OORAGK  HOMTB  TALENT. 


LOVEll'S  SERIFS  OF  SCMH  30'J(S. 


KEY 

TO 

;ELEMENIS  OF  ALGEBRA; 

K  DESIG:,'-  U3E    OK 

CANADIAN  GRAMfflAt;  L¥j  COHOH  SCHOOLS, 


C    "TAI.VING 

yOLUTIOXS   TO   XEARLY   ALL    THE     I' P.OHLE.MS, 

TOrrETHER    WITH 

NOMEEOUS  EXPLANATORY  SSMASSS. 

-  !  13YJ0I1N  HERBERi'  GANGSTER,  m.a.,  m.d, 

TATMEMATICAL  MASTKE  AND  LECTirCBrc  IN  CHEMISTRV  AND  K/.TVKAL 
rUlLOaWUT  in  the  NOKMAL  SCHUO^  a-Oi;  UWER  CANADA. 


MONTRi^AL: 

PRINTED  AND  PUBLISHED  BY  JOILV  LOVELL 

A>D   FOB  8ALB  AT  iUE  BOOKSTORKS. 

.     '  1ST2. 


lIOMJt  JNDlTBTnY. 


Digitized  by  tlie  Internet  Arcliive 

in  2009  witli  funding  from 

Ontario  Council  of  University  Libraries 


http://www.archive.org/details/keytoelementwestOOsang 


LdVELL'S  SERIES  OF  SCHOOL  BOOI^S. 

KEY 

TO 

ELEMENTS  OF  ALGEBRA; 

DB3IQNBD    FOR    THE    DSB    0? 

Canadiai  Gramniar  aiil  Cciimiob  Sciools. 

CONTAINING 

FULL  SOLUTIONS  TO  NEARLY  ALL  THE  PROBLEMS, 

TOaETHER    WITH 

NUMEROUS  EXPLANATORY  REMARKS, 
BY 

JOHN  HERBERT  SANGSTER,  M.A.,  M.D., 

MATHEMATICAL  MA8TKR  AND   LECTURER  IN  CHEMISTRY  ANP  NATDRAL 
PHILOSOPHY    IN  THE   NORMAL    SCHOOL    FOR    PPPKK  CANADA. 


MONTREAL: 

PRINTED  AND  PUBLISHED  BY  JOHN  LOVELL, 
AND  FOR  SALE  AT  THE  BOOKSTORES. 
1872. 


Entered,  according  to  the  Act  of  the  Provincial  Parliament,  in 
the  year  one  thousand  eight  hundred  and  sixtj-four,  by 
John  Lovell,  in  the  Office  of  the  Registrar  of  the  Province 
of  Canada. 


PREFACE. 


The  following  pages  contain  solutions  to  all,  or  nearly 
all  the  problems  and  exercises  given  in  the  Author's 
Elements  of  Algebra.  In  many  cases,  two  or  more  solutions 
of  the  same  problem  are  offered,  so  as  to  afford  the  student 
additional  illustrations  of  the  best  and  neatest  modes  of 
working ;  and  of  the  application  of  artifices  employed  by  the 
experienced  algebraist  in  order  to  obtain  a  required  result. 
On  this  account,  also  nearly  every  operation  has  been 
given  at  full  length. 

The  Author  hopes  that  the  Key  will  prove  serviceable 
to  the  many  who  are  privately  prosecuting  the  study  of 
Algebra,  or  endeavouring,  without  the  aid  of  a  living 
teacher,  to  prepare  themselves  for  entrance  into  our  Univer- 
sities ;  and  that  it  may  likewise  be  of  advantage  to  those 
teachers  whose  school  duties  are  so  many  and  varied  as  to 
render  them  unable  to  devote  to  the  subject  that  time  and 
study  which  long  and  intricate  algebraic  solutioas  hi 
general  require. 

ToEONXO,  October,  1864. 


CONTENTS. 


Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 


1. 
IX. 

XII. 

XVIII. 

XIX. 

XX. 
XXI. 


PAGE 

5 

7 
7 


9 

..  11 

..  13 

XXII 13 

XXIII 17 

XXIV 18 

XXV 19 

XXVI 20 

XXVII 21 

XXVIII. 21 

XXTX 23 

XXX 24 

XXXI 25 

XXXII 26 

XXXIII 31 

XXXIV 39 

XXXV 42 

XXXVI 45 

XXXVIII 49 

XXXIX 50 

XLI 52 

XLII 53 


Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 
Exercise 


PAOU 

xrnn 55 

XLIV 59 

XLV 63 

XL  VI 65 

XLVII 65 

XLVIII 67 

XLIX 69 


L. 

LI. 

LII. 

LIII. 

LIV. 

LV. 

LVI. 


..  71 

..  ^i 

..  7(; 

..  7H 

..  8H 

..  10  1 

..  lOH 

LVII 109 

LVIII 113 

LIX 115 

LX 119 

LXI 123 

LXII 127 

LXIII 129 

LXIV 132 

LXV 135 

LXVI 138 

LXVII 142 

LXVIII 143 


Miscellaneous  Exercises 150 


KEY 


ELEMENTS  OF  ALGEBRA. 


Exercise  IV. 

