(C  +  3J3-4)x2 

+  (D+3C-£)x8+ 

.:A  =  6,  A  =  5, 

£  +  34  =  -2,        £=-17, 

C  +  SB-A  =  0,  C=56, 

2)  +  3C-J?=0,  D=-185, 


Hence        tJ*Q  2g  .=  5-17x+  56x2-  185x»+ 

l  +  3x  — x2 

4  /j. 6  x2 

7.  Expand    __   —        2  to  four  terms  in  ascending  powers  of  x. 

***  ,  4o""?f  »  =  4  +  £s+Cx2  +  flx»  + 

1  —  2  x  +  3  x2 

then  4x-0x2  =  -4  +  (B-2A)x+  (C-  2J5+  3-4) x2 

+  (D-2C  +  35)x8+ 

.-.4  =  0,  -4  =  0, 

5-2.4  =  4,  B=4, 

C-2JB  +  34  =  -6,  C=2, 

2)-2C+3J3=0,  D=-8, 


HenCe  1lX2g+X3x2  =  4g  +  2x2"8g8- 


518  ALGEBRA. 


8.  Revert  the  series  y  =  x  +  z2  +  xfi  + 

Le€  x  =  Ay  +  By*+  Cy*  +  Dy*  + 

then  y  =  Ay  +  By2  +  Cy*  +  Dy*  + 

+  -/tV*+  2ABy*+  (B2  +  24C)y«  +  - 

+  A*y*  +  2A*By*+ 

+  4*y*  + 

.:A  =  1,        A  =  1, 
B  +  A2=0,         J5=  -1, 
C+2AB+A*  =  0,        C=l, 
D+£*  +  2.AC+342JB+4*  =  0,        D=-l, 


Hence  x  =  y  — ^H-y8  — y4-}- 

9.  Revert  the  series  y  =  x  —  2x"  +  3x*  — 

Let  x  =  ^ly  +  By«H-Cy»+Dy*H- 

then  y  =  Ay  +  J5y2  +  Cy'+Dy4^- 

-  24*y2 - 4ABy» -  (2 IP  +  4  AC) y*  + 

+  3-/t*y»  +  942By*  + 

-4^V  + 

.-.4  =  1,        4=1, 
J3-242  =  0,        J5=2, 
C-44J3  +  34*  =  0,        C=5, 
D-2£*-44C+942J5-44*  =  0,        2)  =14, 

Hence 


Let 
then 


Hence 


x  =  y  +  2y*+6y»+14y*  + 

jvert  the  series  y  =  x  —  J x*-h  Jx6—  |x7+ 

x  =  4y  +  J3y2  +  Cy3  +  Dy<+.Ey*  + 

y  =  Ay+Btf+Cy*  +  I>y*  +  Ey*  + 

-\A*y*- 

-^25y4-^J52y6- 

+  i^V+  

.-.4  =  1, 

4  =  1, 

JB=0, 

£=0, 

~      -4s 

c-T  =  o, 

c=h 

D-42J3  =  0, 

D=0, 

^-^452  +  ^^  =  0, 

J?=-J. 

&  =  y  +  Jy8  -12^+  - 

teachers'  edition.  519 


X2  X8 

11.  Revert  the  series  y  =  x  +  ^-^  +  ^  x  2x  g  + 

Let  x  =  4y  +  By*+Cy*  +  Dy*  + 

then  y  =  4y  +  By2  +  Cy8  +  ify4  + 

+%iA*p+ABp  +  ($B2  +  AC)y*+  •• 
+  J^V8  +  -i^2By4+ 

+  A-^V+ 

.-.  il  =  1,        A  =  1, 

C  +  AB+iA*  =  0,         C=i, 
D+iB*  +  AC+$A*B+v\A*  =  0,        D=-}, 


Hence  x  =  y  — Jy2+ Jy8  — J^+ 

12.  Find  the  fractions  in  the  form  — ; ; — ;  whose  expansions 

,        . .  p-r  qx-r  rx* 

produce  the  series : 

l  +  3x+2x2-x8- 

3  +  2x  +  3x2  +  7x8  + 

*-A*  +  H*-Hf*  + 

(*•)        l*hl    a=l  +  3g  +  2x*-x8- 

v        p  +  Q'x  +  rx2 

a  +  6x  =  j>+  (3p  +  g)x  +  (2p  +  3  g  +  r)  X* 

+  (-p  +  2g  +  3r)x8  + 

.\a  =  p,  6  =  3p  +  g, 

0=2p  +  3g  +  r,         0=-p  +  2g  +  3r. 

Eliminating  r  from  the  last  two  equations, 
0=7p+7g. 

r  =  P, 

o  =  Pi 
b=2p. 
Hence  a  +  6x     =     p  +  2px 

p+gx  +  r     p— px  +  px2 
=     l  +  2x 

1-X+X2' 


J 


520  ALGEBRA. 


<**•)         fl  +  ^       =  3  +  2x+3x2  +  7x«  + 

*    '  p  +  qx  +  rx* 


a  +  bz  =  Zp  +  (2p  +  3q)z  +  (3p  +  2  g  +  3r)x2 

+  (7p  +  3g  +  2r)x»  + 

.-.  a=3p,  .&  =  2p  +  3g, 

0  =  3p  +  2g  +  3r,  0=7p  +  3g  +  2r. 

Eliminating  r  from  the  last  two  equations, 

0=  —  \bp  —  5g. 

.-.g=-3p, 

r  =  l>, 

a  =  3p, 

6=-7p. 

_  o  +  te  3p  — 7imc 

Hence  — : — z  =  — ^ f — i 

p  +  gx  +  rx2     j>  —  3  jix  +  px* 

3-7x 

l-3x  +  x«' 


<tt)y+fl*y^-A«  +  H*-Ht*  + 

a  +  te=  }P  +  (~A*  +  }«)* 
+  (HP-A«+fr)«P 
+  (-«iP  +  «9-Ar)x» 
+ 

o  =  Hp-A«  +  fr,        0=-HtJ»  +  ttff-A'- 
Eliminating  r  from  the  last  two  equations, 
0  =  A%l>  +  fi9. 

r=-fp, 
a  =  }p, 
6  =  0. 

Hence  <*  +  ***      - iP 

p  +  qz  -f  rx2     p  +  Jpx  —  Jpx* 
_  3 

4  +  x— 5x2' 


teachers'  edition.  521 


7x  +  1 

13.  Resolve  - — j-tt- ^  **>**>  partial  fractions. 

(x  +  4)(x  — 5)  ' 

it                     7g  +  l               A            B 
1               (x+4)(x-5)      x+4      x-5 
then                             7x  +  1  =  A  (x-  5)  +  B(x  +  4) 
=  (,4  +  B)x-5A  +  4B. 
.•.^i  +  B=7, 
-5A  +  4B=1, 
4  =  3, 
5=4. 
7x  +  l        _     3,4 
Hence  - — .    .v  . rr  =  — r~7  + 


(x  +  4)(x-5)      x  +  4      x-5 

14.  Resolve  - — r~zn — ttt  ^°  partial  fractions, 
(x  +  3)  (x  +  4) 

'  6  A  B 

61  (x  +  3)(xX4)      x  +  3     x  +  4' 

then  6  =  A(x  +  4)  +  B(x  +  3) 

=  (4  +  B)x  +  4A  +  3B. 
.-.  A  +  B=0, 
4A  +  3B  =  6, 
4  =  6, 
B=-6. 
6  6  6 


Hence 


(x  +  3)(x  +  4)      x  +  3      x  +  4 


5X \ 

15.  Resolve  — — into  partial  fractions. 

(2  X  —  1)  (X  —  O) 


Let 


5x-l  A  B 


(2x-l)(x-5)      2x-l      x-5 
then  5x-l  =  4(x-5)  +  B(2x-l) 

=  (4  +  2B)x-(5A  +  B). 
.-.  4  +  2B=5, 
54  +  B=l, 

A=-h 
B=f. 
5x-l  =     — »      |      t 

(2x-l)(x-6)      2x-l      x-5 

__        8 1 

3  (x-5)      3(2x-l)' 


Hence 


522  ALGEBRA. 


x  — 2 
16.  Resolve    t  __  g    _  1Q  into  partial  fractions. 


Let 


x-2  A      ,      B 


xa  — 3x— 10      x  — 6      x  +  2 
then  x  —  2=  4  (x  +  2)  +  JS(x  —  5) 

=  (A  +  B)x+2A-bB. 
.:A  +  B=\, 
2A-6£=-2, 

*=f 

x*-3x-10     x-6      x+2 

=        3        +  — * 

7  (x  -  5)  T  7  (x  +  2) 

17.  Resolve    ,_.  into  partial  fractions. 
3  A     ,     Bx+C 

Let  -r r-  = r  +  " 


x»-l      x-1      x2  +  x+l 
then  3  =  A  (x2  +  x  +  1)  +  (Bx  +  C)  (x  -  1) 

=  (4  +  £)x2  +  (A  -B+C)z  +  A-  C. 
..A  +  B  =  0,  A  =  l, 

A-B+C  =  0y  B=-l, 

^1-C=3,  C=-2. 

3-4  =  3, 

3     __     1  x  +  2 


Hence 


x8— 1     x— 1     x2+x+l 


x9  —  x  —  3 
18.  Resolve  —r-z — —  into  partial  fractions, 
x  (x^  —  4) 

T«t       x2-x-3_;l         B  C 

1M        x(x2-4)  ~  x      x  +  2i"x-2' 

then         x2-x-3  =  4(x2-4)  +  £x(x-2)  +  Cx(x+2) 

=  (J4  +  B+C7)x2+(2C-2JB)x-4^l. 

.-.4  +  B+C=l,  4=|, 

2B-2C=1,  B=i, 

44  =  3,  C=-f 

x*-x-3       3,3  1 

Hence  — — — jr  =  - — h  - 


x(x2-4)       4x      8(x  +  2)      8(x-2) 


teachers'  edition.  523 


19.  Resolve  — — r-^r  into  partial  fractions. 
x2  (x  +  5) 

3x2-4       A  ,  B  ,      n 

Let  -r~. — r-=r  =  —  +  ~Z  +  ' 


X2(X  +  5)        X        X2       X+5 

then  3x2-4  =  4x(x+6)  +  £(x+6)  +  Cx2 

=  (4+C)x2  +  (64  +  B)x  +  6B. 
.-.4  +  0=8,  B=|, 

&A  +  B=0,  -A  =  A. 

5D=-4,  C=fr 

3x2-4         4  4     ,         71 

Hence  -z-t — r-77  =  ^ t~z  + ; 


x2(x  +  6) 
7x*-x 

(X 

-l)2(x  +  2) 
7X2  — x 

26x      6x2  '  25(x  +  6) 


20.  Resolve  ^^^a/^o-ox  ^^  P*1^  fractions. 

.^     ~  A  B  C 

^t  (x-l)2(x+2)      x-1  ",'(x-l)2't"x+2, 

then  7x2-x  =  4(x-l)(x  +  2)  +  B(x  +  2) 

+  C(x-1)2 
=  (i+Cf)xa+(i  +  B-2C)x 
-24  +  2B+C. 
.-.4  +  0=7,  5=2, 

A  +  B-2C=-1,  A  =  ±£, 

-2A  +  2B+C  =  0,  C=^. 

35  =  6, 

7x2-x                 11        ,        2  10 

Hence zrr-t — r^7  =  tt; t:  +  7 TT*  +  ; 


(x-l)2(x  +  2)      3(x-l)      (x-1)2     3(x  +  2) 


O  a»2 7  X  +  1 

21.  Resolve ^^r into  partial  fractions. 

T  «.      2x*-7x+l        A      ,     Bx+C 
Let     rn =  — rr  +  " 


x8+l  x+1     x2-x+l 

then       2x2-  7x  +  1  =  A  (x2-  x  +  1)  +  (Bx  +  Cj  (x  +  1) 

=  (4  +  B)x2  +  (-  -4  +  B  +  C)x  +  -4  +  C. 
.-.4  +  B=2,  il  =  V, 

-4  +  B  +  C=-7,  B=-|, 

4+C=l,  C=-J. 

2x2-7x+l  10  4x+7 


Hence 


x»+l  3(x+l)     3(x2-x  +  l) 


524  ALGEBRA. 


Exercise  CXXVII. 

1.  Find  the  fiftieth  term  of  1,  3,  8,  20,  43, 

Series       =1           3  8           20           43 

lstdiff.     =           2  6           12            23 

2d  diff.     =                 3  7           11 

3d  diff.     =  4           4 

4th  diff.     =  0 

.\  a  =  1,  a\  =  2,  a8  =  3,  a*  =  4,   a*  =  0. 

Substituting  in  formula,  we  have : 

40  x  4ft 
Fiftieth  term  =  l  +  49x2+gx3 

49  X  48  X  47      . 
+    1X2X3        * 
=  1  +  98  +  3628  +  73696 
=  77323. 

2.  Find  the  sum  of  the  series  4,  12,  29,  55, to  20  terms. 

Series       =4  12  29  55 

lstdiff.     =  8  17  26 

2d  diff.     =  9  9 

3d  diff.     =  0 

.-.  a  =  4,  di  =  8,  da  =  9,   as  =  0. 
Sum  to  20  terms 

=  20(4+76+613} 
=  11860. 

3.  Find  the  twelfth  term  of  4,  11,  28,  55,  92, 

Series       =4  11  28  55  92 

lstdiff.     =  7  17  27  37 

2d  diff.     =  10  10  10 

3d  diff.     =  0  0 

.-.  o  =  4,  a\  =  7,  02  =  10,  a8  =  0. 

Twelfth  term  =  4  +  11x7  +  U  *  10  x  10 

=  4  +  77  +  660 
=  631, 


teachers'  edition.  525 

4.  Find  the  sum  of  the  series  43,  27,  14,  4,  —  3, to  12  terms. 

Series       =43  27  14  4  -3 

lstdiff.     =  -16        -13        -10       -7 

2d  diff.     =  3  3  3 

3d  diff.     =  0  0 

.-.  a  =  43,   a\  =  —  16,   a2  =  3,  a8  =  0* 
Sam  to  12  terms 

=  12J43-¥X16  +  I!I^X3J 

=  12(43-88  +  66) 
=  120. 


5.  Find  the  seventh  term  of  1,  1.236,  1.471,  1.708, 

Series       =     1  1.236  1.471  1.708 

lstdiff.     =       0.236  0.236  0.237 

2d  diff.     =  0.001  0.001 

3d  diff.     =  0 

.-.  a=l,   ai  =  0.236,   aa  =  0.001,  a8  =  0. 

Seventh  term  =  1  +  6  x  0.236  +  ^-^  x  0.001 

1  X   L 

=  1  +  1.410  +  0.016 
=  2.426. 


6.  Find  the  sum  of  the  series  70,  66,  62.3,  68.0, to  16  terms. 

Series       =     70  66  62.3  68.9 

lstdiff.     =  -4        -3.7  -3.4 

2d  diff.     =  0.3  0.3 

3d  diff.     =  8 

.-.a  =70,  ai=— 4,  a*  =  0.3,  a8  =  0. 
Sam  to  16  terms 

=  16J70-¥X4  +  rxW3-x0.3} 
=  15{70-28 +  9.1} 
=  766.5. 


526  ALGEBRA. 

7.  Find  the  eleventh  term  of  343,  337,  326,  310, 

Series       =     343  337  326  310 

lstdiff.     =  -6  -11  -16 

2d  diff.     =  -6  -6 

3d  diff.     =  0 

.-.  a  =  343,  «i  =  —  6,  02  =  —  5,  a*  =  0. 

10  X  9 
Eleventh  term  =  343  —  10  X  6  -  1  v  0  X  6 

1  x  & 

=  343 -.60-225 
=  58. 


&  Find  the  sum  of  the  series  7  x  13,  6  x  11,  5  x  9, to  9 

terms. 

Series       =91  66  45  28 

lstdiff.     =  -25        -21        -17 

2d  diff.     =  4  4 

3d  diff.     =  0 

.*.  a=  91,  ai  =  —  25,  a%  =  4,  as  =  0. 

Sum  to  9  terms       =  9  j  91  -  f  X  25  +     ^g  X  4  j 

=  9(91-100  +  4*) 
=  265. 

9.  Find  the  sum  of  n  terms  of  the  series  3x8,  6  X  11,  9  X  14, 

12  x  17, 

Series       =24  66  126  204 

lstdiff.     =  42  60  78 

2d  diff.     =  18  18 

3d  diff.     =  0 

.\  a  =  24,  ai  =  42,  a*  =  18,  a8  =  0. 
Sum  to  n  terms 

=  n{24  +  21n-21  +  3n*-9n  +  6} 
=  3n8+12n2  +  9n 
=  3n(n+  l)(n  +  3) 


teachers'  edition.  527 


10.  Find  the  sum  to  n  terms  of  the  series  1,  6,  15,  28,  45, 
Series       =1  6  16  28  45 

lstdiff.     =  69  13  17 

2d  diff.     =  4  4  4 

3d  diff.     =  0  0 

.-.  a  =  1,  a\  =  6,   02  =  4,   a8  =  0. 
Sam  to  n  terms 


-•('♦srJ«+fcri$rH 

=  n(l  +  fn-f  +  fn2-2n  +  !) 
=  n(fn2+in-i) 

=  £(4n2  +  3n-l) 
o   • 

=  £(4n-l)(n  +  l). 

Exercise   CXXVIII. 

1.  Determine  the  number  of  shot  in  the  side  of  the  base  of  a 
triangular  pile  which  contains  286  shot. 

Let  n  denote  the  required  number ; 
then  n(n  +  l)(n  +  2)=286t 

n(n  +  l)(n  +  2)  =  1716 

=  11  x  12  X  13. 
.-.  n=ll. 

2.  The  number  of  shot  in  the  upper  course  of  a  square  pile  is  169, 
and  in  the  lowest  course  1089.    How  many  shot  are  there  in  the  pile  ? 

In  the  complete  pile,  n  =  33. 

Number  of  shot  in  complete  pile, 
33X34X67 
1X2X3    -12'629' 
In  the  part  of  the  pile  that  is  lacking, 

n=12. 
Number  of  shot  in  part  lacking, 
12  x  13  x  26  _ 

1X2X3    -650' 
12,529  -  660  =  11,879. 
There  are  11,879  shot  in  the  pile. 


528  ALGEBRA. 

3.  Find  the  number  of  shot  in  a  rectangular  pile  having  17  shot 
in  one  side  of  the  base  and  42  in  the  other. 

The  required  number  is 

|(n+l)(3n'-n+l) 

=  ^(17  +  1)  (126-  17  +  1) 
=  6610. 


4.  Find  the  number  of  shot  in  five  courses  of  an  incomplete  tri- 
angular pile  which  has  15  in  one  side  of  the  base. 

Number  in  complete  pile, 

15X16X17  • 

1X2X3    "~w- 
Number  in  part  lacking, 

•  10X11X12_^ 

1X2X3    ~m 

680-220  =  460. 

The  number  of  shot  in  the  pile  is  460. 

5.  The  number  of  shot  in  a  triangular  pile  is  to  the  number  in  a 
square  pile,  of  the  same  number  of  courses,  as  22  :  41.  Find  the 
number  of  shot  in  each  pile. 

The  number  of  courses  is  in  both  cases  the  same  as  the  number  of 
shot  in  a  side  of  the  base.     Hence 

n(n+l)(n  +  2)    n(n+  1)  (2n+  1) 

1X2X3        '         1X2X3  "  -  «* 

n+2  :2n+l=22  :  41 
w  =  20. 
#  w(n+  l)(n+2)_20X  21  X  22 
1X2X3  1X2X3 

=  1640. 
n(n+  1)  (2n+  1)  =  20  X  21  X  41 
1X2X3  1X2X3 

=  2870. 
The  number  of  shot  in  the  triangular  pile  is  1540 ;  in  the  square 
pile,  2870. 


529 

6.  Find  the  number  of  shot  required  to  complete  a  rectangular  pile 
having  15  and  6  shot,  respectively,  in  the  sides  of  the  upper  course. 

In  the  first  missing  course  n  =  5  and  n'  —  14.  Hence  the  number 
of  shot  required  is      j(5+ i}  (42 -6  +  1)  =  190. 

7.  How  many  shot  must  there  be  in  the  lowest  course  of  a  tri- 
angular pile,  so  that  10  courses  of  the  pile,  beginning  at  the  base, 
may  contain  37,020  shot  ? 

Number  of  shot  in  the  complete  pile  is 
n(n  +  l)(n  +  2) 
6 
Number  in  lacking  part  is 

(n  -  10)  (n  -  9)  (n  -  8) 

Hence     *<"+  ^n  +  2>  -  <»"  10>  <"  ^  ^V  =  37,020, 

n  (n  +  1)  (n  +  2)  -  (n  -  10)  (n  -  9)  (n  -  8)  =  222,120, 
n*  +  3  n2  +  2  n  -  n«  +  27  n2  -  242  n  +  720  =  222,120, 
30n2-240n=  221,400, 
na-8n  =  7380, 
n  =  90. 
Therefore,  the  number  of  shot  in  a  side  of  the  lowest  course  is  90, 
and  the  number  in  the  lowest  course  is 

1  +  2  +  3  +  +  90  =  46  X  91  =  4096. 

8.  Find  the  number  of  shot  in  a  complete  rectangular  pile  of  15 
courses,  which  has  20  shot  in  the  longest  side  of  its  base. 

Here,  n  =  15,  n'  =  20  ;  and  the  required  number  is 
if-  X  16  X  (60  -  15  +  1)  =  1840. 

9.  Find  the  number  of  shot  in  the  bottom  row  of  a  square  pile 
which  contains  2600  more  shot  than  a  triangular  pile  of  the  same 
number  of  courses. 

n(n+l)(2n+l)      n(n+l)(n  +  2)_ 

1X2X3  1X2X3       -^ow» 

n(n+l){2n+l-(n  +  2)}  =  2C00< 

»(n+l)(n-l)  =  16,600, 
(n-l)n(w+l)  =  24x  26  X  26, 
n  =  25. 
The  required  number  is  25. 


530  ALGEBRA. 


10.  Find  the  number  of  shot  in  a  complete  square  pile  in  which 
the  number  of  shot  in  the  base  and  the  number  in  the  fifth  course 
above  differ  by  225. 

n*-(n-  5)2  =  225, 
10n-25  =  225, 
n=25, 
n  (n  +  1)  (2  n  +  1 )  =  25  X  26  X  51 
1X2X3  1X2X3 

=  5525. 
The  required  number  is  5525. 

1L  Find  the  number  of  shot  in  a  rectangular  pile  which  has  600 
in  the  lowest  course  and  11  in  the  top  row. 

The  difference  between  the  number  of  shot  in  the  length  of  any 
course  and  the  number  in  the  width  is  the  same  for  all  courses. 
Hence,  ri  —  n  =  11  —  1 

=  10, 
nn'  =  600. 
.-.  n'  =  30, 
n  =  20, 
£  (n  +  1)  (3n' -  n+  1)  =  \°  X  21  x  71 
=  4970. 
The  required  number  is  4970. 

Exercise    CXXIX. 

1X4    2X5 


1.  Sum  to  n  terms,  and  to  infinity,  the  series 
1 


3X6 


r+;  l     •     l 


1X42X63X6  n  (n  +  3) 

=  i/i_l\  +  l/l_l\  + 1/1 L_\ 

8\1      4/^3V2      5/T         3\n     n  +  3/ 

=  1/1  +  1  +  1 1 1 1_\ 

8U      2      8      n+1      n  +  2      n  +  3/ 

■.1/11  8n»+12n+ll       \ 

3V6       (n+l)(n+2)(n  +  3)>J 

lln8  +  48n2  +  49n 
18(n+l)(n  +  2)(n  +  3) 
=  sum  to  n  terms. 
The  sum  to  infinity  =  i  (t  +  i  +  *) 


teachers'  edition.  531 


2.  Sum  ton 

1 

terms,  a 

term  is 
1 

md  to  infinity,  the  serie 

1 

1 

1 

S  1X3X5' 

C 

+  4' 

4)  +  Bn(nH 

2X4X6' 

3X5X71 
The  general 

n(n+2)(n+4) 
~A  \      B      \ 

iMlt                    . 

7i  (n 
then 

+  2)(n 

+  4)      n   '  n  +  2  '  n 
l=4(n  +  2)(n  + 

M) 

+  Cn(n+2) 
=  (4  +  B+C)n2+  (64  +  4B  +  2  C)n 
+  84. 
/.i  +  5+C  =  0,  4  =  4, 

64  +  4B+2C  =  0,  ^=~i, 

84  =  1,  C=J. 

XT  1  1  1  1 

Hence   — : — r-zrz — ttt  =  s tt — r-^:  + 


n(n  +  2)(n  +  4)      8n     4(n+2)      8(n  +  4) 
=  1/1__     2 


Hence 


8vn      n  +  2 
1.1..  1 


n4-4/ 


1X3X52X4X6  n  (n  +  2)  (n  +  4) 

=   lr/l_2      IX      /1_2      lx      

8|Al      3^5>/TV2      4T6r 

+     (I 2-+-M1 

T  An      »  +  2      n  +  4/J 

=      1(1  +  1  +  1  + +  i\ 

8\V2^3^  n/ 

~~4V3  +  4  +  5  + +  nT2/ 

+  1/1  +  1+1  + +  _L_\ 

=      l<l  +  i\-l/i  +  i\-l/-J^+_L\ 
8U      2/      8V3^4/      8Vn+l      n  +  2/ 

8Vn+3Tn  +  4/ 

_      11      1/       2n+3 2n+7       \ 

96      8\.(n+l)(n  +  2)      (n  +  3)(n+4)>> 


11  2n2+10n+ll 


96      4  (n  +  1)  (n  +  2)  (n  +  3)  (n  +  4) 
=  sum  to  n  terms. 
Sum  to  infinity         =  i  (i  +  i)  ~  i  (i  +  i) 
=  ii 


532  ALGEBRA. 


3.  Sum  to  n  terms,  and  to  infinity,  the  series  g>  .  x         , 


0  X  8  X  10 
The  general  term  is ;  — 


2n(2n  +  2)(2n  +  4)      8n(n+l)(n  +  2) 


Let      — - — 7-rrz — r-^7  =  —  +  - 


n  (n  +  1)  (n  +  2)      n      n  +  1      n  +  2 
then  l  =  4(n  +  l)(n  +  2)+Bn(n  +  2)+Cn(>i+l) 

=  (A+B  +  C)n*+(SA+2B  +  C)n  +  2A. 
.•.i  +  B+C  =  0,  4  =  *, 

34  +  2B+C  =  0,  B=— lf 

24  =  1,  C=*. 

XT  1  1  1        ,  1 

Hence  — : — r~rrz — r^r  =  « rr  +  : 


n  (n  +  1)  (n  +  2)      2n      n  +  1      2  (»  +  2) 

=  1  /I 2  1     \. 

2  \n      n+1      n  +  2/ 

and        1 =  JL/1 2_  +  _L\. 

8n(n+l)(n  +  2)      16\n      n+1      n  +  2/ 

XT                     1,1,  ,  1 

Hence    ftw,v><0+,v>/,w0  + + 


X4X64X6X8  2n(2n  +  2)(2n+4) 

-s[(i-:+i)+(S-i+i)+ 

\n     n+1      n  +  2/J 

=      1/1  +  1  +  1+ +  1\ 

WU      2      8T  n) 

_  1/1  +  1  +  1  + +  _1_\ 

8  \2T3T4T  n+1/ 

+f(s+H+ *£i) 

~      16  Vl)  ~  16  (2)  -  16  (n  +  l)  +  16  Vn+V 


1  /l 


/l- 


) 


16  V2      (n+1)  (n  +  2) 
=  na  +  3n 

32  (n  +  1)  (n  +  2) 
=  sum  to  n  terms. 

Sum  to  infinity    =^(i-J) 

=  A- 


teachers'  edition.  533 


4.  Sum  to  n  terms,  and  to  infinity,  the  series  .  ^  Q  ^  .1  Q  w  .  w  _, 

1ft  iX(JX4    oX4X6 


4X6X6 

o  «.  _1_   "I 

The  general  term  is 


Let 


(n  +  1)  (n  +  2)  (n  +  3) 
3n+l  4      ,      £ 


(n+l)(n  +  2)(n+3)      n+1      n+2n+3 
then  3n+l  =  ^4(n+2)(n+3)+£(n  +  l)(n  +  3) 

+  C(w+l)(w  +  2) 
=  (A+B  +  C)n2  +  M+ 4B+3C)n 
+  64  +  3B  +  2C. 

...4+b+c=o,  ^  =  -1, 

6X  +  4£+3C=3,  5=5, 

64  +  3£  +  2C=l,  C=-4. 

3n+l  -1,6  4 


Hence 


(n  +  1)  (n  +  2)  (n  +  3)      n  +  1      n  +  2      n  +  3 


^  4,7,  ,  3n+l 


2X3X43X4X6  (n  +  1)  (n  +  2)  (n  +  3) 

V  2^3      4/^V      3^4      h)^ 

+   f-l  +  -5 M 

V  n      n+1      n  +  2j 

=-  G+S+i+ +m) 

+*(!+H+ +sii) 


=  _l+4      ^ 4_ 

2     3     n+2      n  +  3 
_      5  3n  +  5 

6      (n  +  2)(n  +  3) 
_  6n*+7n 

6  (n  +  2)  (n  +  3) 
_  n(6n  +  7) 

6(n  +  2)(n  +  3) 
=  sum  to  n  terms. 

Sum  to  infinity  =  f . 


J 


534  ALGEBRA. 

5.  Sum  to  n  terms,  and  to  infinity,  the  series  »  » 

-  lXJXo     JXoX4 

fxlx  5* 

Each  term  of  the  series  is  8  times  the  corresponding  term  in  the 
series  of  Ex.  3.     Hence  the  sum  to  n  terms 

n2  +  3n 
4(n+l)(n  +  2)' 
and  the  sum  to  infinity  =  £. 


Exercise  CXXX. 

L  If  6,  7,  8,  3,  2  are  the  digits  of  a  number  in  the  scale  of  r, 
beginning  from  the  right,  write  the  algebraical  value  of  the  number. 
The  algebraical  value  of  the  number  is 

2r*  +  3r8+8r2+7r  +  6. 

2.  Find  the  product  of  234  and  125  when  r  is  the  base  of  the 
scale. 

234  X  126=  (2r*  +  3r  +  4)  (r«+  2r  +  5) 
=  2r*  +  7r»  +  20  r2  +  23r  +  20. 

3.  In  what  scale  will  the  common  number  756  be  expressed  by 
530? 

Let  r  be  the  base  of  the  scale ;  then 
5r*  +  3r  =  756. 
.-.  r=12. 

4.  In  what  scale  will  540  be  the  square  of  23  ? 
Let  r  be  the  base  of  the  scale ;  then 

(2r  +  3)2  =  5r*  +  4r 
—  r2  +  8r+9  =  0.       - 
.-.  r  =  9. 

5.  Show  that  1234321  will,  in  any  scale,  be  a  perfect  square,  and 
find  its  square  root. 

In  any  scale 

1,234,321  =  r»  +  2t*  +  3r*+  4r»  +  3r*  +  2r  +  l 
=  (r3  +  r2  +  r  +  l)2 
=  llll2  in  the  scale  of  r. 


TEACHERS7   EDITION. 


535 


6.  In  what  scale  will  212,  1101,  1220  be  in  arithmetical  progres- 
sion? 

Let  r  be  the  base  of  the  scale  ;  then 

2  X  2  X  1101  =  212  +  1220, 
2(r8  +  r2+l)  =  (2»a  +  r+2)  +  r8  +  2r3  +  2r, 
r8-2r2-3r  =  0, 
r(r  +  l)(r-3)  =  0. 
,-.r  =  3. 

7.  Multiply  31.24  by  0.31  in  the  scale  of  6. 

31.24 
0.31 
3124 
14432 


20.2444 


8.  Find  the  least  multiplier  of  13,168  which  will  make  the  product 
a  perfect  cube. 

13,168  =  16  X  823  =  2*  X  823. 

Hence  the  required  multiplier  is  2*  X  8232  =  1646*  =  2,709,316. 


Exercise   CXXXI. 
1.  Solve  the  following  equations  by  constructing  their  loci : 


2z  +  3y  =  8) 
3x  +  7y  =  7j 


(i.)  Ifx  =  0,  1,2,  3,  4,  5,  -1, 
-2,-3,-4, 
y  =  2|,  2,  1J,  },  0,  -{, 

H,  4,  4f ,  5}. 

(ii.)  Ifx  =  0,  1,  2,  3,  4,  5,  -1, 
-2,-3,-4, 

y — 1»  y»  7»      7»      ?» 

-1},  If,  l?,2f,2?. 

The  loci  are  straight  lines,  as 
represented  in  the  figure,  and 
they  meet  at  the  point  x  =  7, 
y=-2. 


536 


ALGEBRA. 


2.  Solve  the  following  equations  by  constructing  their  loci : 

3x  — 5y  =  2   ) 
2x+7y=22j 


(i.)  Kx  =  0,  1,  2,  3,  4,5,  -1, 
-2,-3,-4, 

*=-!.*.  ti  If,  2,2?, 
-1,  -If,  -2J,  -2|. 

(ii.)  Ifx  =  0,  1,  2,  3,  4,  5,-1, 
-2,-3,-4, 

y  =  3},  2f,  2*,  2?,  2,  If, 
3f,  3^,  4,  4|. 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  the? 
meet  at  the  point  x  =  4,  y  =  2. 