1.  18-  1  =  1  -  1  =  0 

2.  3^-3  X  3  =  2'7-9  =  18 

3.  1x2  +  3x4  =  2+  12  =  14 

4.  =l2x  2'' -(3-1)  =  1x4-2  =  4-3-2 

5.  V2T3TI  =  V9  =  3 

6.  0  •.■  7tt  =  0 

7.  6  X  ( I  -  3^)  =  6  X  (9  -  1)  =  6  x  8  =  48 

8.  (22x4^-3x0)'  =  (4x  16)^  =  (^64)2=  4^=  16 
0.  (1+  2)  X  (4  -  0)2  =  3  x  4-  =  3  x  IG  =  48 

10.  4  {I  -  (4  -  3)/'  =  4  (1  -  1)''  =  4  X  0*'  =  4  X  0  =  0 

11.  6c'/  =  2  X  3  X  4  =  24 

12.  (4^  -  2  X  3)2  (3^  -  2  X  3  X  4)^'  -  (16  -  6)2  (27  -  24)^ 
-  102  X  33=  2700 

1  +J      ^_  6+12  +  1         3  3        6 

^^-    i  +  1  =   2   -  ^  '"  '  rn  "  i  +"1  "  III  "  y  "  T  "  ^  =  *' 
&c. 

14.  14  X  1  -  (3  X  2  +  3)  =  11  -  9  •=  5 ;  4^  -  2  (2  +  3)  =  16  -  10 
c  e,  and  5  <  6 


K..Y   TO 


[Ex.  IV. 


15.  Each  -  0  •.•  OT,  one  factor  of  each,  is  equal  0 

16.  Vl  X  4x  27-4(2  +  4)3  =  Vl08  -4x6x3  =  V36  =  6 
^(2  +  3)  X  (16  +  9)  =  ^h  X  25  =  ^l25  -  5,  and  6  >  5 


17. 


1  X  4x27-2x4      108-8      100 


10 


1+2+3+4 
=  2  X  5  =  10 

1  X  9  +  0  -  (4  -  3)= 


10 


=  10,  and  2  X  (2  +  3)  +  0 


15 


9-1^ 


SJt  (16  +  '.•)  +  2  (3  +  4)      ^'2  X  25  +  2  X  7     -^64       4 
anfl  {4  X  3  -  (i  +  3  +  2  +  I)}  =  12  -  10  =  2 

19.  (2  -  2)(3  +  8  -  3)  +  j2  +  (12  -  6)}  -  4  (6  -  6)  -  {18  -  (9  +  l)j 
+  {8-(3  +  4)x  1}2;  =0x8+(2  +  6)-4x0-(18-I0)  +  (8-7)x2; 
=0+8-0-8+1x2^1x2=2 

20.  (9  -  1)(4  -  0)  +  0  +  3  {1  +  3  (4  -  3)}  =  8  X  4  +  0  +  3  (1  +  3) 
=  32  +  3  X  4  =  32  +  12  =  44. 

21.  ((1  -  2)  +  (3  +  4j}2  +  {(3  +  0)  -  (2  -  l)f  -  {(0  +  4)  +  (4  -  3)}' 
=  (-  1  +  7)2-4-(3-  l)^-(4+  1)2;  =  6^+  2''-5^=  36+8-25=  19 


li. 


22.  V(l  +  3)  X  4 +  ^9  X  (1  +  2) +  {2  (4 +  6)2+ (28 -12)}' -(24+1)'^ 
--  V4x  4  +  ^9"x~3  +  (2  X  102+  ^g)i  _  (05^2  ^  ^y^  ^  ^77  +  ^210 
-  (V25)3  =  4  +  3  +  6  -  5*  =  13  -  125  =  -  112 

7  X  VO  +  3  V4  -  (8  +  12)      36  -  28  +  {64  X  (1  +  3;}^ 


23. 


J  X  6  +  0 


\  ■^{(2-l)+l}{4-(2  +  0)}~'^^'*~^*^ 


0  +  6-20      36-28+V256        -        14      24 

24.  \\1{\  +  2)j  -  i  {6(3  +  1)}  +  \  {(3  -  2)(2  +  3)}  +  1  {(4  +  3) 
(l  +  6-6  +  4)2}  =  i(2x3)-i  (6x4)  +  }  (1  x5)  +  ;  (7  x  25) 
=  2  -  6  +  1  +  25  =  22 

3(1  +  2-3)'+!  1  {(3 +  6X2-2  +  2)}    {(l  +  12)'-(27+10)-(3+4)}^ 


25. 


{(9  +  2)-V-l]('i  +  3  +  4-0) 


0+V36-1 


J 


(4+ 12 -8)(4  +  3)      3x0»+ll(9x2)      (169-37-7)'      8x7 
■*■  771+4)  "  (Tl-2)(K-0)  "*"  ~      ^  "^ 


11x18      (^125)'^      56  5« 


6-  1 
2+5+1=8 


7x8 


«,x.  IX.  ill.]  ALGEBRA.  7 

EXEBCISE  IX. 

1.  «  +  j/t-c  +  6  +  5-Tn-a-c  +  c  +  3-5c-m=14-m-5c-« 
3.  a-b-c-b  +  c  +  a-c  +  b+a-a-h-c=2o-2b-  2c 

3.  3a  -  4  -  61/  +  X  -  ort  +  4  +  6y  -  3a  +  4  -  6  -  X  -  5a  -  2 

4.  6  +  (  -  {  -  (  -  {  -  («.)  }  )  }  )  =  6  -  {  -  (  -  {  -  ^m)  }  )  } 
-6  +  (-{-(m)})  =  6-{-  (m)  j  =  6  +  (ffi)  =  6  +  m 

5.  2a  -  3r  +  4'f  -  Si  +  (//i  +  3f/)  +  5a  -  (  -  4  -  J)  -  3a  +  (4a  -  5i  -  4 ) 
-  2a  -  3c  +  4'i  -  5(i  +  7/1  +  3a  +  5a  +  4  4-  rf  -  3>i  +  4a  -  5(f  -  4 
=  lla  -  3c  -  5d  +  m 

6.  m^  -  c^  +  a^  +  7?.2  -  2a2  +  20^  -  m^  -  S/n^  +  c^  +  a"''  -  c^  +  3m^ 
>::  2a2  -  m^  -  c3 

■r.  1  +  1-1  +  1+1-1  =  2 

8.  u^+2x-a^+a^+  2x^-2in^-i-  rn^-;-c2+2x  +  m^ +3a~  +  3x  +  3r'>' 
=  5a^+  7a:  +  2x2  +  ^m^. 

9.  d^bc  +  3c2  +  Sa^ic  -  y/i  -  t  +  4u-6c  +  c  -  30^  -  7n  =  8a^bc  -2m 

10.  3a-2a-l+a-2+a+l-a-2a+2+a+l  =  a+I 

11.  -a-6-c  +  o-c-c+a  + 2a -36 -2c- 36 -a-6-c-a 
=  a  -  86  -  6c 

12.  a77i-c-7  +  5-  7am  +  c +  3a+  5am, -4jm  -  6  + c- 9- 3c -4a 
=  -  o  -  5am  -  2c  -  17. 


Exercise  XII. 
1.  3am  -  3x  +  3y  +  5ax  +  15ay  +  2a77i  -  2my  +  4crx  +  4x 
=  5ani  +  x  +  3?/  +  9ax  +  I5ny-  2my  =  Ham  +x  +  Dax  +  .T!/+  \bay-2my 

-  bam  +  ( 1  +  9a)x  +  (3  +  ]  5o  -  2m)y 

2.  am  —  mx  +  TTiy  +  3/nx  +  3ax  +  4a  —  4i/  +  3ay  +  3x7/  =  a7/i  +  2w.r 
+  mxj  +  3ax  +  4a  —  4i/  +  3jy  +  3x7  =  4a  +  am  +  27nx  +  3ax  +  Sxy 

-  4v  +  7717/  +  Say  =  (4  +  tti)  o  +  (2r7i  +  3a)  x  +  (3x  -  4  +  ?«  +  3a)  y 

3.  7a  +  76  -  7c  -  56  -  5x  +  56c  -  3m  +  3a  +  3c 

=  10a  +  26  -  4c  -  5x  -  56c  -  37a  =  10a  -  5x  -  56c  +  26  -  4c  -  3m, 
=  5  (2a  -  X  -  6c)  +  2  (6  -  2c)  -  3m 


8  KEY   TO  [Ex.  xn,  xviii. 

4.  ax  +  mx  -  ^amxy  -  3fi:i/+  2ay^ ~  2cm.M^ ■{■  ax  ^  ai/  +  cxy  +  axy 

-  by'  -fy' 

=  2ax  +  mx  -  Snmxy  -  2cxy  +  Say'  -  2cmy^  4-  otj/  -  fcy'  -  /"i/^ 
=  2'^/a;  +  7nx  -  Samxy  -  2cxy  +  axy  +  Bay'  —  2cmy^  -  by^  -fy^ 
=  (2«  +  7«)  a:  -  (3f/?;i  +  2c  -  a)  a:?/  +  (3a  -  2cm  -  b  -/)  y' 

5.  3«]/  -  3by  +  3cy  —  2inx  +  ex  ~  3amx  —  3amy  +  Zamz  —  (3amx 
+  3amy  +  3amz  +  2cx  +  2cz  +  acy  -  acz)  =  Say  +  Sby  +  Scy  -  2mx 
A-  ex  —  Sarnx  —  Sumy  +  Samz  —  Samx  —  Samy  —  Samz  —  2cx  —  2cz 

-  aey  +  aez 

=  Say  -  Sby  +  Scy  -  2mx  +  ex  -  Qamx  -  Gamy  -  2cz  -  acy  +  acz 

-  Say  —  Sby  +  Scy  —  Gamy  —  acy  —  2inx  +  ex  -  6omx  —  2cz  +  aez 
=  (3a  -  Sb  +  3c  -  &am  -  ac)  y  -  (2m.  -  c  +  6am)  x  -  (2  -  a)  cz 

6.  1  lamy  +  l\bmy  -  Saxy  +  Sbxy  -  Sexy  -  (2aep  +  2aexy  -  Scm 
+  6cx^  -  Sci/  -  Soy  -  Sac) 

=  liamy  +  llbmy  -  Saxy  +  Sbxy  -  Sexy  -  2acp  -  2(icxy  +  Scm 

-  Gcxy  +  Scy^  +  Say  +  Sae 

=  llamy  +  llbmy  +  Scy'  +  Say  -  Saxy  +  Sbxy  -  2acxy  -  9exy 

-  2(tcp  +  Srm  +  3ac 

=  {11  (a  +  b)m  +  3   (CJ/  +  a)\  y  -  \S(a  -  b)  +  (2a  +  9)e]  xy 
+  S  (m  +  a)  c  -  2acp 


EXERCISK  XVIII. 