3.  Solve  the  following  equations  by  constructing  their  loci : 


2x—   9y=  11 
3x-12y=15 


(i.)  Ex  =  0,  1,  2,  3,  4,6,-1, 
-2,-3,-4, 

(ii.)  Ifx  =  0,  1,  2,  3,  4,  5,-1, 
-2,-3,-4, 

-i»  -o,  -f,  -i, 

The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  1,  y  =  —  1. 


teachers'  edition. 


537 


4.  Solve  the  following  equations  by  constructing  their  loci : 


4x  —  2y  =  20 
6x  =  9y 


(i.)  Ifx  =  0,  1,  2,  3,  4,5,-1, 
-2,-3,-4, 

y=-10,  -8,-6,-4, 
-2,  0,  -12,  -14, 
- 16,  - 18. 

(ii.)  Ifx  =  0,  1,  2,  3,  4,  6,-1, 
-2,-3,-4, 

y  =  Q,  f,  *,M,y,  -f, 

The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  7J,  y  =  5. 


5.  Solve  the  following  equations  by  constructing  their  loci : 

2x  — 3y  =  4 
3x  +  2y  =  3S 


(i.)  Ifx  =  0,  1,  2,3,4,5,-1, 
-2,-3,-4, 

V  =  -  *•  -  f  0,  f  J,  2, 

-2,-},  -^,-4. 

(ii.)  If  x  =  0,  2,  4,  6,  8,  10,  -  2, 
-4,-6,-8, 

y=16,  13,   10,   7,   4,   1, 
19,  22,  25,  28. 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  8,  y  =  4. 


538 


ALGEBRA. 


6b  Solve  the  following  equations  by  constructing  their  loci : 

2x  +  3y=7) 
4x-5y  =  3J 


(L)  lfx  =  0,  1,  2,  3,  4,5,-1, 
-2,-3,-4, 

y  =  i,  f*  i,l,  -h  -i. 

(ii.)  If  x  =  0,  1,  2,  3,  4,  6,  - 1, 
-2,-3,-4, 

*=-!»*»  i.  {.¥.  ¥» 


The  loci  are  straight  lines,  as  represented  ki  the  figure,  and  they 
meet  at  the  point  x  =  2,  y  =  1. 


7.  Solve  the  following  equations  by  constructing  their  loci : 


;-9y=ll) 
:-4y=7  ) 


2x  — 9y  =  11) 
3x- 


(i.)  If  x  =  0,  1,  2,  3,  4,  5-1, 
-2,-3,-4, 

*=-¥>-!. -fc-fc 

(U.)  If  x  =  0,  1,  2,  3,  4,  5,-1, 
-2,-3,-4, 

y=-i,-i,-i,M>2> 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  1,  y  =  —  1. 


teachers'  edition. 


539 


8.  Solve  the  following  equations  by  constructing  their  loci : 


3x  — 4y  =  —  5 
4x  —  6y  =  1 


(i.)  Ifx  =  0,  1,  2,3,4,6,-1, 
-2,-3,-4, 

-ii  -1,  ~b 

(ii)  If  x  =  0,  1,  2,  3,4,  5,  -1, 
-2,-3,-4, 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  29,  y  —  23. 


9.  Solve  the  following  equations  by  constructing  their  loci : 


x  — 2y  =  4 
2x  —  y=  6 


(i.)  If  x  =  0,  2, 4, 6,  8,-2,-4, 
-6, 

y  =  -2,-l,  0,1, 2,-3, 
-4,-6. 

fii.)  If  x  =  0,  1,  2,  3,  4,  6,  -1, 
-2,-3,-4, 

y=-5,  -3,  -1,1,3,6, 
-7,-9,-11,-13. 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
meet  at  the  point  x  =  2,  y  =  —  1. 


540 


ALGEBRA. 


10.  Solve  the  following  equations  by  constructing  their  loci : 

§-*  =  5 

x     y 

x     y 

(i.)  Ifx  =  0,i,i,i,l,|,2,-}, 

-*.  -t,-l, 
y  =  0,f  4,-4,-2,-J, 

—  f  >     A> —  rr> —  f  > 
-*•    . 

(ii.)  If*=0,i,i,|,l,|,2,-}, 

y=o,  i,  f,  -v,  -!, 

The  loci  are  hyperbolas,  consisting  each  of  two  infinite  branches, 
as  represented  in  the  figure ;  and  they  intersect  at  the  two  points 
s  =  0,  y  =  0,  and  x  =  —  1,  y  =  —  \. 

XL  Solve  the  following  equations  by  constructing  their  loci : 


x     y 

§_?  =  4 

x      y 

• 

(i.)lf*  =  0,i,  J,  1,2,  3, 4,-1, 

-2,-3, 

y  =  0,  oo,  1,  |,  $,  £,£, 

h  J.  A- 

.    (ii.)  Ifi6  =  0,$,l,2,3,-l,-2, 
-3,-4, 

y  =  0,  oo,-2,  -4,  -|, 

-?.-A.-i.-A- 

The  loci  are  hyperbolas,  as  represented  in  the  figures,  and  they 
intersect  at  the  points  x  =  0,  y  =  0,  and  x  =  ±,  y  =  1. 


teachers'  edition. 


541 


12.  Solve  the  following  equations  by  constructing  their  loci : 


x2  +  2/*=104) 
x  +  2/=12   J 


(i.)  Kx=0,±2,i4,±6,±8, 
±10, 

y=±10.2,  ±10,  ±9.4, 
±8.2,  ±6.3,  ±2. 

(ii.)  If  x=0,  2,  4,  6,  8,  10,  12, 
y=12,  10,8,6,4,2,0. 


The  first  locus  is  a  circle,  the  second  a  straight  line,  as  repre- 
sented in  the  figure,  and  they  intersect  at  the  two  points  x  =  2, 
y  =  10,  and  x  =  10,  y  =  2. 


13.  Solve  the  following  equations  by  constructing  their  loci : 

x  —  y  =  10 


x  —  y  =  10  ) 

2+y2=178) 


(i.)  If  x  =  0, 2, 4, 6,  8, 10, 12,  IS, 

y=-10,  -8,  -6,  -4, 
-2,0,2,3. 


(ii.)  Ifx  =  0,  ±2,  ±4,  ±6,  ±8, 
±  10,  ±  12,  ±  13, 

y=  ±13.3,  ±13.2,  ±12.7, 
±11.9,  ±10.7,  ±8.8, 
±5.5,  ±3. 


The  first  locus  is  a  straight  line,  the  second  a  circle,  as  repre- 
sented in  the  figure,  and  they  intersect  at  the  two  points  x  =  13, 
y  =  3,  and  x  =  —  3,  y  =  —  13. 


542 


ALGEBRA. 


14.  Solve  the  following  equations  by  constructing  their  loci : 


xy—  12  =  0 
x  —  2y  =  5 


(i.)  If  x  =  0,    ±2,    ±4,    ±6, 

±8,  oo, 

y  =  co ,    ±6,   ±3,   ±2, 
±|.fc 

(ii.)  Ifx=0,  1,  3,  6,  7,9, 

y  =  -f,-2,-l,0,l,2. 


The  first  locus  is  a  hyperbola,  the  second  a  straight  line,  as 
represented  in  the  figure ;  and  they  intersect  at  the  two  points 
x  =  8,  y  =  J,  and  x  =  —  3,  y  =  —  4. 


^15.  Solve  the  following  equations  by  constructing  their  loci : 

x  +  y=13) 
xy  =  I 


(i.)  If  x  =  0,  2,  4,  6,  8,  10,  12, 
y  =  13,  11,  9,  7,  6,  3,  1. 

(ii.)  Hi  =  0,  ±2,  ±4,  ±6,  ±9, 
±  12,  ±  18, 

y  =  co,   ±18,    ±9,   ±6, 
±4,  ±3,  ±2. 


The  first  locus  is  a  straight  line,  the  second  a  hyperbola,  as  repre- 
sented in  the  figure ;  and  they  intersect  at  the  two  points  *  =  4, 
y  =  9,  and  x  =  9,  y  =  4. 


teachers'  edition. 


543 


16.  Solve  the  following  equations  by  constructing  their  loci : 
-4x2=12      ) 

j  +  y=_10) 


3y*-4x2  =  12 
2x- 


(i.)  Ifx  =  0,±2,±4,±6,±8, 
±10, 

y  =  ±  2,    ±  3.0,    ±  5.2, 
±  7.2,  ±  9.4,  ±  11.7. 

(ii.)  Ifx  =  0,  4,  -2,  -4,  -6, 

-8,-10, 

y=  —10,  —18,-6,  —2, 
2,  6,  10. 


The  first  locus  is  a  hyperbola,  the  second  a  straight  line,  as 
represented  in  the  figure  ;  and  they  intersect  at  the  two  points 
x  =  —  3,  y  =  —  4,  and  x  =  —  12,  y  =  14. 


17.  Solve  the  following  equations  by  constructing  their  loci : 

4  5      1 

5  +  y      12  +  x  L 

2x  +  5y  =  36         J 


(i.)  Ifx=0,  2,  4,  6,  -2,-4, 
-6, 

y  =  ¥,  V,  ¥,  ¥.  ».  *. 
-1- 

(ii.)  If  x  =  0,  2,4,6,  -2,  -4, 
-6, 


The  loci  are  straight  lines,  as  represented  in  the  figure,  and  they 
intersect  at  the  point  x  =  2,  y  =  ^. 


544 


ALGEBRA. 


18.  Solve  the  following  equations  by  constructing  their  loci : 


x  3y  27 
J_  1  =  11 
4x  +  y      72 


(i.)  If  x  =  0,  3, 6,  9, 13.6, 18, 26, 
y=  0,    3.2,    9,    22.5,  oo, 
-  46,  34.6. 

(ii)  lfx  =  0,  if,  3,  6,  9,  -3, 
-6,-9, 

y  =  0,  oo ,  14.4,  9,  8,  4.2, 
6.1,  5.5. 


The  loci  are  both  hyperbolas,  only  one  branch  of  the  first  being 
shown  in  the  figure.  They  intersect  at  the  two  points  x=0, 
y  =  0,  and  x  =  6,  y  =  9. 


Exercise  CXXXII. 


1.  Construct  the  locus  of  the 
equation, 

x*  +  3x-10  =  0. 

Let    x2  +  3x  — 10  =  y, 
thenif  x  =  0,  1,  2,  4,  —2,  -4, 
-6,-6, 

y  =  -  10,    -  6,    0,    18, 
—  12,  -  6,  0,  8. 

The  locus  is  represented  in  the  figure.     The  roots  are  2  and  —  5. 


TEACHEBS'    EDITION. 


545 


2.  Construct  the  locus  of  the  equation,  x8  —  2  x2  +  1=0. 


Let    x8  —  2x2+l  =  y, 
then  if  x  =  0,  1,  2,  3,  -  1,  -  2, 
y=  1,0, 1,10,  -2,-15. 

The  locus  is  represented  in  the 
figure.    The  roots  are 

1+  V5       J  1-  V6 


V 


2 


>  and  • 


3.  Construct  the  locus  of  the  equation,  x4  —  20  x*  +  04  =  0. 

Let  x4  —  20x2  +  64  =  y, 
then  if  x=0,  ±2,  ±3,  ±4,  ±5, 
y  =  64,  0,  -  35,  0,  189. 

The  locus  is  represented  in  the 
figure,  except  that  the  ordinates 
are  ^  their  computed  length. 
The  roots  are  ±  2  and  ±  4. 


4.  Construct  the  locus  of  the  equation,  x2  —  4  x  4- 10  =  0. 


Let    x9  — 4x  +  10  =  y, 
then  if  x  =  0,  1,  2,  3,  4,  5,  —  2, 
y=10,  7,  6,  7,  10,15,22. 

The  locus  is  represented  in  the 
figure.     The  roots  are  imaginary. 


y 


546 


ALGEBRA. 


5.  Construct  the  locus  of  the 
equation, 

x*-6x*  +  4=0. 
Let    x4  —  5x2  +  4=y, 
then  if  x  =  0,  ±  1,  ±  2,  ±  3, 
y  =  4,  0,  0,  40. 

The  locus  is  represented  in  the 
figure.    The  roots  are  ±  1  and 

±2. 


Exercise   CXXXIII. 

1.  Determine  whether  —  5  is  a  root  of  the  equation 

a^  +  Cx'-lOx8--  112x2-2O7x-110  =  O. 

1  +  6  -  10  -  112  -  207  -  110|-5 
-5-    5+    75+185  +  110 
+  1-15-37-22  +  0 
Hence  —  5  is  a  root. 

2.  Determine  whether  1  is  a  root  of  the  equation 

x6 -  8x*  +  7x»  +  x2- 3x  +  2  =  0. 

1-8+7  +l-3  +  2[l 
1-7+0+ 1-2 
-7+0+1-2+0 
Hence  1  is  a  root. 


3.  Determine  whether  —7  is  a  root  of  the  equation 
x*  +  21x+ 7x»+ 147  =  0. 

l  +  7  +  0  +  21  +  147|-7 
-  7  +  0  +    0-147 
0  +  0  +  21  +  0 
Hence  —  7  is  a  root. 


teachers'  edition.  547 

4.  Determine  whether  —  8  is  a  root  of  the  equation 

x*  +  8x4  -  7  x2  -  54x  + 16  =  0. 

l  +  8-7-54+16|-8 
-8  +  0  +  56-16 
0-7+    2  +  0 
Hence  —  8  is  a  root. 

5.  Determine  whether  2  is  a  root  of  the  equation 

x4-4x8-3x2-2x-8  =  0. 

1-4-3-*  2-   8 [2 
+  2-4-14-32 

-  2  -  7  -  16  -  40 
Hence  2  is  not  a  root. 

6.  Determine  whether  —7  is  a  root  of  the  equation 

xs  +  l4x2  +  65x+  112  =  0. 

l  +  14  +  65+112|-7 

-  7-49-112 
+    7  +  16  +  0 

Hence  —7  is  a  root. 

7.  Determine  whether  6  is  a  root  of  the  equation 

2x*  -  4x»  -  62  x2+114x- 180  =  0. 

2-   4-62+114-18016 
+  12+48-    84  +  180 


+    8-14+    30  +  0 
Hence  6  is  a  root. 

8.  Determine  whether  —  5  is  a  root  of  the  equation 
x*-7x-2x2-15  =  0. 

1  +  0-   2-     7-    15|-5 

-  5  +  25  -  115  +  610 

-  5  +  23  -  122  +  595 
Hence  —  5  is  not  a  root. 


548  AXJ5EBRA. 

9.  Determine  whether  —  0.3  is  a  root  of  the  equation 

x*  +  2.3x"*  +  3.6x2  +  4.9x  +  1.2  =  0. 
1  +  2.3  +  3.6  +  4.9  +  1.2 1-0.3 
-0.3-0.6-0.9-1.2 
+  2     +3     +4     +0 
Hence  —  0.3  is  a  root. 

10.  Determine  whether  *  is  a  root  of  the  equation 

X»-ix2-^X-i  =  0. 

5  +  i+i 

+1+ i+° 

Hence  J  is  a  root. 

Exercise  CXXXIV. 

1.  Find  the  equation  whose  roots  are  2,  6,  and  —  7. 
The  equation  is  (x  —  2)  (x  —  6)  (x  +  7)  =  0 

or  x8  — x2  — 44x  +  84  =  0. 

2.  Find  the  equation  of  which  the  roots  are  1,  4,  —  1,  and  —  3. 
The  equation  is 

(x  -  1)  (x  -  4)  (x  +  1)  (x  +  3)  =  0 
or  (x2-l)(x2-x-12)  =  0 

or  x*-x«-13x2  +  x+12  =  0. 

3.  Find  the  equation  of  which  the  roots  are  2,  3,  —  2,  —  3,  and  —  6. 
The  equation  is 

(x-  2)  (x  -  3)  (x  4-  2)  (x  +  3)  (x  +  6)  =  0 
or  (x2  -  4)  (x2-  9)  (x  +  6)  =  0 

or  x*  +  6x*  -  13x» -  78x2  +  36x  +  216  =  0. 

4.  Find  the  equation  of  which  the  roots  are  0.2,  \,  and  —  0.4. 

The  equation  is 

(x  -  0.2)  (x  - 1)  (x  +  0.4)  =  0 
or  (*-i)(*-i)(x+f)  =  0 

or  (6x  -  1)  (8x  -  1)  (5x  +  2)  =  0 

or  200x8+  15x2  —  21x+  2  =  0. 


teachers'  edition.  549 


5.  Find  the  equation  of  which  the  roots  are 

5,  3  +  V^l,  and  S-^l. 
The  equation  is 

(x  -  6)  (x  -  3  -  y/^1)  (x  -  3  +  V^l)  =  0 
or  (x  -  5)  (x2  -  6  x  +  10)  =  0 

or  x8  -  11  x2  +  40x  -  50  =  0. 