1.  \(a  -b)  +  e}{(a  -b)  -  e}  =  (a- b)' -  c' =  &C. 
lrj-(b-  e)}\a  +  (b-  e)]  =  a' -  (b  -  c)'  =  &c. 

[a  +  (b  +  e)][(t  -  (b  4-  c)}  -  a' -  (b  +  e)'  =  Ac. 

2.  {4  f  (3a  -  2c)}{4  -  (3a  -  2c)}  =  16  -  (3rt  -  2c)'  =  Ac. 
\2:i  -  (X  -  Sm')\{2a  +  (x-Sm')}  =  4a''  -  (x  -  Sm')'  =  Ac. 
12.1!/  +  (2a  -  3y)}\2xij  -  (la  -  Sy)\  =  Ax-y'  -  (2a  -  3y)'  =  Ac. 

3.  {(2a  -  3r)  +  (2x  -  3v)}  {(2a  -  3r)  -  (2.r  -  3;/)}  =  (2a  -  3c)» 
-  (2x  -  Sy)-  =  Ac. 

j(a  +  3,/)  +  (2c+  4;n)}{(a  4-  3rf)-(2c  +  4/n)}  =  (a  +  3,/)'- (2c  + 4w)' 
-Ac 


£i.  xvin,  XIX.]  ALGEBRA.  9 

4.  {(3a  -  m^)  -  (2  -  xy)}  {{3a  -  m^)  +  (2  -  xy))  =  (3a  -  m^f 

-  (2  -  xyy  =  Ac. 

}(2a2  -  3x2)  +  (n.  2/2)}/(2a2  _  3x^)  -  (1  +  J/^)}  =  {2a^  -  3x^)2 

-  (1  +1/2)2=  4c. 

5.  (5aZ>  +  6a2  -  66^  -  (43^  -  I6ab  +  IGi^)  -  4(9  -  fl^)  -  4(4a2 

-  4a J  +  b^)  -  5ab  +  6a^  -  66^  -  ia^  +  IQab  -  166^-  36  +  4a2-  16a^ 
•h  16a6  -  46^  =  &c. 

6.  (24ax!/  -  16a2  -  9xV)  +  3(40^  +  4axy  +  x-y^)  -  1(x^y^  -  9a^) 
+  4(4a^  -  I2axy  +  9xV)  =  2iaxy  -  I6a^  -  9xY  +  Ua^  +  Uiixy 
+  3xV  -  7x"^y^  +  tiSa^  +  lea^  -  48axi/  +  Sijx'^y''  =  &c. 

7.  (1  -x'^)(l  +x2)(l  +x4)  +  &c.  to  7  terms  =  (1  -  x*)(l  4- x*) 
(1  t-x")  +  &c.  to  6  terms  =  (1  -x")(l  +x'')(l  +x^^)  +  &c.  to  5 
terras  =  (1  -xi6)(l  +x'«)(l  +  x^-^)  +  &c.  to  4  terras  -  (1  -  x^-) 
(I  +X«)(1  +X«*)  rr  (1  _  x'^*)(l  +x6'»)  ^  1  -xi2« 

8.  Product  of  first  two  terms  =  a^  -  x'V;  of  first  three  terms 
=  a*  —  x*y*  ;  of  first  four  terms  -  a'  ~  x"?/',  and  so  on. 

Now  the  index  of  each  term  in  the  product  of  the  first  two 
factors  =  2  =  2^  =>  2^-1 

Index  of  each  term  in  the  product  of  the  first  three  factors 
=  4  =  2^=  2^-1 

Index  of  each  term  in  the  product  of  the  first  four  factors 
t  8  =  2^=  2*-!,  and  so  on 

Therefore  the  index  of  each  term  in  the  product  of  n  such 

factors  -  2"  "  ^  .-.  (a  -  xy){a  +  xy)(a?  +  x'^y'^)   ....   to  n  terms 

oi  -  1  a"  '  1 

=<  a-  -  (xi/)^ 


Exercise  XIX. 

4.  (a»+  6=')(«"'-i3)  =  &c. 

5.  {c^f  -  {x^f  =  (a?  -  x»)(a-5  +  a^x*  +  x^ )  =  &c. 
t.  (a^  +  /rt^x^)  (,{2  _  j/i-x^J  =  ^5, 


10 


KEY  TO 


[Ex.  XIX 


8.  (2a) '  -f-  x'  =  (2a  +  x){(2a)*  -  (2a)'x  +  (2a)V  -  2ai'  +  x*\ 
=  &c. 

9.3*-  (2c)4  =  {3*  +  (2c)2}{32  -  (2c)2}  =  (9  +  4c2)(3  +  20(3  -  20 

10.  (ZmY  -  (2c)5  =  (3/?!.-  20{(3m)*  +  (3ot)»(20  +  (3m)^(20' 
+  (3/«.)(2c)3  +  (20*{  "■  Ac. 

11.  (a')*+(xO^-  (a^  4-a:')(ai*-a"x'  +  x")  =  &c. 

12.  (a*)*  +  (/n*)'=  (a*+ ?/i*)(a'S-ai^/n*+ a'ni»-a*/7i"+m"',> 

13.  (c8)S+  (a;8)3=  &<.. 

14.  (xi<')»+  (7?i'0)^=  (xif  +  rn'O)  (x2"-x"'m'<'  +  m")  =  4'c 

15.  Oi2*+  c2*)(tti2+  ci-)(a^+  c6)(u»+  c3)(a»-c^; 

=  {{d-y  +  (c^y'lK"*)'  +  (c*)'j[(a')'  +  (c2y*}(a3  +  c»)(a»  -  r")  =  &c. 