Exercise  CXXXV. 

1.  Form  the  equation  whose  roots  are  2,  4,  and  —  3. 

2  +  4-3  =  3, 
2  X  4  +  2  X  (-  3)  +  4  (-  3)  =  -  10, 
2  X  4  X  (-  3)  =  -  24. 
Hence  the  equation  is 

x8-3x2-10x  +  24  =  0. 

2.  Form  the  equation  whose  roots  are  2,-1,  and  —  7. 

2  -  1  -  7  =  -  6, 
2  x  (- 1)  +  2  x  (-  7)  +  (-  1)  X  (-  7)  =  -  9', 
2  x  (-  1)  X  (-  7)  =  14. 
Hence  the  equation  is 

x«  +  6x2-9x-14  =  0. 

3.  Form  the  equation  whose  roots  are  2,  0,  and  —  2. 

2  +  0-2  =  0, 
2  X  0  +  2  X  (-  2)  +  0  X  (-  2)  =  -  4, 
2  X  0  X  (—  2)  =  0. 

Hence  the  equation  is 

oj8  —  4  x  =  o. 

4.  lorm  the  equation  whose  roots  are  6,  6,  and  6. 
6  +  6  +  6  =  18, 
6X6  +  6X6  +  6X6  =  108, 
6X6X6  =  216. 

H<ice  the  equation  is 

x8  -  18x2  +  108x  -  216  =  0. 


560  ALGEBRA. 

5.  Form  the  equation  whose  roots  are  2,  1,  —  2,  and  —  1. 

2  +  1-2-1  =  0, 
2X  1  +  2  X  (-2)  +  2X  (-1)+1  X  (-2)4-1  X(-1) 
+  (_2)X(-l)=-5, 
2  X  1  X  (-  2)  +  2  X  1  X  (-  1)  +  2  X  (-  2)  X  (-  1) 
+ 1  x  (-  2)  x  (-  1)  =  0. 
2  x  1  x  (-  2)  x  (-  1)  =  4. 

Hence  the  equation  is 

x*-5x*  +  4  =  0. 

6b  Form  the  equation  whose  roots  are  2,  $,  —  2,  —  }. 

2  +  l-2-i  =  0, 
2  X  \  +  2  X  (-  2)  +  2  X  (-  i)  +  i  x  (-  2)  +  i  x  (-  i) 

+  (-2)x(-i)  =  -¥, 
2  X  J  X  (-2)  +  2  X  J  X  (-  i)  +  2  X  (-  2)  X  (-  J) 
+  Jx(-2)X(-i)  =  0, 
2Xi(~2)X(-i)  =  l. 

Hence  the  equation  is 

&*-¥«*+ 1  =  0. 

Exercise  CXXXVI. 

1.  Find  the  roots  of  the  equation  x2  + 11  x  +  24  =  0. 

x2  +  1 1  x  +  24  =  (x  +  3)  (x  +  8)  =  0. 
Hence  x  =  —  3  or  —  8. 

2.  Find  the  roots  of  the  equation  7  x2  +  161  x  +  714  =  0. 

7  x2  +  161  x  +  714  =  7  (x2  +  23  x  +  102) 
=  7(x  +  17)(x+6). 
Hence  x  =  —  17  or  —  6. 

3.  Find  the  roots  of  the  equation  x*  —  4  a2  x2  +  3  a4  =  0. 

x*-  4a2 x2  +  3a*  =  (x2-  a2)  (x2-  3a2) 

=  (x  +  a)(x-a)(x+  VSa)(x- V3a). 
Hence  x  =  —  a,  +  a,  —  V3a,  or  +  V3a. 


TEACHERS*    EDITION.  551 

4.  Find  the  roots  of  the  equation  x&  +  4  x8  +  8  x2  +  32  =  0. 
X6  +  4x8  +  8a:2  +  32  =  (x8  +  8)  (x2  +  4) 

=  (x  +  2)  (x2  -  2  x  +  4)  (x2  +  4) 
Hence  x  =  -  2,  1  ±  V11^,  ±  2  V-11!. 

5.  Find  the  roots  of  the  equation  12  x2  —  5  x  —  2  =  0, 

12x2-6x-2=(3x-2)(4x+l). 
Hence  x  =  J  or  —  ±. 

6.  Find  the  roots  of  the  equation  4x*  —  9  x2  +  6  x  —  1  =  0. 
4x*  -  9x2  +  6x  -  1  =  (2x2)2  -  (3x  -  l)2. 

=  [2x2+(3x-l)][2x2-(3x-l)] 
=  (2x2  +  3x-l)(2x2-3x+l) 
=  (2x2+3x-l)(x-l)(2x-l). 

„                                   t    ,         -3±  Vl7 
Hence  x  =  1,  £,  or 

4 

7.  Find  the  roots  of  the  equation  49  x2  —  112  bx  +  64  62  =  0. 
49x2-  112 6x  +  64 62  =  (7x  -  86)2. 

„  86    86 

Hence  x=  —  » 

7       7 

8.  Find  the  roots  of  the  equation  x6  —  64  =  0. 

x8-64=(x8  +  8)(x8-8) 

=  (x  +  2)  (x2  -  2x  +  4)  (x  -  2)  (x9  +  2x  +  4). 
Hence  x  =  -  2,  1  ±  V11^,  2,  or  -  1  ±  \T=H. 

9.  Find  the  roots  of  the  equation  3  x8  —  x2  +  3  x  —  1  =  0. 
3x8  -  x2  +  3x  -  1  =  3  (x8  +  x)  -  (x2  +  1) 

=  (x2+l)(3x-l). 
Hence  x  =  J  or  ±  V—  1. 

10.  Find  the  roots  of  the  equation  x  —  27  x4  =  0. 

x-27x*  =  x(l-27x8) 

=  x(l-3x)(l  +  3x  +  9x2). 

„                                  A   ,        -1±  V^ 
Hence  x  =  0,  J,  or 


552  ALGEBBA. 


Exercise   CXXXVII. 

L  Solve  the  equation  x8  +  3x2  —  25x  —  12  =  0. 

x«  +  3x2- 26x- 12  =  0 

-4+3 

7-28  -28  +  7  = -4. 

Hence         x8  +  3x2- 25x-  12  =  (x-  4)  (x2  +  7x  +  3), 

-7±  V37 
and  x  =  4,  or, 

2.  Solve  the  equation  x8  —  4x2  —  8x  +  8  =  0. 

x«-4x2-   8x  +  8  =  0 
2+4 
-6     -12  -12 +  (-6)  =2. 

Hence  x8-  4x2-  8x  +  8  =  (x  +  2)  (x2-  6x  +  4), 

and  x  =  —  2,  or  3  ±  sfc. 

3.  Solve  the  equation  x8  —  7  x2  +  19x  —  21  =  0. 

x»-7x2+19x-21  =  0 
-3+7 

-4      +12  12 +  (-4)  =  -3. 

Hence  x8- 7x2+ 19x-21  =  (x-3)(x2-4x  + 7), 

and  x  =  3,  or  2  ±  V^. 

4.  Solve  the  equation  x8  —  8x2  +  21  x  —  18  =  0. 

x8-8x2  +  21x-18  =  0 
-2+9 

-6+12  12  +  6  =  2. 

Hence  x8-8x2  +  21x- 18=  (x  -2)(x2-6x  +  9) 

=  (x-2)(x-3)2 
and  x  =  2,  3,  3. 

5.  Solve  the  equation  x8  —  26  x  —  6  =  0. 

x8  +  0x2-26x-5  =  0 

5-1 

-  5     -  25  -  25  +  (-  6)  =  5. 

Hence  x8  —  26x  —  5  =  (x  +  5)  (x2  —  5x  —  1), 

5±V29 
and  x  =  —  5,  or • 


teachers'  edition.  553 

6.  Solve  the  equation  x8  —  3  x2  —  54  x  —  104  =  0. 

x8-3x2-54x-104=0 
4     -26 
-7      -28  -28-r(-7)  =  4. 

Hence  x8-  3x2-54x  -  104=  (x  +  4)  (x2-  7x-  26), 

,  .        7±Vl53 

and  x  =  —  4,  or - • 

2 

7.  Solve  the  equation  x8  +  9x2  +  2x  —  48  =  0. 

x8  +  9x2  +    2x-48  =  0 
-2      +24 
11      -22  -22-rli=-2. 

Hence  x8  + 9x2  + 2x-48  =  (x-2)  (x2+llx  +  24) 

=  (x-2)(x  +  3)(x+8), 
and  x  =  2,  —  3,  or  —  8. 

8.  Solve  the  equation  x8  —  2  x2  —  25  x  +  50  =  0. 

x8-2x2-25x+50=0 
-2      -25 
0  0 

Hence  x8-  2x2-25x  +  50=  (x-2)(x2-25) 

=  (x-2)(x-5)(x  +  5), 
and  x  =  2,  +  5,  —  5. 

9.  Solve  the  equation  x8  —  3  x2  —  61  x  +  63  =  0. 

x8-3x2-61x  +  63  =  0 
- 1     -  63 

-2+2  2-r(-2)  =  -l. 

Hence  x8  -  3  x2  -  61  x  +  63  =  (x  -  1)  (x2  -  2  x  -  63) 

=  (x-l)(x  +  7)(x-9), 
and  x  =  1,  —  7,  or  9. 

10.  Solve  the  equation  x8  —  37  x  —  84  =  0. 

x8  +  Ox2-  37  x-  84  =  0 
4     -21 
-4      -16  -16-r(-4)  =  4. 

Hence  x8  -  37  x  -  84  =  (x  +  4)  (x2  -  4  x  -  21) 

=  (x  +  4)(x  +  3)(x-7), 
and  x  =  —  4,  —  3,  or  7. 


554  ALGEBRA. 


Exercise  CXXXVIIT. 

L  Solve  the  equation  x4  —  2x8  —  13x2  +  38x  —  24  =  0. 
m  +  p  =  —  2, 
ti  +  mp  4-  9  =  —  13, 
np  +  mq  =  38, 
Tig  =  —  24. 
By  trial  it  is  found  that  we  may  take 

n  =  2,  q  =  —  12,  m  =  —  3,  p  =  1. 
Hence  x4-  2x»-  13x2  +  38x-  24 

=  (x2  -  3  x  +  2)  (x2  +  x  - 12) 
=  (x-l)(x-2)(x-3)(x  +  4), 
and  x=l,  2,  3,  or  —4. 

2.  Solve  the  equation  x4  —  6x«—  2x2  +  12x  +  8  =  0. 

x*_5x8_2x2  +  12x+8 

=  (x2  -  x  -  2)  (x2  -  4x  -  4) 

=  (x-M)(x-2)(x2-4x-4). 
Hence  x  =  —  1,  2,  or  2  ±  2  V2. 

3.  Solve  the  equation  x4  —  4  x8  —  8  x  +  32  =  0. 

X4_4x8_8x  +  32 

=  (x2-6x  +  8)(x2  +  2x  +  4) 
=  (x-  2)  (x  -  4)  (x2  +  2x  +  4). 
Hence  x  =  2,  4,  or  —  1  ±  V^. 

4.  Solve  the  equation  x4  — 12  x*  +  50  x2  —  84  x  +  49  =  0. 

x4  -  12x»  +  50 x2  -  84x  +  49 

=  (x2-6x+7)(x2-6x+7). 
Hence  x=3±V2,  3±V2. 

5.  Solve  the  equation  x4  — 11  x2  +  18x  —  8=0. 

x4-llx2  +  18x-8 

=  x2-2x  +  l)(x2+2x-8) 
=  (x  -  1)  (x  -  1)  (x  -  2)  (x  +  4). 
Hence  x  =  1,  1,  2,  or  —  4. 


teachers'  edition.  555 

6.  Solve  the  equation  x*  —  10x2  —  20x  — 16  =  0. 

x*-10x2-20x-16 

=  (x2-  2x - 8)  (xa  +  2x  +  2) 
=  (x  +  2)  (x-  4)  (x2  +  2x  +  2). 
Hence  x  =  —  2,  4,  or  —  1  ±  V^T. 

7.  Solve  the  equation  x*  —  7x»  +  23  x2  —  47x  +  42  =  0. 

x*-7x»  +  23x2-47x  +  42 

=  (x2-5x  +  6)(x2-2x  +  7) 
=  (x-2)(x  —  3)(x2-2x  +  7). 
Hence  x  =  2,  3,  or  1  ±  V— 6. 

a  Solve  the  equation  x*  + 2  x8- 9  x2  —  8x  +  20  =  0. 
x*  +  2x8-9x2-8x  +  20 

=  (x2-4)(x2  +  2x-6* 

=  (x-  2)  (x  +  2)  (x2  +  2x-  6). 

Hence  x  =  2,  —  2,  or  —  1  ±  VS. 

9.  Solve  the  equation  x*  -  4x»  —  102 x2  -  188x  -  91  =  0. 

x*  -  4x»  -  102 x2  -  188x  -  91 

=  (x2  +  8  x  +  7)  (x2  -  12  x  -  13) 
=  (x  +  1)  (x  +  7)  (x  +  1)  (x  -  13) 
Hence  x  =  —  1,  —  1,  —  7,  or  13. 

10.  Solve  the  equation  x*  -  11  x*  4-  46x2  -  117  x  +  46  =  0. 

x*  -  llx8  +  46X2  -  117x  +  45 

=  (x2  -  7x  +  3)  (x2  -  7x  +  16). 
7±V37        2±Vi:li 


Hence  x  =  ■ 


2 


Exercise   CXXXIX. 

1.  Determine  the  signs  of,  the  roots  of  the  equation  x*  +  4  x8 
-  43  x2  —  68  x  +  240  =  0,  all  the  roots  being  real. 
+     +     -     -     + 
No.  of  variations,         2. 
No.  of  permanences,    2. 
Hence  there  are  two  positive  and  two  negative  roots. 


656  ALGEBRA. 


2.  Determine  the  signs  of  the  roots  of  the  equation  x8  — 22x* 
+  166  x  —  360  =  0,  ail  the  roots  being  real. 

+     -     +     - 
No.  of  variations,         3. 
No.  of  permanences,    0. 
Hence  there  are  3  positive  roots. 

3.  Determine  the  signs  of  the  roots  of  the  equation  x4  +  4  x3 

—  36  X2  —  78  x  +  360  =  0,  all  the  roots  being  real. 

+     +     -     -     + 
No.  of  variations,         2. 
No.  of  permanences,    2. 
Hence  there  are  2  positive  and  2  negative  roots. 

4.  Determine  the  signs  of  the  roots  of  the  equation  x8  — 12  aJ* 

—  43x  —  30  =  0,  all  the  roots  being  real. 

+     -    -    - 
No.  of  variations,         1. 
No.  of  permanences,    2. 
Hence  there  is  1  positive  and  2  negative  roots. 

6.  Determine  the  signs  of  the  roots  of  the  equation  x*  —  3x* 

—  6x8  +  16x*  +  4x  —  12  =  0,  all  the  roots  being  real. 

+     --     +     +     - 
No.  of  variations,         3. 
No.  of  permanences,    2. 
Hence  there  are  3  positive  and  2  negative  roots. 

&  Determine  the  signs  of  the  roots  of  the  equation  x*-f  12  x2 
+  47  x  —  60  =  0,  all  the  roots  being  real. 

+     -     +     - 
No.  of  variations,         3. 
No.  of  permanences,    0. 
Hence  there  are  3  positive  roots. 

7.  Determine  the  signs  of  the  roots  of  the  equation  x4  — 2x* 

—  13  x2  +  38  x  —  24  =  0,  all  the  roots  being  real. 

+     -     -     +     - 
No.  of  variations,         3. 
No.  of  permanences,    1. 
Hence  there  are  3  positive  and  1  negative  roots. 


teachers'  edition.  557 

8.  Determine  the  signs  of  the  roots  of  the  equation  x5  —  x4 
—  187  x8  —  359  x8  +  186  x  +  330  =  0,  all  the  roots  being  real. 

+     ---     +     + 
No.  of  variations,         2. 
No.  of  permanences,    3. 
Hence  there  are  2  positive  and  3  negative  roots. 

9.  Determine  the  signs  of  the  roots  of  the  equation  x6  —  10  x5 
+  19  x4  +  110  x3  —  536  x2  +  800  x  —  384  =  0,  all  the  roots  being 
real.  +     _     +     +     _     +     _ 

No.  of  variations,         5. 
No.  of  permanences,     1. 
Hence  there  are  5  positive  and  1  negative  roots. 

JO.  Determine  the  signs  of  the  roots  of  the  equation  x7  —  10  x6 
+  22x*  +  32  x*  -  131  x8  +  50  x2  +  108  x  -  72  =  0. 
+     -     +     +     -     +     +     - 
No.  of  variations,         5. 
No.  of  permanences,    2. 
Hence  there  are  5  positive  and  2  negative  roots. 