16.  (a»^>'+(m«V=  &c. 

17.  (a^*  +  cSOCa'^'  +  c'0(aS'  -  c'^')  =  {(o"')^  +  (c")"} 
{(ffO)'*  +  (c^)''jf(«^)*-  (c^)'}  =  («'*+  c'«)(o3ti-  ai«c^«+  c3«) 
(a-i  +  cO  ((i^*  -  a^c^  +  c^')  (a9  -  c')  (a^*  +  a^c'^  +  c'^) 
^  {(a6)8+  (c6)3]{(a»)3  +  (c3/j{(a'*)*  -  (c3/}(a3  6  -  a^'c'*  +  c»«) 
(ai"  -a^c^  +c'0(a'*  +  a-^ca  +  c'«)  =  (a^  +  c6)(ai^  -  a«c6  +  c^^) 
(a'-r  c3)(a6  -  aV  +  c'^){u^  -  c'^)(a^  +  aV  +  c6)(a36  -  u'^ci' 
+  c^'^X^i^^  -  a^c^  +  c^^XO'^^  +  «^c9  +  c'^)  =  {(aY  +  (c'fl 
(a»  +  c*)  (a'  -  c»)  (a' 6  -  a^^c^*  +  c^S)  (a'»  -  a^c^  +  c") 
(a'«  +  (;9c9  +  ci8)(ai2  _  a^cS  +  c^'Xa^  -  aV  +  r«)(a6  +  aV 
+  cO  =  &c. 

18.  (//i^'*)3  +  (c*0'  =  ("i"'  +  c*")  (m.96  -  m*^c*^  +  c^e) 
=  {('/i""0*+(f-''^)'}("i''*  -"i*'c*»  +C'''6)  =  &c. 

19.  (a^)'  +  (mry  =  &c. 

20.  (a^'m-^'y-  (p*-)'^  (a"m2'-;>iO(«"'*"»'*  +  a«T7n»"p«' 
+  ;>"-')  =  {(t^«'')^-(i5'')^}(a''^'n**+«"'»i*'j'^'+/>'0  =  («"»'' -;»0 

-(jj8)3|(ai''ffii«+uPm''p''+pi')(o''*m«*+o'^'m«  "!)»"+;>'••)=  &c. 


Ex.  XX.]  ALGEBRA.  11 

EXEKCISE    XX. 

1.  a  —  x  +  x-a  —  a-i-a  +  a  —  x  +  a  —  X  —  a=a  —  2x 

2.  3(a2_  x'O  -  2(a2-  iax  +  4x0  -  (12ax  -  9a''-  Ax^)  -  4(9x2-  ^2) 
"  3a^-  3x2-  2a^4.  gax  -  Sx^  -  12ax  +  9a^  +  4x2  _  32^,2  ^  ^^2 
■■■■  14a2  -  43x2  _  4g^ 

(4)  (5) 

^m  +  j-p  +  g  a  +  x)a"-x"(a"'i-a"-2x  +  u"-'x2-&c. 

o^m  +  a'xJ'  +  s  r^"^x-x™ 

-a^x^-P-x™*'  -fl"-ix-a"-2;c3 


(7)  a«-2x2  +  c»  3x8 

1  -    i)  1  (1  +  1  +  1  +  1,  &c.  -a"-«x«-x'',&c. 


1-1 
+  1 
1  -  1 


1 

1  -  1 


1,  &c. 

8.  (^^  +  x9)  (a»  -  x^)  =  {{a^f^  {x^f]  \{s?f  -  (x^^j  ^  (,^3^  ^S) 
((t"  -  x=*)(a^  -  a^  x'  +  xS)(a6  4.  ^s  x'  +  x^)  =  &c. 

9.  x2  ?n2  (a*  x'  -  4a'''  xp  +  4p2)  =  m^  x^  (a^  x  -  2^)2. 

^(j    ylQjTF)  ^^-  _  {ii{3  +  4}^)'  +  6}  -  {12(3  +  4)J^} 
12-0  0  + V4  (12  +  1) -3  -  (2 +  3+  1) 

^/5^      {(14  -  1)2  +  6}  -  (34  +  1)  _  ^  _  (169^6)  -  85 
"^12  -^bl^^-Q  12  V49-6 

1       175-85       1 

(11) 

1      2•^-3^24•l  1  +  2  +  3  +  4. 

1  -£+J 1-2-3 

r-fT+  i4j  +  i  1  +  2  +  3  +  4 

~3-4-«-4-2  -2-4-6-8 


12 


KEY  TO 


[Ex.  XX. 


1+2+3+2+1 


_3_6-9-12 


l+n  +  0-2  +  0  +  0  +  1  1  +  0-4-8-17-  12 

a;6  4. 0x5  4. 0x4-  2x^  +  0^2+  Ox+  1    „«  +  0a*6 - ^c^b'^- Sa'b^-llab*-  126* 
=  x^  -ix^'+l  =  a«  -  Aa?b^-  8a%^-  Hob*-  126' 

(12) 
x*  +  Ox^  -  (a^  -  b  +  c)  x'  +  (a6  +  etc)  a;  -  6c  ^  x^  +  ox  -  c 
1  1  +  0  -  (fl^  -  6  +  c)  +  (a6  +  ac)  -  be 
—  n     —  a  +   a^  —  ab 

+  c  +  c  -  ac  +  6c 

1-0+6  =  x^  -  ox  +  6. 