Exercise   CXL. 

1.  Find  the  successive  deriva-  2.  Find  the  successive  deriva- 

tives of  the  polynomial  x2  +  2  x        tives  of  the  polynomial  x8  —  3  x2 
+  3.  +7x  +  25. 

F     (x)  =  x2  +  2x  +  3,  F     (x)  =  x8-3x2+7x  +  25, 

F«   (x)  =  2x+2,  F'   (x)  =  3x2-6x+7, 

F«  (x)  =  2,  F»  (x)  =  6x  -  6, 

Fm(x)  =  0.  F*"(x)  =6, 

i?iv  (X)  =  0. 

3.  Find  the  successive  derivatives  of  the  polynomial  x4  +  2  x3 
-5x2  +  64.  F     (x)  =  a.4  +  2x8-5x2+  64, 

F*   (x)  =  4  x8  +  6  x2  -  10  x, 
F"  (x)  =  12  x2  +  12  x, 
F»«(x)  =  24x+  12, 
F"  (x)  =  24, 
F*  (x)  =  0. 


558  ALGEBRA. 


4.  Find  the  successive  derivatives  of  the  polynomial  x6  +  x4  —  6  x8 
+  3x2-4x  +  27. 

F    (x)  =  xfi  +  x4-6x8  +  3x2-4x  +  27, 

F»   (x)  =  5x4  +  4x8-18x2  +  6x  —  4, 

F"  (x)  =  20 x8  +  12 x2  -  36x  +  6, 

F»"  (x)  =  60  x2  +  24  x  -  36, 

F"(x)  =  120x+24, 

F*  (x)  =  120, 

F"  (x)  =  0. 

5.  Find  the  successive  derivatives  of  the  polynomial  x4  —  3  ax8 
+  6  6x2  —  9  ex  +  mn. 

F     (x)  =  x*  —  3  ax8  +  6  6x2  —  9  ex  +  mn, 
F«   (x)  =  4x8-9ax2+12te-9c, 
F"  (x)  =  12x2-18ax+12  6, 
F»"  (x)  =  24  x  —  18  a, 
F*v  (x)  =  24, 
F*  (x)  =  0. 


Exercise   CXLL 

1.  Find  all  the  roots  of  x8  —  8  x2  +  13  x  —  6  =  0. 

F  (x)  =  x*-8x2  +  13x-6, 
F»(x)  =  3x2-16x+13 

=  (3x-13)(x-l). 
<f>  (x)  =  x  —  1. 
.-.  F(x)  =  (x-l)*(x-6). 
The  roots  are  1,  1,  and  6. 

2.  Find  all  the  roots  of  x8  —  7x2  +  16  x  — 12  =  0. 

F  (x)  =  x8  -  7x2  +  16x  -  12, 
Fi(x)  =  3x2-14x+  16 
=  (3x-8)(x-2). 
0  (x)  =  x  —  2. 
.-.  F(x)  =  (x-2)2(x-3). 
The  roots  are  2,  2,  and  3. 


TEACHERS'    EDITION. 


559 


3.  Find  all  the  roots  of  x4  —  6x*  —  8x  —  3  =  0. 
F  (x)  =  x4  —  6x2  —  8x  —  3, 


F«(x)  = 
4  +  0-12-8 
4  +  8+    4 

=  4x*-12x-8. 
1  +  0-    6-   8-    3 
4 

-  8  -  16  -  8 

-  8  -  16  -  8 

4  +  0-24-32-12 
4  +  0  -  12  -   8 

12)- 12-24-12 

-1-2-1    -4x-8 
.\  0(x)=x2  +  2x+l 

=  (x  +  l)2. 

.-.  F(x)  =  (x  +  l)«(x-3). 

The  roots  are  —  1,  —  1,  —  1,  and  3. 

4.  Find  all  the  roots  of  x«  —  2x2  —  15x  +  36  =  0. 

F  (x)  =  x*  -  2x2  -  15x  +  36, 
^(x)  =  3x2-4x-15 
=  (3x  +  5)(x-3). 
0  (x)  =  x  —  3. 
.-.  F(x)  =  (x-3)2(x  +  4). 
The  roots  are  3,  3,  and  —  4. 

5.  Find  all  the  roots  of  x*  —  7x8  +  9x2  +  27x  —  64  =  0. 

F  (x)  =  x*-7x8+9x2  +  27x-64, 
F'(x)  =  4x*-21x2+  18x  +  27. 


4- 

4- 

-21  +  18  +  27 

-24  +  36 
3-18  +  27 
3-18  +  27 

.-.*(x)  = 

.-.F(x)  = 
The  roots  are 

1-7+      9+27-64 
4 

4  -  28  +    36  +  108  -  216 
4-21+    18+    27 

-    7  +    18  +    81-216 
4 

-28+    72  +  324-864 
-  28  +  147  -  126  -  189 

-75)-    75  +  460-676 

1-      6+      9 
=  x2-6x  +  9 
=  (x-3)2. 
=  (x  -  3)»(x  +  2). 
3,  3,  3,  and  —  2. 

-7 


4x  +  3 


560 


ALGEBRA. 


6.  Find  all  the  roots  of  x*  —  24x2  +  64x  —  48  =  0. 
F  (x)  =  ar*  -  24x2  +  64x  -  48, 
Fi(x)  =  4x8-48x+64. 


4)4  +  0-48  +  64 


1  +  0- 
1-4  + 

12-16 
4 

4- 
4- 

16+16 
16+16 

1-0-14  +  64-48 

1  +  0-12  +  16 

-  12)  -12  +  48  -  48 


1-    4+    4 


x  +  4 


.-.  *(x)  =  x2  —  4x  +  4 

=  (x-2)2. 
.'.F(z)=  (x-2)»(x  +  6). 
The  roots  are  2,  2,  2,  and  —  6. 


7.  Find  all  the  roots  of  x*  -  10x8  +  24 x2  +  lOx  -  25  =  0. 

F  (x)  =  x*  -  10x8  +  24 x2  +  lOx  -  25, 
F*  (x)  =  4x*  -  30 x2  +  48x  +  10. 


1  -  10  +    24  +    10  -    25 

4 


4 
9 

-30+      48+      10 

36 
36 

-270+    432+      90 
-200+    100 

-70+332+      90 
9 

-630  +  2988+    810 
-  630  +  3500  -  1750 

-512)-    512  +  2560 

4-40+    96+    40- 
4-30+    48+    10 


100 


10+    48+    30-100 


-20+    96+    60-200 

-  20  +  150  -  240  -    50 

-6)-    54  +  300-150 

9-    50+    25 


-5 


4x  — 70 


.-.  0  (x)  =  x  -  5, 
F{x)=  (x  -  5)2(x2  -  1). 
The  roots  are  6,  5,  1,  and  —  1. 


teachers'  edition. 


561 


a  Find  all  the  roots  of  x6 -11x4+19x8 +115x2-200x-  500  =  0. 
F  (x)  =  x*  -  11  x*  +  19x»  +  116x2  -  200x  -  500, 
F'  (x)  =  6x*  ~  44x»  +  67 xa  +  230x  -  200. 


5-44  +  57  +  230-200 
5-40+25  +  260 

1-11+    19+    115-   200-     600 
6 

-  4  +  32-   20-200 

-  4  +  32-    20-200 

5-55+    96+    576-1000-  2600 
5-44+    67+    230-   200 

-11+   38+   345-   800-  2500 
6 

-65+190+1725-4000-12500 
-65  +  484-   627-2530+  2200 

—294)  -  294  +  2362  -  1470  — 14700 

1-       8+        5+       50 

-11 


5x-4 


.-.  0  (x)  =  x*  -  8x2  +  5x  +  50, 
0/(x)  =  3x2-  16x+  5. 


3 
3 

-16  +  6 

-15 
-1  +  5 
-1  +  6 

.-.  H.C.F. 

1-8+      5+50 
3 

3  -  24  +    15  +  150 
3  -  16  +      5 

X 

2  -    8  +    10+150 
3 

—  24+    30  +  450 

-  24  +  128  -    40 

-8 

-48)-    98-490 
1-      6 

of  0  (x)  and  0'(x)  =  . 

3x-l 
c-6. 

*(x)  =  (x-5)»(x  +  2), 
.-.  F(x)  =  (x-5)8(x  +  2)«. 
The  roots  are  5,  6,  6,  —  2,  and  —  2. 


562 


ALGEBRA. 


9. 


Find  all  the  roots  of  x6  —  2a*  +  3x3  —  7 x2  +  8x  —  3  =  0. 
F  (x)  =  x*-2x*+3x*-  7x2  +  8x-3, 
F>  (x)  =  6x*  -  8x*  +  9x2  -  14x  +  8, 
F»(x)  =  20x*  -  24 x2  +  18x  -  14. 
Find  H.C.F.  of  F*  (x)  and  F"  (x). 


>20- 

24  + 

18- 

14 

10- 

7 

12  + 

9- 

7 

70- 

70- 

84  + 
280  + 

63- 
220 

49 

206  — 
7 

167- 

49 

1442- 
1442- 

1099-  343 
6974  +  4632 

4876) 

4875- 

4876 

1- 


5-  8+  9- 
2 

14+8 

10  -  16  +  18  - 
10-12+  9- 

28  +  16 

7 

-  4+  9- 
6 

21  +  16 

-  20  +  46  - 

-  20  +  24  - 

106  +  80 
18+14 

3)21- 

87  +  66 

7- 
7- 

29  +  22 

7 

— 

22  +  22 
22  +  22 

—  2 


lOx-206 

7x-22 


.-.  H.C.F.  of  F1  (x)  and  F»  (x)  =  x  —  1. 
Hence  Fl  (x)  contains  (x  —  l)2  as  a  factor. 
Bat  1  is  a  root  of  F(x)  =  0. 

.-.  F(x)  =  (x  -  1)»  (x2  +  x  +  3). 

™u         x          ,    ,    ,        ,  -  1  ±  V-  11 
The  roots  are  1,  1,  1,  and 

2* 
10.  Find  all  the  roots  of  x4  +  6x«  +  xa  —  24x  +  16  =  0. 
F  (x)  =  x*  +  6x«  +  x2-  24x  +  16. 
Fi  (x)  =  4x»  +  18x2  +  2x  -  24. 


2)4+  18  +  2-24 
2  +"  9  +  1-12 
2+    6-8 


3  +  9-12 
3  +  9-12 


1  + 
2 


6+    1-24+    16 


2  +  12  + 
2+    9  + 


2-48  + 
1-12 


3+    1-J 


6+    2-72+    64 
6  +  27+    3-    36 


-  26)  -26-76+100 


1+3-     4 


/.  0(x)  =  x2  +  3x-4 

=  (x-l)(x  +  4). 
.-.  F(x)  =  (x-l)2(x  +  4)*. 
The  roots  are  1,  1,  —  4  and  —  4. 


Sx  +  3 


teachers'  edition.  563 


Exercise   CXLII. 


1.  Put  the  equation  2x*  +  § x2  —  x  +  J  =  0  in  the  form/(x)  =  0. 

Put  -  for  x ; 

o 

then  the  equation  becomes 

2x»  +  4x2-  36s  +  36  =  0, 

or  x«+  2x2  -18x+  18  =  0. 

2.  Put  the  equation  3x*  +  5x2  —  \x  —  8  =  0  in  the  form/(x)  =  0. 
Put  ~  for  x; 

then  the  equation  becomes 

3x8  +  30X2  -  I26x-  1728  =  0, 
or  x8  +  10x2-   42x-   576  =  0. 

3.  Put  the  equation  5x*  —  x8  —  *f  x2  —  ^x  +  1  =  0  in  the  form 
/(*)  =  0. 

put  Is***; 

then  the  equation  becomes 

6x*  -  30x8 -  6750x2  -  90000x  -h        30*  =  0, 
or  x*—   6  x8  -  1350  x2-  18000  x  +  162000  =  0. 

4.  Put  the  equation  x6  +  *x*  +  jx8  -  *x2  +  x  -  3  =  0  in  the 

form/(x)  =  0. 

*  x 

Put  g  f  or  x  ; 

then  the  equation  becomes 

x6  +  3x*  +  24x8-72x2  +  1296x  -23,328  =  0. 

5.  Put  the  equation  x*  —  2x2  +  \x  —  14  =  0  in  the  form/(x)  =  0. 

x 
Put  -  for  x  ; 

then  the  equation  becomes 

x*-8x2  +  4x-224  =  0. 


564 


ALGEBRA. 


Exercise   CXLIII. 


L  Diminish  the  roots  of  the 
equation  x*  —  11  x2  +  31  x  —  12 
=  0  by  1. 

1-11  +  31-12(1 

+    1-10  +  21 
1-10  +  21+    9 

+    1-9 
1-    9+12 

+    1 
1-    8 
The  required  equation  is 
y«-8y2+12y  +  9  =  0. 

2.  Diminish  the  roots  of  the 
equation  x*  —  6  x8  +  4  x2  +  18  x 
-  5  =  0  by  2. 

1-6  +  4  +  18-   5[2 
+  2-8-   8  +  20 


3.  Diminish  the  roots  of  the 
equation  x8  +  10x2  +  13x  — 24 
=  0  by  -  2. 

1  +  10  +  13-241  —  2 
-   2  —  16+    6 


1  + 


8—   3-18 
2  —  12 


1+    6  —  15 

-   2 
1+    4 


The  required  equation  is 
y8  +  4y2-15y-18  =  0. 

4  Diminish  the  roots  of  the 
equation  x8  —  9x2+ 22x  — 12  =  0 
by  3. 


1-4-4+10  +  16 

+  2-4-16 
1-2-8-   6 

+  2  +  0 
1  +  0-8 

+  2 
1  +  2 

+  3-18+12 
1-6+    4+    0 

+  3-   9 
1-3-    6 

+  3 
1+0 

The  required  equation  is 

The  required  equation  is 

^  +  2^-8^-6^+15  =  0. 

1^  —  6^  =  0. 

5.  Diminish  the  roots  of  the  equation  x*  +  x8  —  16 x2  — ^x  +  48 
=  0  by  4. 

1+    1-16-    .4  +  48)4 
+    4  +  20+    16  +  48 


1  + 

+ 

6+    4+    12  +  96 
4  +  36+160 

1  + 
+ 

9  +  40  +  172 
4  +  62 

1  +  13  +  92 
+    4 

1  +  17 

The  required  equation  is  y4  +  17  y3  + 

2y2  +  172y  +  96  =  0. 


teachers'  edition.  565 

6.  Diminish  the  roots  of  the  equation  x4  +  2x8  —  25  x2  —  26  x 
+  120=0  by  0.7. 

1+2-25       -         26+         120|0.7 

+  0.7+    1.89-16.177-29.5239 
1  +  2.7  -  23.11  -  42.177  +  90.4761 

+  0.7+    2.38-14.511 
1  +  3.4-20.73-56.688 

+  0.7+    2.87 
1  +  4.1-17.86 

+  0.7 
1  +  4.8 
The  required  equation  is 

y*  +  4.8 y*  -  17.86  jfl  -  56.688  y  +  90.4761  =  0. 

7.  Diminish  the  roots  of  the  equation  x4  —  x2  —  3  x  +  4  =  0  by  0.3. 

1  +  0     +1-3     +4|0.3 

+  0.3  +  0.09  +  0.327  -  0.8019 
1  +  0.3  +  1.09  -  2.673  +  3.1981 
'  + 0.3 +  0.18 +  0.381 
1  +  0.6  +  1.27  -  2.292 

+  0.3  +  0.27 
1  +  0.9  +  1.54 

+  0.3 
1  +  1.2 
The  required  equation  is 

y*  +  1.2y»  +  1.54  y2  -  2.292y  +  3.1981  =  0. 

#  8.  Diminish  the  roots  of  the  equation  x6  +  x4  +  3  xa  —  2x—  16  =  0 
by  0.5.  1+1+0       +3        _2  -16|0.5 

+  0.5  +  0.75  +  0.375  +  1.6875-   0.15625 


1  +  1.6  +  0.76  +  3.376  -  0.3125  -  16.16626 

0.5+1       +0.875  +  2.125 
1  +  2     +1.76  +  4.26    +1.8125 

+  0.5+1.25+1.50 
1  +  2.5  +  3       +  5.76 

+  0.5+1.5 
1  +  3     +4.6 

+  0.5 
1  +  3.5 
The  required  equation  is 

y6  +  3.6^  +  4.5y»  +  5.76y2  +  1.8125y  -  16.15626=  0. 


5G6  ALGEBRA. 

9.  Diminish  the  roots  of  the  equation 

a*-  3x* -  2x»  +  3x»-  7x  +  12  =  0  by  -  1. 
1-3-    2+    3-7  +  12J-1 
-1+    4-    2-1+    8 


1 

-4  + 

2  + 

1- 

-8  +  20 

-1  + 

6- 

7  +  6 

1 

-5  + 

7- 

6- 

-2 

-1  + 

6- 

13 

1 

-6+13- 

19 

-1  + 

7 

1 

-7  +20 

-1 

1-8 
The  required  equation  isy6  — 8y*  +  20y»  — 19y«— 2y +  20  =  0. 