13.  (asi  -  jftS")  -  (0^24.  „^32')(a32_  „,a2_)  ^  (^32 4.  m*0(a'6+  wi'^'' 
(a' 6  -  m'6)  =  &c. 

14.  {a}+  c^)  X  4">  factor  --  a-^  +  c-  ;  (o  -  c)  x  B'*  factor  =  <t"  -  c" 
and  (fl  +  c)  X  last  factor  =  a"  +  c".  Hence  required  value 
^  („u  _  tii)(aii  +  c")("^''  +  c^-)  =  ("^^  -  c22)(a2^  +  c^)  =  a"  -  r^* 

15.  0  +  6  =  0  and  o  =  i  .-.  6  =  -  i  ;  o  +  6  +  c  =  0,  and  o  +  6  =  0 

.-.  c  =  0 ;  i  {i  -  f  -  Dij  =  Mi  -  (  -  j)}  =  Hi  +  i)  =  i  X  i  -  J 

IG.  fz3_  i,i_  2a^b  +  3a62+  3a^6  +  3a6'^  +  a^  +  b^  ^  2a8  +  t>a6' 

-  2a  (rt^  +  36^) 

17.  3  (a-7»)^-  2  (4^2-  9/71^)  -  (lOwi^  +  6am)  +  7  (n^  -  /n') 
+  (100771.-  4?7i^)  =  3a^-  Go77i  +  3m'^-  8a'^  +  ISm^  -  lOm^-  60771 
+  7a''  -  7771*  +  10o7rt  -  4771^  =  2a^  -  2am  -  2o  (o  -  tti) 

18.  =77i[(77i-2o)(77l-26)  +  (77l-26)(/7J-2c)  +  (/7I  -  2r)(77l  -  2fl)} 

=  TTij (6  -  0  +  f )(o  -  i-  +  c)  +  (o  -  6  +  c) (o  +  6  -  c)  +  (o  +  6  -  c) (6  -  0  +  c)} 

=  TTijc^  _  (6  -  0)2  +  a2  -  (6  -  cf  +  62  -  (o  -  c)'\ 

=  77i(c2  -  62  +  2a6  -  a2  +  0^  -  62  +  26c  -  c^  +  6^  -  o^  +  2ac  -  c^) 

=  ?7)(2rt6  +  2ac  +  26c  -  o2  _  ftz  _  c-) 

=  7n(2a6  +  2oc  +  26c  -  o^  -  6^  -  c^  -  4a6  +  4a6) 

^  771  ( 2oc  -  206  +  26c  -  o2  _  i2  _  c3  +  4aft) 

=  77i{(c  -a-  b)(u  +  6  -  r)  +  4a6j  =  77i{(c  -a-  b)(a  +  6  +  c  -  2r)  +  4a6} 

=  77. 1  (f  -  o -  6)(77i  -  2f )  +  4o6}  =  771  [77i(c  -  o  -  6)  -  2c(r  -  0  -  6)  +  4a6} 

s  8o6c  +  )7(2(c  -  o  -  6)  +  406771  -  2c77i(c  -  o  -r  6)  -  8a6c 


£x.  zx,  xxiii  ALGEBRA.  i 

=  8abc  +  m^(m  -2a-  2b)  +  Aabni  -  2f/n(«i  -2a-  26)  -  8abc 
=  Babe  +  771  (;/i^  -  2am  -  2bm  +  4a6)  -  2c{m^  -  2am  -  2bm  +  4a6) 
=  8a6c  +  (m  -  2c)(m*  -  2am  -  2bm  +  4a6) 
-  Sabc  +  (m  -  2c) (;n  -  26) (m  -  2a) 


Exercise  XXI. 

1.  36  X  Qab'^m,  and  4am^  x  6ab'^in. 

2.  Sa^/R''  X  7a*,  Sa'^m^  x  Gam,  and  Sa'^m'  x  Sm*. 

3.  aiy(8ax  +  17/n  -  3a7n^x),  and  xy(5  +  3a  -  14a^x). 

4.  (x=*  -  TTix^)  +  (2x  -  2mx),  and  (x^  +  4r  +  4)  4-  («x  +  2a)  ; 
that  isof  x^(l -7/i)  +  2x(l -7/i);  and  Cx  +  2)''  +  a{x  +  2)  ;  that  is 
of  (x^  +  2x)(l  -  771),  and  (x  +  2)(x  +  2  +  a)  ;  that  is  of  x(x  +  2) 
(1  -m),  and  (x  +  2)(x  +  2  +  a). 

5.  Tlia'.  is  of  3i!-(a  -  x)(a  +  x),  and  'ia^x^(^a  -  x)^ ; 

C.  That  is  of  3w^(a='-/7i3)(a  +  v;i);  4/7i='(a"'-77i-)^,  a,adiin^(a''- m^) 
(u  -  ///)  ;  that  is  of  377i'  (a*  -  ni^^a^  +  am  +  vi^)  ;  4/7,*  (a-  -  vi^)'-, 
ami  477i^  (a^  -  771^)  (a  —  771) 

7.  That  is  of  (x  -  7)(x  +  3)  ;  (x  -  7)(x  -  5),  and  (x  -  7)(x  +  VI) 

8.  That  is  of  a^(x  -  \y\  and  a^{x  _  l)(x  -  2) 

9.  That  is  of  (x  +  4)(x  -  1)  ;  (x  -  1)^,  and  (x  -  l)(x  +  1) 


Exercise  XXII. 

(1)  (2) 

X*-x-6)x2-5i-14(I    2xS-12x2+21x-10)x<-8x'+2Ix2-20x  +  4 
x2-x-6  2 


-4x-8  2x*- 16x*  +  42x"-40x  +  8(x - 

2x*-  12x*+  21x^-  lOx 


4(x  +  2) 
x''  +  2x 


-    4x''+21x^- 30x +  8 
a:  +  2)x'-x-C(x-3  _    4^»  +  24x'' -  42x  +  20 


-    3x^+  12x-  12 
-3x-6  -3(1^-    4x  +  4) 

-  3x  -  6  ^ 


14 


KEY  TO 


ffer.  xxii. 


x*-4i+4)2x»-  12x2+  2ix-  10(2x-4 
2x'  -  8x^  +  8x 

-4x*+  13i-  10 
-4x''+  16x-  16 

-  3x  +  6 

-  3(x  -  2) 
X  -  2)x*  -  4x  +  4(x  -  2 

x«-2x 


-2x  +  4 
-2x  +  4 


3,  (a»  -  ax)  -  (7a  -  7x),  and  (a»  -  a^x)  -  (3a  -  3x) 
fl(a  -  x)  -  7(a  -  x),  and  «^(a  -  x)  -  3(a  -  x) 

(a  -  7)(ff  -  x),  and  (a^  -  3)(a  -  x) 

4.  x(x'^  +  X  -  12),  and  x'^(x  +  4)  +  5(x  +  4) 

x(x  +  4)(x  -  3),  and  (x^  +  5)(x  +  4). 


5,  a^ 


ab  -2b'^)it'-  3ab  +  2i^(l 
d^-ab  -  26" 


-  2ab  +  46* 

-  26(a  -  26) 


a  -  2b)(i'^  -  ab  -  2b'\a  +  b 
d^  -  2ab 

ab  -  2b'' 
ab  -  2b' 


6.  a^-5ab  +  4b^)a^  -  a'^b  +  Zab'^  ~  3}^{a  +  46 
a«  -  ba%  +  Aah^ 

4a-6^a6^  -  36* 
4o''«6  -  20o62  +  166" 


19u6--  196* 
1962(a-6) 
a  -  b)d^  -  bab  +  46'-(a  -  45 
a*-  ab 


-  4«6  +  46» 

-  4o6  +  46* 


ts.  ii^ii.j  ALfitlBRA.  1.3 

7.  Rejecting  the  factor  2  from  the  first  quantity        fAx'^-  98 
15x4-9x*+47x*-21x  +  28)60x6-36x«+48i*-45x='+42x2-45x  +  12 
60x6  _  se^s  +  ]  88x*  -  84x'*  +  1 1 2x=* 
-  140x*  +  39x*  -  70  x^  -  45x  +12 
3 


-420x*+  117x»- 210x2-  1352  +  36 
-  420x*  +  252x'  -  ISlGx^*  +  588x  -  784 
^1 35x»"+~ll  0  Gx^"^?  2  S^-TS  2  0 
135x3 -1106x»+723i-820)15x*-9x'+47x'- 21  c  +  2g(j-  +  205 

9 

135x*  -  Slx^  4-  423x^  -  189x  +  252 
135x*  -  1106x3  +  723x2  -  820x 

1025x3-  300x^+  631x  +  252 
27 


27675X*  -  8100x2  ^  j  7o37x  +  6804 
27675x»  -  226730x=' +  14821 5x-  168100 
~       218630x2-  131178X  +  1749i,4 
43726(5x2 -3x  + 4) 
6x'-3x  +  4)135x''-  1106x2+  723x  -  820(27x  -  205 
135x«-   81x2+ lOSx 

-  1025x2  + 6 I5x  -820 

-  1025x2 +  615X  _g20 

8.  That  is  of  26(3a'  -  3a^y  -  y^  +  ay^),  and  36C4a2  +  y^  -  5ay) 
TJxat  isof  26{(3a»-  3a?y)  +  (aj/2-  i/S)},  and  36{(4a2_  ^ay)  -  (ay-y^) 
26{3a2(a  -  y)  +  y\a  -  y)},  and  36{4a(a  -  y)  -  y(a  -  y)] 
2&(a  -  ]/)(3a2  +  y'^),  and  3b(a  -  y)(4a  -  y)  ■ 

Otherwise, 
4a*  -  bay  +  y^)  3a»  -  3a2y  +  ay^  -  y' 
4 
12a*  -  12a2y  +  4ay2  _  4y»  (3a  +  3y 

12a»-15o2y  +  3ay^ 

3a^y  +    ay'  -  4i^ 
4 
12a2y  +  4ay'^  -  16j/» 


16  Key  to  [Ei.  xxtt 

19y^(a-s/) 
a  -  y)  4a*  -  5ay  +  y^  (4a  -  y 
4a*  -  4ay 


-  ay  +  y^ 

-  ay  +  y' 


G.  C.M.^bia^  y) 


y.  a*  +  12a  -  28)  a»  +  Sa^'  +  2  7a  -  98  (a  -  3 
a'+  1205=-  28a 
-  3a^  +  55a  -  98 
-3a*-36a+    84 