10.  Diminish  the  roots  of  the  equation 

7fi  -x*  +  2x*-  3z*  +  4x*-r-  6x  +  6  =  0  by  0.2. 

1-1     +2      -3        +4  -6  +6|0.2 

+  0.2  -  0.16  +  0.368  -  0.5264  +  0.69472  -  0.861056 

1  -  0.8  +  1.84  -  2.632  +  3.4736  -  4.30528  +  5.138944 
+  0.2  -  0.12  +  0.344  -  0.4676  +  0.60320 


1  -  0.6  +  1.72  -  2.288  +  3.0160  - 

-3.70208 

+  0.2  -  0.08  +  0.328  -  0.3920 

1  -  0.4  +  1.64  -  1.960  +  2.624 

+  0.2  -  0.04  +  0.320 

1-0.2+1.6  -1.64 

+  0.2  +  0 

1  +  0  +1.6 

+  0.2 

1  +  0.2 
The  required  equation  is 
tf  +  0.2  tf  +  1.6 y*  -  1.64 y«  +  2.624  y*  -  3.70208y  +  6.138944  =  0. 


TEACHERS     EDITION. 


567 


Exercise 

1.  Find     the     two    commen- 
surable roots  of  the  equation 

a;4-4x8-8x  +  32=0. 
Try  4  ;  then  2. 

1-4  +  0-8  +  32 

+  4  +  0  +  0-32 

1+0+0-8        0 

+2+4+8 
1+2+4      0 
The  commensurable  roots  are 
2  and  4. 

2.  Find  the  one  commensurable 
root  of  the  equation 

x8-6x2+10x-8  =  0. 
Try  4.  * 

*    1-6  +  10-8 
+  4-   8  +  8 
1-2+    2      0 
The  commensurable  root  is  4. 

3.  Find    the    four    commen- 
surable roots  of  the  equation 

x*  +  2x8  -  7x2  -  8x  +  12  =  0. 


Try 

1  ;    then  2 

;    then   - 

-2; 

then  - 

-3. 

1  +  2- 

7- 

8+12 

+  1  + 

3- 

4-12 

1  +  3- 

4- 

12        0 

+  2+10+12 

1  +  6  + 

6 

0 

-2- 

6 

1  +  3 

0 

-3 

1      0 

The 

>  roots  are  1, 

2,  -2, 

and 

-3, 

CXLIV. 

4.  Find  the  one  commensurable 
root  of  the  equation 

x8  +  3x2-30x+36  =  0. 

Try  3. 

1  +  3-30  +  36 

+  3+12-36 

1  +  6-18        0 

The  commensurable  root  is  3. 

5.  Find  the  two  commensurable 
roots  of  the  equation 

x*-  12x«  f  32  x2  +  27x- 18  =  0. 

Try  6  ;  then  —  1. 

1-12  +  32  +  27-18 
+    6-36-24+18 

1-6-4+3'  0 
-1+7-3 

1-7+3        0 

The  commensurable  roots  are 
6  and  —  1. 

6.  Find  the  two  commensurable 
roots  of  the  equation 
x4-9x8+17x2  +  27x-60  =  0. 

Try  4  ;  then  6. 


1-9+17  +  27-60 
+  4-20-12  +  60 

1-6- 

+  6  + 

3  +  16 
0-16 

0 

1  +  0-    3        0 

The  commensurable  roots  are 
4  and  5. 


568 


ALGEBRA. 


7.  Find  the  five  commensurable  roots  of  the  equation 

x*  -  6x*  +  3x»  +  17  x2  -  28x  +  12  =  0. 
Try  1 ;  then  1 ;  then  2  ;  then  3  ;  then  —  2. 

1-6  +  3+17-28+12 
+  1-4-    1  +  16-12 

1-4-1  +  16-12  0 
+  1-3-   4+12 


1-3- 

+  2- 

-4+12 
-2-12 

1-1-6 
+  3  +  6 

0 

1  +  2 
-2 

0 

1      0 
The  roots  are  1,  1,  2,  3,  and  —  2. 

8.  Find  the  four  commensurable  roots  of  the  equation 

x*  -  10x8  +  35x2  -  50x  +  24  =  0. 
Try  1 ;  then  2 ;  then  3 ;  then  4. 


+ 

10  + 

1- 

35-50  + 
9  +  26- 

24 
24 

+ 

9  +  26-24 
2-14  +  24 

0 

+ 

7  +  12 
3-12 

0 

+ 

4 

4 

0 

0 

The  roots  are  1,  2,  3,  and  4. 

9.  Find  the  three  commensurable  roots  of  the  equation 

x* -  8x*  +  11x3  +  29 x2-36x  -46=0. 
Try  3 ;  then  6 ;  then  —  1. 

1-8+  11  +  29-36-45 
+  3-15-12  +  51  +  46 


1-6- 

+  6  + 

4+17  +  15       0 
0-20-15 

1  +  0- 
-1  + 

4-3        0 
1+    3 

1-1-    3        0 
The  commensurable  roots  are  3,  5,  and 


teachers'  edition.  569 

10.  Find  the  one  commensurable  root  of  the  equation 

x5  —  x4  —  6  x3  +  9  x2  +  x  —  4. 
Tryl.  1-1-6  +  9+1-4 

+1+0-6+3+4 
1+0-6+3+4      0 
The  commensurable  root  is  1. 


Exebcise   CXLV. 

1.  Compute  the  value  of  x4  —  6  x8  +  26  x2  —  4  x  +  7  when  x  =  6. 

-6  +  26-     4+      7|5 
+  5+    0+130  +  630 
+  0  +  26  +  126  +  637 
The  required  value  is  637. 

2.  Compute  the  value  of  x8  —  4  x2  +  6  x  —  22  when  x  =  —  7. 

-  4+    5-    22|-7 

-  7  +  77-574 
-11  +  82-596 

The  required  value  is  —  596. 

3.  Compute  the  value  of  x5  —  2x4  +  3X3  +  x2  —  28  when  x  =  2. 

-2  +  3  +  1+   0-28(2 
+  2  +  0  +  6+14  +  28 
0  +  3  +  7  +  14  +  0 
The  required  value  is  0. 

4.  Compute  the  value  of  x5  +  7  x8  —  2  x2  —  49  when  x  =  —  3. 

0+7-2+     0-   49|-3 

-  3  +    9-48+150-450 

-  3  +  16  -  50  +  150  -  499 
The  required  value  is  —  499. 

5.  Compute  the  value  of  x5  — 14  x8  +  473  when  x  =  6. 

0-14+     0+     0+   473(6 
+  6  +  36  +  132  +  792  +  4752 
+  6+  22  +  132  +  792  +  6225 
The 'required  value  is  5225. 


570  ALGEBRA. 

6..  Compute  the  value  of  x«  —  2x5  +  3x*  +  2x8  +  x2  —  7x  — 96 
when  x  =  —  2. 

-2+    3+    2+    1-    7-    96|— 2 
-2+    8-22  +  40-82+178 
-4+11-20  +  41-89+    82 
The  required  value  is  82. 

7.  Compute  the  value  of  x6  — x6  — 2x*  +  x8  — 6x  +  14  whenx=3. 

-1-2+    1+    0-     6+    14[3 
+  3  +  6+12  +  39+117  +  333 
+  2  +  4+13+39+111  +  347 
The  required  value  is  347. 

8.  Compute  the  value  of  x6  —  4x*  +  2  x2  —  7 x  +  16  when  x  =  10. 

-   4+    0+     2-       7+       16 1 10 
+  10  +  60  +  600  +  6020  +  60130 
+    6  +  60  +  602  +  6013  +  60146 
The  required  value  is  60,146. 

9.  Compute  the  value  of  x7  — x6  — 2X6  — 3x*  +  2x»  +  x2  — x  +  4 
when  x  =  —  2. 

-1-2-   3+    2+    1-    1+     4|-2 
-2  +  6-    8  +  22-48  +  94-186 
-3  +  4-11  +  24-47  +  93-182 
The  required  value  is  —  182. 

10.  Compute  the  value  ofx7  —  6x6  +  6x8  +  3x— 1  when  x  =  4. 

0-    6+    0+      6+     0+       3-         1(4 
+  4  +  16  +  44  +  176  +  728  +  2912  +  11660 
+  4  +  11  +  44  +  182  +  728  +  2915  +  11659 
The  required  value  is  11,669. 

Exercise  CXLVI. 

1.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  — x2— 2x+  1  =  0. 
If           x  =  -  2,  -  1, 0,  +  0.4,  +  0.5,  +  1,  +  2, 

/(x)  =  -  7,  +  1,  +  1,  +  0.104,  -  0.125,  -  1,  +  1. 
Hence  the  roots  are  —  1.  +,  0.4  +,  and  1.  +  . 


teachers'  edition.  571 

2.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  — 5x  — 3  =  0. 

If  x  =  -  2,  -  1  -  0.7,  -  0.6,  +  0,  +  1,  +  2,  +  3, 

f(x)  =  -  1,  +  1,  +  0.167,  -  0.216,  -  3,  -  7,  -  6,  +  9. 
Hence  the  roots  are  — 1.  +  ,  —  0.6+ ,  and  2.+. 

3.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  — 6x2  +  7  =  0. 

If  x= -2, -1,0,4-1,  +2, +  3, +  4, +  6, 

/(x)  =  -  21,  +  1,  4-  7,  4-  3,  -  5,  -  11,  -  9,  +  7. 

Hence  the  roots  are  —  1.4- ,  1.4-,  and  4.+. 

4.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8— 7x  +  7  =  0. 

If  x  =  -  4,  -  3,  4- 1,  +  1.3,  4- 1.4,  4- 1.6,  +  1.7, 

f(x)  =  -  29,  +  1,  4-  1,  +  0.097,  -  0.066,  -  0.104,  +  0.013. 
Hence  the  roots  are  —  3.  +,  1.34- ,  and  1.6+ . 

5.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  +  2xa  -  30x  +  39  =  0. 

If  x  =  -  8,  -  7,  +  1,  +  2,  +  3,  +  4, 

fix)  =  -  89,  +4,  +  12,  -  6,  -  6,  +  16. 

Hence  the  roote  are  —  7.  +  ,  1.  +  ,  and  3.+. 

6.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  — 6xa  +  3x  +  5  =  0. 

If  x=  -  1,-0.7,-0.6,  +  1,  +  2,  +  6,  +  6, 

f(x)  =  -  6,  -  0.383,  +  0.824,  +  3,  -  6,  -  6,  +  23. 

Hence  the  roote  are  —  0.6+ ,  1.+,  and  5.+. 

7.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  +  9  x2  +  21  x  +  17  =  0. 

If  x  =  -  6,  -  4,  -  3,  -  2,  -  1, 

f(x)  =  -  3,  +  1,  -  1,  -  3,  +  1. 

Hence  the  roots  are  —  4.+,  —  3.  +  ,  and  —  1.  +  . 


572  ALGEBRA. 

8.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x8  —  15x*  +  63x  —  50=  0. 

If  x  =  0,  +  1,  +2, +  6,  +  7, +  8, 

/(x)  =  -  50,  -  1,  +  24,  +  4,  -  1,  +  6. 

Hence  the  roots  are  1.  +  ,  6.  +  ,  and  7.+. 


9L  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x*  —  8x8  +  14x2  +  4x  —  8  =  0. 

If  x  =  -  1,  -  0.8,  -  0.7,  +  0.7,  +  0.8,  +  2,  +  3,  +  5,  +  6, 

/(x)  =  +  11,  +  2.2+,  -0.9+,  -0.8+,  +  0.4+,  +8,-5, 
-  13,  +  88. 

Hence  the  roots  are  —  0.7+,  0.7+,  2.  +  ,  and  5.  +  . 


10.  Determine  the  first  significant  figure  of  each  root  of  the  equa- 
tion x*  — 12x2+ 12x  — 3. 

If  x=  -  4,  -  3, 0,  +  0.4,  +  0.5,  +  0.6,  +  0.7,  +  2,  +  3, 

/(x)  =  +  13,  -66,  -  3,  -  0.09+ ,  +  0.06+ ,  0.0+ ,  -0.2, 
-11,  +  6. 

Hence  the  roots  are  —  3.  + ,  0.4+ ,  0.6+ ,  and  2.  + . 


teachers'  edition. 


573 


Exercise  CXLVII. 

1.  Compute  to  six  decimal  places  the    root    of   the    equation 
x8  +  10  x2  +  6x  —  120  which  lies  between  2  and  3. 


+  10 
+  2 
+  12 
+  2 
+  14 
+  2 
+  16 


+    0.8 
+  16.8 


+  0.03 
+  18.43 
+  0.03 
+  18.46 
+  0.03 
+  18.49 


18 


+  6 

+  24 
+  30 
+  28 
+  68 


+  13.44 
+  71.44 
+  14.08 

+  85.52 


+  0.8 
+  17.6 
+  0.8 

+  18.4 

+  0.5529 
+  86.0729 
+  0.5538 
+  86.6267 

+  86.627 
+  0.055 
+  86.682 
+  0.055 

+  86.837 


+  86.84 
+  86.8 
+  87 


-12012.833066+ 
+    60 
-    60 


+  67.152 

-  2.848 

+  2.682187 

-  0.266813 

+  0.260046 

-  0.005767 
""+  0.005208 

-  0.000559 
+  0.000522 

-  0.000037 


J 


674 


ALGEBRA. 


2.  Compute  to  six  decimal    places    the  root    of   the   equation 
z ■*  +  x2  +  x  —  100  =  0  which  lies  between  4  and  5. 


+  1 
+_§ 
+  6 
+__§ 
+    9 

+  J 
+  13 


+  0.2 
+  13.2 
+  0.2 
+  13.4 
+  0.2 
+  13.6 


+  0.06 
+  13.66 
+  0.06 
+  13.72 
+  0.06 
+  13.78 


+  14 


+  1 
+  20 

+  21 
+  36 

+  67 


+  2.64 
+  69.64 
+  2.68 
+  62.32 


64.08 

64.1 

64 


-10014.264429  + 
+  84 
-  16 


+  11.928 
-  4.072 


+  3.788376 
-  0.283624 


+   0.256076 
-  0.027548 


+  0.8196 
+  63.1396 
+  0.8232 
+  63.9628 

+ 
+ 
+ 

0.025632 
0.001916 
0.001282 
0.000634 
0.000576 
0.000058 

+  63.963 
+  0.056 
+  64.019 
+  0.056 
+  64.075 

teachers'  edition. 


575 


3.    Compute  to  six  decimal    places  the  root  of   the   equation 
x4  —  2  x3  +  21  x  —  23  =  0  which  lies  between  1  and  2. 


—  2 

+  1 

-1 

+  1 

0 

+_! 

+  i 
+  i 

+  2 


+  0.1 
+  2.1 
+  0.1 
+  2.2 
+  0.1 
+  2.3 
+  0.1 
+  2.4 


+  0.05 
+  2.45 
+  0.05 
+  2.50 
+  0.05 
+  2.55 
+  0.05 
+  2.60 


+  0 
-1 
-1 

+  0 

-1 

+  1 

0 


+  0.21 
+  0.21 
+  0.22 
+  0.43 
+  0.23 
+  0.66 


+  0.1225 
+  0.7825 
+  0.1250 
+  0.9075 
+  0.1275 
+  1.0350 


+  1.04 


+  1. 


+  0.021 
+  19.021 
+  0.043 
+  19.064 


+  0.039125 
+  19.103125 
+  0.045375 
+  19.151500 


+  19. 
+  0. 


15150 
00728 


+  19. 
+  0. 


15878 
00728 


+  19.16606 


+  19. 
+  0. 


1661 
0004 


+  19. 
+  0. 


1665 
0004 


+  19.1669 
+  19.167 


-23 1 1.157450  + 
+  20 
-  3 


+  1.9021 
-  1.0979 


+  0.95515625 
-  0.14274375 


+  0.13411146 
-  0.00863229 


+  0.00766660 
-  0.00096569 


+  0.00095835 


-  0.00000734 


576 


ALGEBRA. 


4  Compute  to  six  decimal  places  the  root  of  the  equation  x*  —  ox8 
+  3  x*  +  36  x  —  70  =  0  which  lies  between  2  and  3. 


-  6 

+  2 

-  3 
+_2 

-  1 
+_2 

+  1 
+  2 
+  3 


+  0.6 

+  3.6 

+  0.6 

+  4.2 

4-  0.6 

+  4.8 

+  0.6 

+  6.4 


4-  0.04 
+  6.44 
+  0.04 
+  6.48 
+  0.04 
6.62 
0.04 


+ 
+ 


4-  6.66 
4-10 


+  3 
-6 
-3 
-2 
-6 
4-2 

-3 


4-2.16 
-0.84 
4-2.62 
4-1.68 
+  2.88 
+  4.56 


4-  0.2176 
4-4.7776 
4-0.2192 
4-4.9968 
4-  0.2208 
4-  6.2176 


4-6.22 
+  0.05 
+  6.27 
+  0.06 
+  6.32 
+  0.06 
+  5.37 
+  6 


+  36 
-  6 
+  29 
-10 
+  19 


-  0.604 
+  18.496 
+  1.008 
+  19.604 


+  0.191104 
+  19.695104 
+  0.199872 
+  19.894976 


+  19.89498 
+  0.02635 
+  19.92133 
+  0.02660 
+  19.94793 


+  19.9479 
+  0.0036 
+  19.9614 
+  0.0035 
+  19.9649 
+  19.966 
+ 19.95 


-  7012.645751+ 

+  58 

-12 


+  11.0976 
-  0.9024 


+  0.78780416 
-  0.11459584 


+  0.09960665 
-  0.01498919 


+  0.01396598 

-  0.00102321 
+  0.00099775 

-  0.00002646 
+  0.00001995 

-  0.00000661 


TEACHEB8'    EDITION. 


577 


5.    Compute  to  six  decimal   places  the  root  of   the  equation 
x4  —  12  x2  +  12  x  —  3  which  lies  between  —  3  and  —  4. 


0       - 

12       -  12 

-  3 1 3.907378- 

h 

+  3       + 
+  3       - 

9       -  9 
3        -  21 

-63 
-66 

+  3       + 
+  6       + 

18 
15 

+  45 
+  24 

+  66.0241 
-  0.9769 

+  3 
+  9 

+ 

27 
42 

+  48.249 
+  72.249 

-  0.97590000 
+  0.92662260 

+  3 

+  12 

+ 
+ 

11.61 

+  69.427 

-  0.05027740 

63.61 

+  131.676 

+  0.03984378 

+  0.9 
+  12.9 
+  0.9 

+  13.8 

+ 
+ 
+ 
+ 

12.42 
66.03 
13.23 
79.26 

+  131.67600 
+   0.55580 
+  132.23180 
+   0.55678 

-  0.01043362 
+  0.00929908 

-  0.00113454 
+  0.00106280 

+  0.9 

+ 
+ 
+ 
+ 

79.26 

+  132.78858 

-  0.00007174 

+  14.7 
+  0.9 
+  15.6 

0.14 
79.40 

0.14 
79.64* 

0.14 
79.68 

+  132.7886 
+   0.0240 
+  132.8126 

+  16.60 
20 

+ 
+ 
+ 

+   0.0240 
+  132.8366 
+  132.837 

+  80 
+  100 

+   0.007 

+ 132.844 

+   0.007 

132.851 

132.85 

378 


ALGEBRA. 


6.    Compute  to  six  places  of  decimals  the  root  of  the  equation 
x5  +  2  x4  +  3  x8  +  4  x-  +  5x  —  54321  =  0  which  lies  between  8  and  .9. 


+  2 

+  8 
+  10 
+  8 
+  18 
+_8 
+26 
+  8 
+  34 
+  _8 
+42_ 
+  0.4 
+42.4 
+  0.4 
+42.8 
+  0.4 
+  43.2 
+  0.4 
+43.6 
+  0.4 


3 

80 


+  83 
+  144 
+  227 
+  208 
+  435 
+272 
+  707 


+44.0 


+  4 
+  664 
+  668 
+  1816 

+  2484 
+  3480 
+  5U04 


+   16.90 


+  723.96 
+   17.12 

+  741.08 
+   17.28 


+  289^584 
+  6263.584 
+  296.432 
+6550.016 
+  303.344 

+6853.360 


+  758.30 
+   17.44 


+  775.80 


+6853.4 
+       7.8 


+6861.2 
+       7.8 


+  780 


+6869.0 

+ 7JS 

+6876.8 


+6880 
+  7000 


+         5 
+  5344 

+  6349 
+  19872 

+25221 


+  2501.4336 
+  27722.4336 
+  2620.0064 
+30342.4400 


+30342.440 

+ 68.612 

+30411.052 
+  68.690 
+30479.742 


+30479.74 
+       27.52 


+30507.26 
+       27.62 


+30534.78 


+30534.8 

+ 2JS 

+30537.6 

+ 2JJ 

+30540.4 
+30540 


-5432118.414454  + 

+42792 

—  11529 


+  11088.97344 
—     440.02656 


+ 


+  304.11052 
-  135.91604 


+  122.02904 
= 13.88700 


12.21504 


-    1.67196 


.  1.62700 


0.14496 
0.12216 


-    0.02280 


TEACHERS7    EDITION. 


579 


7.  Compute  to  six  places  of  decimals  the  root  of  the  equation 
x4  —  69  x2  +  840  =  0,  which  lies  between  4  and  6. 


0 
+    4 

+   4 

±_i 
+  8 
+  4 
+  12 
+  4 
+  16 


+  0.8 
+  16.8 
+  0.8 
+  17.6 
+  0.8 
+  18.4 
+  0.8 
+  19.2 


+  0.09 
+  19.29 
+  0.09 
+  19.38 
+  0.09 
+ 19.47 
+  0.09 
+ 19.56 


+  20 


-  59 
+  16 

-  43 

+  32 

-  11 

+  48 

+  37 


+  13.44 

+  60.44 

+  14.08 

+  64.62 

+  14.72 

+  79.24 


+  1.7361 

+  80.9761 

+  1.7442 

+  82.7203 

+  1.7523 

+  84.4726 


+ 
+ 


84.47 
0.16 


+  84.63 
+  0.16 


84.79 
0.16 


+  84.95 


+  85. 
+  100. 


0 
-172 
-172 
-  44 
-216 


+  40.352 
- 175.648 
+  51.616 
- 124.032 


+  7.287849 
- 116.744151 
+  7.444827 
- 109.299324 


- 109.29932 
+  0.67704 
- 108.62228 
+  0.67832 
- 107.94396 


- 107.9440 
+  0.0765 
-107.8675 
+  0.0765 
-  107.7910 


-107.791 
+  0.008 
-107.783 
+  0.008 
- 107.775 
- 107.78 


+  84014.898989+ 

-688 

+  162 


- 140.5184 
+  11.4816 


-  10.50697359 
+   0.97462641 


-  0.86897824 
+  0.10664817 


-  0.00097002 


+  0.00005259 


-  0.09708075 
+   0.00856742 


-  0.00754481 
+   0.00102261 


580 


ALGEBRA. 


8.  Compute  to  six  places  of  decimals  the  real  root  of  the  equation 
z*  -  36499  =  0.  » 


0              0- 

36499|32.865378+ 

+  30          +900 
+  30          +900 

+  27000 
-  8499 

+  30 

+  60 

+  1800 
+  2700 

+ 

5768 
2731 

i 

+  30 
+  90 

+  184 
+  2884 

+ 

2519.552     1 
211  448 

+   2 
+  92 

+  loo 
+  3072 

+ 

194.005666    « 
17.442344    \ 

+   2 

+  94 
+   2 

+   77.44 
+  3149.44 

4-    7ft  Oft 

+ 

16.199170 
1.243174 

+  96 

+  3227.52 

+ 

0.972108 

+   0.8 

4-     A  QA7« 

— 

0.271066 

+  96.8 
+   0.8 
+  97.6 
+   0.8 

+  98.4 

+  3233.4270 
+    5.9112 
+  2339.3388 

+ 
+ 

0.240828 
0.030238 
0.027520 

+  3239.339 
+   0.495. 

0.002718 

+   0.06 
+  98.46 
+   0.06 

+  3239.834 
+    0.495 
+  3240.329 

+  98.52 
+   0.06 
+  98.68 

+  3240.33 
+   0.03 
+  3240.36 

+  99 
+  100 

+   0.03 
+  3440.39 

+  3440.4 
+  3440 

TEACHERS'    EDITION. 


581 


9.  Compute  to  six  decimal  places  the  real  root  of  the  equation 
x8- 242970624  =  0. 


0 
600 


+  600 
+  600 
+  1200 


+  360000 
+  360000 
+  720000 
+  1080000 


-242970624[624 
+  216000000 
-   26970624 


+    22328000 
-     4642624 


-t-  6UU 

+   36400 
+  1116400 

+  4642624 

o 

+  1800 

+   20 
+  1820 
+   20 
+  1840 
+   20 

+  1860 

+  36800 
+  1163200 

+   7456 
+  1160656 

+  1864 


10.  Compute  to  six  decimal  places  the  positive  real  root  of  the 
equation  x*  —  707281  =  0. 


0 

+  20 


+    9 
+  89 


0 
400 


0 
8000 


-707281|29 
+  160000 


+  20 

+  400 
+  800 
+  1200 
+  1200 
+  2400 

+  8000 
+  24000 
+  32000 

-  647281 

+  20 
+  40 

-647281 
0 

+  20 
+  60 

+  28809 
+  60809 

+  20 
+  80 

+  801 
+  3201 

582 


ALGEBRA. 


11.  Compute  to  six  places  of  decimals  the  real  root  of  the  equation 
&  -  147008443  =  0. 


0 
+  40 
+  40 

0 
+  1600 
+  1600 
+  3200 
+  4800 
+  4800 
+  9600 
+  6400 
+  16000 

0 
+  64000 
+  64000 
+  192000 
+  256000 
+  384000 
+  640000 

0 
+  2560000 
+  2660000 
+  10240000 
+  12800000 

-147008443143 
+  102400000 
-  44608443 

+  40 
+  80 

+  44608443 
0 

+  40 

±l5X 

+  2069481 
+  14869481 

+  40 
+  160 

+  49827 
+  689827 

+  40 
+  200 
+   3 

+   609 
+  16609 

+  203 


TEACHERS7    EDITION. 


583 


12.  Compute  to  six  places  of  decimals  the  positive  root  of  the 
equation  x2  —  551791  =  0. 


0 
700 


+  700 
+  700 
+  1400 


+ 40 

+  1440 

+ 40 

+  1480 


+ 2 

+  1482 

+ 2 

+  1484 


+ 08 

+  1484.8 

+ 08 

+  1485.6 


0.02 


+  1485.62 
+  0.02 
+  1485.64 


0.006 


+  1485.646 
+  0.006 
+  1485.652 


+  1485.65 
+  1495.7 
+  1486 


-5517911742.826359  + 
+  490000  ~ 
-  61791 


+  57600 
-   4191 


+   2964 
-   1227 


1187. 


39. 


84 
16 


29.7124 


9.4476 


8.913876 


0.533724 


0.445695 


088029 
074285 


013744 
013374 


0.000370 


584 


ALGEBRA. 


'  13.  Compute  to  six  places  of  decimals  the  root  of  the  equation 
z2  —  17  x  +  70.3  =  0  which  lies  between  7  and  8. 


-17 

+  7 
-10 
+  7 
-   3 


+  0.1 

-  2.9 
+  0.1 

-  2.8 


-  2.793 

-  2.79 


+    0.003 

-   2.797 

+    0.003 

-   2.794 

+    0.0005 

-   2.7936 

+    0.0006 

-   2.7930 

+  70.3|7.103575+ 

-70.    

+   0.3 


-   0.29 
+    0.01 


-   0.008391 
+    0.001609 


-  0.00139675 
+  0.00021225 

-  0.00019551 
+  0.00001674 

-  0.00001395 
+  0.00000279 


teachers'  edition. 


5S5 


14.  Compute  to  six  places  of  decimals  the  root  of  the  equation 
x8  +  9  x2  +  24  x  +  17  =  0  which  lies  between  —  4  and  —  6. 


-9              +24             - 

17|4.532088+ 

+  4             — 
-5              + 

20 
4 

+  16 
-  1 

+  4 
—  1 

4 

0 

+ 

0.875 
0.125 

1 

+  4 
+  3 

+ 

+ 

1.75 
1.76 

+ 

0.116577   1 
0.008423   1 

X  ft  Fl 

2.00 
3.75 

"T  U.O 

+  3.5 

+ 

+ 

+ 

0.008063768  i 
0.000359232  | 

+  0.5 

+  4.0 
+  0.5 
+  4.5 

T 
+ 
+ 
+ 

0.  Ioo9 
3.8859 
0.1368 
4.0227 

+ 
+ 

0.000323288 
0.000035944 
0.000032328 
0.000003616 

+  0.03 
+  4.53 
+  0.03 
+  4.66 

+ 
+ 
+ 
+ 

0.009184 
4.031884 
0.009188 
4.041072 

+  O.Od 
+  4.69 

+ 
+ 

4.0411 
4.041 

+  0.002 
+  4.692 
+  0.002 
+  4.694 
+  0.002 
+  4.596 

586 


ALGEBRA. 


16.  Compute  to  six  places  of  decimals  the  root  of  the  equation 
x4  —  8x8  +  14xa  +  4x  —  8  =  0  which  lies  between  0  and  —  1. 


+    8 
+    0.7 

+    8.7 
+    0.7 

+    9.4 
+    0.7 

+  10.1 

+    0.7 

+  10.8 

+    0.03 

+  10.83 
+    0.03 

+  10.86 
+    0.03 

+  10.89 
+    0.03 

+  10.92 

+  10 


+  14 
+  0.09 
+  20.09 
+  6.58 
+  26.67 
+  _7J07 
+  33.74 


+  0.3249 
4-  34.0649 
+  0.3258 
+  34.3907 
+  0.3267 
+  34.7174 


+  34.72 
+  0.02 
+  34.74 
+  0.02 
+  34.76 
+  0.02 
+  34.78 


-  4 
+  14. 


+  10. 
+  18. 


,063 
063 


+  28.732 


+  1.021947 


+  29. 
+  1. 


753947 
031721 


+  30.785668 


+  80. 
+  0. 


78567 
06948 


+  30. 
+  0. 


85615 
06952 


+  30.92467 


+  20. 
+  30. 


926 
9 


-810.7320508+ 
+  7.0441     I 
-  0.9559     i 


+  0.89261841 
-  0.06328159 


+  0.06171030 
—  0.00157129 


+  0.00154625 

-  0.00002504 
+  0.00002472 

-  0.00000032 


teachers'  edition.  587 


Exercise  CXLVIII. 

1.  Solve  x*  +  7x«-7x-l  =  0. 

x*+7x*-7x-l  =  0, 
(x2-l)(x2+7x+l)  =  0. 


-7±3V5 
.-.«=  ±1,  or - 


2.  Solve  x4  +  x8  +  x2  +  x  +  1  =  0. 


x*  +  x*  +  x2  +  x  +  1  =  0, 
x2  +  x  +  l  +  i+-  =  0, 

X        X2  ' 

(*a+J0+(*+;)+1=o. 

(•+i),+(»+i)-1=a 


.  i     -i±Vs 


(1)  *»+i-5^x+l  =  0, 


(2)  x«  +  i^x+l  =  0, 


5=^pi±*V-2V6-10. 


,=  -V5-l±iV2Vg_10 


3.  Solve  s«-3!B»+5iB»-5s»  +  3x-l  =  0. 

afi  -  3x*  +  5x«  -  6x»  +  3*  -  1  =  0, 
(x»  -  1)  (as*  -  3x»  +  6x*  -  3x  +  1)  =  0. 

.-.  x  =  ±  1, 
or  x*  — 8x«  +  6x2  — 3x+l  =  0, 

**-3x  +  6-|+±  =  o, 


588  ALGEBRA. 


(x+l)»_2_3(x+i)  +  6  =  0j 
(x+I)'_3(x+I)  +  4  =  0. 


.   1      Z±^r7 

,x+-  = 

x  2 


(1)      *»-*±^x  +  l  =  0, 


:=3  +  VE7±  jVeV^-w. 

4 


x-h  1  = 


(2)      x2 x+l  =  0, 


g=3_V-7±i  V-6V^7-14. 


4.  Solve  x*  -  5x8  +  6x2  —  5x  +  1  =  0. 
x*  —  5  x8  +  6  x2  -  5  x  +  1  =  0, 
x2-5x  +  6--  +  -  =  0, 

X       X2  ' 

(x2+i.)_5(x+l)+6=0, 

(x  +  l)4_2_6(x+i)  +  6  =  0, 
(x  +  I)4_6(x+i)  +  4  =  0. 

.-.  x  +  -  =  1,  or  4. 

x       ' 

(1)  x2-x+l  =  0, 

1±VI:3 
X  =  — 1 

(2)  x2-4x+l  =  0, 

x=2±V3. 


teachers'  edition. 


589 


5.  Solve    2x4-6x3  +  6x2-5x  +  2  =  0. 
2x*-5x8  +  6x2-6x  +  2  =  0, 

2x2-5x+6--  +  -|  =  0, 

X        X2  ' 

»(a?+s)-6(«+i)+8=a- 

a(x+ly_4-6(*+y+6=0, 

2  (x  +  ~x \2  -  5  (x  +  i  )  +  2  =  0. 


(1) 
(2) 


.-.  x  +  -  =  2,  or  £. 

x        '       * 

xa-2x+l=0, 
x=l. 

x2-£x  +  l  =  0, 


6.  Solve x6  — 4x*  +  x»  +  x2-4x+l  =  0. 

x6  —  4x4  +  x*  +  x2  -  4x  +  1  =  0, 
(x  +  1)  (x*  -  6x8  +  6x2  -  6x  +  1)  =  0. 

.-.x=-l, 
r  x4  — 6x8  +  6x2  — 6x+l  =  0, 

*-6x  +  6-|  +  ±  =  0, 
x2  +  l2_5(x+I)  +  6  =  0) 

(x  +  l)2-5(x  +  i)+4  =  0. 


(1) 
(2) 


.-.  x  +  -  =  1,  or  4. 

x       ' 

x2  -  x  +  1  =  0, 

*  =  — 

---4x4-1  =  0, 

x  =  2  ±  V3. 


500  ALGEBRA. 


7.  Solve  x*  -  10x8  +  26 x*  -  lOx  +  1  =  0. 
a-*-  10x8  +  26x*-10x+l  =  0, 

x2-10x  +  26-  —  +-^  =  0, 
x      x2 

(xJ  +  la)_10(a!+I)  +  26=0) 

.-.  x  +  -  =  4  or  6. 

x 

(1)  x2-4x+l  =  0, 

x  =  2  ±  V5. 

(2)  x2-6x+l  =  0, 

x  =  3±2V2. 

a  Solve  x8  +  mx2  +  mx  +  1  =  0. 

x8  +  mx2  +  mx  +  1  =  0, 
(x  +  1)  {x2  +  (m  -  1)  x  +  1}  =  0. 

.-.  x=  —  1, 
or  x2+  (m  — l)x+l  =  0, 


1  —  m  ±  Vto2  —  2  m  —  3 

x= 


9.  Solve  x5  + 1  =  0. 

x6+l  =  0, 
(x  +  1)  (x*-  x8  +  x«-  x+  1)  =  0. 

.-.x=-l, 

r  x*  —  x8  +  x2  —  x  +  1  =  0, 

(*+£)-(*+i)  +  »=«. 

,  i    i±Vs 

.-.  x+-  =  — 

x  2 

(1)  x2-1-^x  +  l  =  0, 


x  = 


591 


(2)  x2-^-~x+l  =  0, 


x=  *    A        ±  4-V-2V5-10. 
4  * 


10.  Solve  3x*-2x*  +  5x8-5x2  +  2x-3  =  0. 

3x6  _  2a;4  +  6x8  -  5x2  +  2x  -  3  =  0, 

(x-  1)  (3x*  +  x8  4-  6x2  +  x  +  3)  =  0. 

.-.x=l, 

r  3x*  +  x8  +  6x2  +  x  +  3  =  0, 

3x2  +  x+6  +  l  +  -  =  0, 
X      xa 

■(•+s),+  ('+i)  =  °- 

.-.  x  +  -  =  0  or  -  i. 
x 

(1)  X2  +  1  =  0, 

X  =  ±  V17!. 

(2)  x2  +  *x+l  =  0, 


■  1  ±  V-  35 
6 


Exercise   CXLIX. 


1.  Solve  11*  =346. 

2.  Solve  3*  =  10. 

11*  =346, 

3*  =  10, 

x  log  11  =  log  346, 

x  log  3  =  log  10, 

log  346 
log  11 

log  10 
log  3 

2.6391 

1.0000 

1.0414 

0.4771 

=  2.438+. 

=  2.096  + 

592  ALGEBRA. 


3.  Solve  10*  =745. 

a  Solve  146*=  12984. 

10*  =745, 

146*=  12984, 

x  log  10  =  log  745, 
lo?745 
x      log  10 

2.8722 

x  log  146  =  log  12984, 
_  log  12984 
X        log  146 
4.1134 

""1.0000 

2.1644 

=  2.8722+. 

=  1.900+. 
7.  Solve  0.2*  =  0.4 

4.  Solve  7*  =  324. 

0.2*=  0.4, 

7*=  324, 

xlog0.2  =  log0.4, 

x  log  7  =  log  324, 

log  0.4 

log  324 

X     log  0.2 

X"   log7 

9.6021  —  10 

2.5105 

"  9.3010  —  10 

0.8451 

0.3979 

=  2.970  +. 

"0.6990 
=  0.669+. 

5.  Solve  4*  =3.74. 

a  Solve  14.74*  =  8.64. 

4*=  3.74, 

14.74*  =  8.64, 

x  log  4  =  log  3.74, 

x  log  14.74  =  log  8.64, 

log  8.74 
log  4 

log  8.64 

X  "  log  14.74 

0.6729 

0.9365 

""0.6021 

1.1685 

=  0.951+. 

=  0.801  +. 

9.  Solve  x*  =  2.767. 

x*  = 

2.767, 

x  log  X  = 

log  2.1 

'67 

0.4420 

1  log  1  = 

1X0 

=  0, 

1.7  log  1.7  = 

:  1.7  X  0.2304  =  0.3917, 

1.77  log  1.77  = 

:  1.77X0.2480  =  0.4390, 

1.774  log  1.774  = 

:  1.774X0.2490  =  0.4417. 

.-.  x  = 

:  1.774 +. 

teachers'  edition. 


593 


10.  Solves*  =  23.10. 

xx  =  23.10, 

x  log  x  =  log  23.10 

=  1.3636. 

2  log  2  =  2  X  0.3010  = 

0.6020, 

2.9  log  2.9  =  2.9  X  0.4624 

=  1.3410, 

2.92  log  2.92  =  2.92  X  0.4664  =  1.3590, 

2.925  log  2.926  =  2.925  X  0.4661  =  1.3636. 

.-.x=  2.925+. 

11.  Given  P=750, 4=1797.42, 
r  =  6%;  find*. 

f  _  log  A  -  log  P 
log  (1  4-  r) 

4.3383  -  3.7506 

0.0294 
0.5877 
0.0294 
=  20. 

3.2547  -  2.8751 

0.0253 
0.3796 


0.0253 
=  15. 


12.  Given  P=  780,4=1559.! 
r=8%;  find*. 

.log  A  —  log  P 

log  (1  +  r) 
_  3.1926 -2.8921 
~         0.0384 

0.3005 
~~  0.0334 
=  9. 


13.  Given  P  =  5630.76,   A- 

21789.22,  r=  7%;  find  t 

.log  A  —  log  P 
log  (1  +  r) 


14.  Given  P  =  300,  A-  615.46, 
r=7%;  find*. 

.  _  log  A  —  log  P 
log  (1  +  r) 
_  2.7122 -2.4771 
"  0.0294 

_  0.2351 
0.0294 
=  8. 


15.  Given  P =84.65, 4=289.47 
r=7±%;  find*. 

log  4  —  log  P 

log  (1  +  r) 
2.4616  - 1.9276 


<  =  • 


0.0314 
0.5340 
~~  0.0314 
=  17. 


594  ALGEBRA. 


Exercise  CL. 

1.  x8  +  12x2  +  45x  +  50  =  0.  +7—49 

Letx  =  y-4.  (Cf.  §  600.)  *-   7+    12 

1  +  12 +  45 +50^  -j^ 

1+    8+13-    2  *  T       y 

-  4  -  16  p  =  12,  q  =  -  63. 


1  +  4- 
-4 


2=  VAf±V64  +  i-9^ 

8i : 


l  +  o  =  V-6/  ±  -*$■ 

..yS-3y-2  =  0.  =  -lor4, 

p=-B,g=-2.  *-2~"3^~3 

=  -1  +  4+7  or  4  —  1+7 


-«/-»«*>/s+$ 


=  10. 


=  1. 

p       m 


=  ^1±V^1  +  1  1-21  +  159-490[10 

31          1T  +10-110 
1-11+   49 

x=z~3?~3  x2-llx  +  49  =  0. 

=  1  +  1-4  llisV^ 

=  -2.  .'.x=10,  or  g 

Divide  the  given  equation  by  ^  x8  — 6x2+ 13x  — 10. 
x  +  2. 


1  +  12  +  45  +  60^2  176  +  13710L2 

1+10  +  25 


Let  x  =  y  +  2. 
1-6  +  13 
+  2-    8+10 


1-4+    5+    0 
x2+10x  +  25  =  0,  +2—   4 

(3+5)2  =  0.  1-2+    1 

.-.x=-2,  -5,  -5.  +2 

1  +  0 
2.  x8  -  21  x2+159x- 490  =  0.  ...  ^8  +  y  =  o. 

Let  x  =  y  +  7.  y  (y2  +  i)  -  o, 

l-21  +  159-490[+7  y  =  0,  oriV^l, 

+    7-98  +  427  x=y+2 


1-14+61-63  =  2,  or  2  ±  V--1. 


TEACHERS'    EDITION. 


595 


4.  x8+3x2  +  9x-13  =  0. 

Let  z  =  y  —  1. 

l  +  3  +  9-13|-l 

-1-2-   7 
1  +  2  +  7-20 

-1-1 
1  +  1  +  6 

-1 
1  +  0 
.-.  y*  +  6y  —  20=0. 
p  =  6,  q  =  —  20. 

2  =  VlO  ±  V8  +  100 


=  Vio±eV§ 

=  1±V3, 

p___m 

X~Z      32      3 

=  1  +  V3 ^"p-1 

1  +  V3 

=  1  +  V3+1-V3-1 

=  1. 

1  +  3+    9-1311 

+  1+    4  +  13 

1  +  4  +  13       0 

x2+4z+13  =  0. 

.-.x=l,  or  -2±3V=T. 

5.  y8  +  48y+  504  =  0. 
p  =  48,  g  =  504. 

2=  V-  252  ±  V4096  +  6350^ 


=  V-  252  ±  260 

=  2, 


y  =  2  — 


3« 


1  +  0  +  48  +  5041-6 

-6  +  36-504 
1-6  +  84 
y2-6y  +  84  =  0. 
.y=-6,  orSisV1^. 


6.  2/8-  21  y-  344  =  0. 
p  =  -  21,  g  =  -  344. 
z  =  Vl72  ±  V343  +  29584 


=  V172  ±  173 

V 
y  =  Z~Sz 

=  1  +  7 
=  8. 

1  +  0-21-344  [8 

+  8  +  64  +  344 
1  +  8  +  43 

y2  +8y  +  43  =  0. 
.•.y  =  8,  or  —  4±3V^3. 

7.  y8~3y  +  2  =  0. 
p=-3,  g  =  2. 


3r 


=  2-8 


2=  V- 1  ±  V- 1  +  1 

=  -1, 

p 
y  =  Z-S~z 

=  -1-1 

=  -2. 

1  +  0-3  +  21-2 

-2  +  4 
1-2+1 

2/2-2y+  1-0. 
.-.y=-2,  1,  1. 


596 


ALGEBRA. 


a  y»-60y  + 071  =  0. 
p  =  -  60,  q  =  671. 

2=  V—  4  Ji±V—  8000  +15J^tl 


"tf 


*/-C71±llV41 


_-ii±Vii 

2 f 

*  =  *-£ 

_-ii  +  Vii  . 


40 


=  -ll  +  V41      11  +  V41 
2  2 

=  —  11. 

1+0-    60  +  6711-11 

-11  +  121-671 
1-11+    61 

ya—  lly  +  61  =  0. 


-11+V41       -y=-^or 


11  ±  V-  123 


Exercise   CLI. 


1.    x8  +  3x-6  =  0. 
p  =  3,  g=-6. 

=  -  0.4000. 
log  tan     e  =  9.60206  (n), 
0=158°  11' 65", 
i  0=79°  6' 58". 
log  tan  ie=  10.71539, 
log  tan  \<p  =  i  log  tan  *  0 
=  10.23846, 
i0  =  59°59'39", 
0  =  119°  51'  18", 
cot  0  =  -  0.57709, 
csc0=  1.1546, 


31  = 


COt0 


Ip         ,    i — 

X2  =  ^  £  COt  0  +  V —  p  CSC  0 

=  -0.57709+  1.9998  V^. 
Xg  =  -y  g  cot  0  —  V^p  esc  0 
=  -  0.57709  -  1.9998  V1^!. 

2.    x*+7x  +  3. 

p=7,g  =  3. 

tan  6- 


sqyls 

14     ll 


=  -1.16418. 


log  14=    1.14613 
logV7=    0.42255 
colog     9=   9.04576  —  10 
colog  V3  =    9.76144—10 
log  tan     0  =  10.37688 


TEACHERS'    EDITION. 


597 


0  =  67°  10'  36", 

1  =33°  35'  18", 

2  , 

log  tan     |=9.82223, 

<t> 
log  tan    |=ilogtan£0 

=  6.94074, 

£=41°  6'  12", 

0  =  82°  12'  24", 
cot  0  =  0.13686, 
esc  0=1.0093, 

log     2  =  0.30103 

log  Vp=  0.42265 
colog  V3=  9.76144  -10 
log  cot  0  =  9.13629 
log  Xi  =  9.62131  -  10 

xx  =  -  0.41813. 
x2  =  -v?  cot  0  +  V—  p  CSC  0. 

log  Vp  =  0.42255 

log  esc  0  =  0.00403 

0.42658 

X2  =  0.20906  +  2.6704  V^I, 
x8  =  0.20906  -  2.6704  V^. 

3.    x«- 7x+ll  =  0. 
33\3 


log    14=1.14613 
log  V7  =  0.42255 
colog  33=8.48149-10 
colog  V3  =  9.76144 -10 
log  sind  =9.81161 

0=40°  23'  40", 
i  =  20°  11'  50", 

log  tan  £=9.56570, 

0  ,  0 

log  tan -=*  log  tan  - 
=  9.85523, 
£  =  35°  37'  21", 
0  =  71°  14'  42". 
log  cot  0  =  9.53091, 
log  esc  0  =  0.02369. 


*=-Vi 


CSC0. 


sin  0- 


log    2  =  0.30103 

logVp  =  0.42255 
colog  V3  =  9.76144  -10 
log  esc  0  =  0.02369 
log  xi  =  0.50871 

Xx  =  -  3.2264. 

«2  =  \/|  CSC  0  +  V— p  COt  0. 

logVp  =  0.42255 

log  cot  0  =  9.53091 

9.95346 

Xa  =  1.6132 +  0.89838  i, 
x8  =  1.6132  -  0.89838  i. 


598 


ALGEBRA. 


4  x«-4x-5=0. 


»•(£<?> 


sin0  = 

2|>      J> 
3g  \3 

16 

15  V§ 

log  16  = 

1.20412 

colog  15  = 

8.82391 

colog  V3  = 

9.76144  - 

10 

log  sin  6  = 

9.78947- 

10 

0  = 

38°  0/  48" 

» 

0  _ 

19°  0/  24" 

. 

log  tan  -  =  9.53713, 

log  tan  |  =  - log  tan  | 

=  9.84571, 

£  =  36°1'48", 

<t>  =  70°  3'  36". 

log  cot  0  =  9.55965, 
log  esc  0  =  0.02685. 

*i=  ~ 2  \f  csc  *  =  ~  Taj086*" 

log  4  =  0.60206 
colog  V3  =  9.76144  -10 
log  csc  4>  =  0.02685 
log  Xi  =  0.39035 

Xi  =  -  2.4567. 
x2  =  -v/^  csc  0  4-  V— p  cot  0. 


log  Vp=  0.30103 

log  cot  0  =  9.55965 

9.86068 

Xa  =  1.2283  +  0.72557  V^lf 
x8  =  1.2283  -  0.72557  V^l. 

5.  x*-5x+4  =  0. 

»-  — •  «?>?> 

2j>  \p 

6     /3 

=  5\5' 

log    6  =  0.77815 

logV3  =  0.23856 

colog    5  =  9.30103 

colog  V5  =  9.65051 


log  sin  $  - 

=  9.96825 

0  = 

=  68°  21'  24", 

e  _ 

3" 

=  22°  47'    8", 

eop-f= 

=  37°  12'  52", 

60o+|  = 

=  82°  47'    8". 

Xi  = 

=Wl-i- 

log    2  = 

logV5  = 

colog  V3  = 

:  0.30103 
=  0.34949 
=  9.76144 

U«2^  = 

=  0.41196 

log  sin -  = 

:  9.58803 

log  Xi  = 
Xi  = 


:  9.99999 
1. 


teachers'  edition.  699 


*=2Vf8to(6oo-f> 

log2>/f  = 

■  0.41196 

logsii 

,(60°-!)  = 

10gX2  = 

=  9.78161 

=  0.19367 

X2  = 

:  1.6616. 

Xz- 

=-2>lh* 

(60O  +  I).     * 

log2iJf  = 

=  0.41196 

logsii 

log  x8  = 

:  9.99665 

0.40851 

Xz  = 

-2.5616, 

1 