91a-  182 
91  (a -2) 
rt-2)  a*+  12a-28(a+  14 
a*- 2a 

14a -28 
14a -28 


10.  86*  (a^  -  3a'b  +  Sat*  -  6*),  and  12a*  (a*  -  2ab  +  6*^ 
That  is  of  86*  (a  -  6)»,  and  12a*  (a  -  6)* 

11.  Rejecting  the  factor  2  from  the  first  quantity  and  mal- 
tiplying  the  second  by  3  fa  +  "7 
3ai+10a*-6a»-24a*+lla  +  6)3a6+12a'-9a*-48ti''+33a*  +  3Ga-27 

SaS  +  10a»  -  6a*  -  24a»  +  1  lu*  +  6a 
2u5  -  3a*  -  24u*  +  22a^  +  30a  -  27  ' 
3 


6a''  -  9a*  -  72a^  +  C6a^  -f  90a  -  81 
6a« +20a<- 12a»-  48a*  +  22a  +  12 
^29u*  -  60a»  +  1  iVu^  +  68a  -y 3 
I9u*+  60u'- 1 14u*-  CSa  +  93)3a*+  10a*-  Ou«-  24a-  +  1  lii  +  G(3;i  +  110 
29 
87a»  +  290u*  -  174a*  -  G96u'''  +  319a  +  174 
87a»  +  180a*  -  342a»  -  204a*  +  279a 


110a*  +  168a»  -  492a»  +  40a  +  174 


j<29  = 


tx    XXII,  XXIII. ]  ALGEBRA.  1  < 

3190a*  +  48T2(i''  -  142G8a^  +  1 160a  +  504t3 
3190«*  +  6600ct^  -  12540a2  _  'r480a  +  10230 
'  -17230^-    n28a''+ 8G40,i  -    5184 

-  1728  (a^  +  a^-  5a+  3) 
fl'  +  a-*  -  5a  +  3)  29a*  4-  GOa^  -  IHa^  -  68a  +  93  (29a  4  31 
29a4  +  29a*  -  145a^  +  87a 

3]~aM-  31a2^rr5"5a  +  93 

31a3  +  31a2-  155a  +  93 

12.  Rejecting  the  factor  2  from  the  first,  and  3c  from  the  second 
-  d^b  -  8ah^  +  66*)  a*  -  3a«^  -  Ba^b^  +  I8ab^  -  8b*  (a  -  26 
a*-    a%-8a%'^+    6ab^ 

~'ia''b  +  12a6*-    86* 

-  2a''6  +  2a^6'^  +  16a6«-  126* 

-  2a^6^  -    4a6^*T~46* 

-  26^  (a*  +  2a6  -  36=*) 
o'  +  2«6  -  26^)  a«  -  d'b  -  8ab''  +  66»  (a  -  36 

a*  +  2a''6  -  2(ib^ 
-  3a^6  -  6a6^  +  66» 
-3a26-6a62+66» 


Exercise  XXIII. 

1.  4  X  -  3  X  a'b^xY  =  -  12a='62x^» 

2.  4  X  3  X  a'xh/z'^  =  IZu^x^z^ 

3.  (a:  -  yy  (x^  -yy^=  [(^x-  ]/)(x'-"  -  y)Y  =  (x»  -  x^y  -  xy  +  /)'' 

4.  (x*  +  XT/  +  y'^)  (x*  -  y*)  =  x^  +  x^y  +  x*?/^  _  x'^y*  -  xy'^  -  y^ 

5.  x\1  -  x)* ;  (X  -  l)(x  +  1 ),  and  4x(l  +  x)  that  is  x^(l  -  x)' ; 
(1  -  x)(l  4-  x),  and  4x(l  +  x)  =  4x^(1  -  x)^  (1  +  x)  =  4x»  -  4x* 

-  4i''  +  -ii'* 

B 


lb  KEY   TO  lEi.  xxu,  xxi\. 

6.  a^  -  L^  contains  a?  -  b^,  and  a*  +  6'  as  factors  ;  .•.  /.  c.  m 
=  36  (a  -  b)  (a"  -  6")  :^  36  (a''  -  a^b  -  ab^  +  b^)  -  36a'  -  3Gaf>b 
-  36a6«  +  366' 

V.  J-  (x  -  3)  ;  (X  -  3)  (x  -  7)  ;  and  x  (x  -  7) 

.-.  /.  c.  III.  =  X  (x'^  -  lOx  +  21)  -  X*-  10x^  +  21* 
8.   (a^  -  X*),  and  (a^  -  ax)  -  (a  -  x) 
a*  -  x^,  and  a  (a  -  x)  -  (d  -  x) 
d^  -  x",  and  (a  -  x)  {a  -  1) 
.-.  /.  c.  Ill  =  (a^  -  x^)  ((/  -  1)  =  a*  -  u^  -  flx*  +  x* 
9    G.  C.  M.  of  two  given  quantities  is  a^  -  7a  +  12 
(i^  -  9a^  +  26a  -  24 

;z^"ir7.r+'i2     - "  ~  ^ 

(<is  -  8^2  +  19a  -  12)  (a  -  2)  =  tt*  -  10a'  +  35a*  -  50a  +  24 
10.  3(a  -  i)(a2  +  a6  +  i^)  ;  4  (^  -  6)*;  5(a  -  6;(a  +  i)(a''  +  b^) 

6(a  -  /))2,  and  \{a  -  6)(a  +  b)f 

Or  of3(a-/))(a2+^6  +  62);  4(a-6)3;  5(a-6)(a  + 6)(ri2  +  6'); 

6(a  -  bf,  and  (a  f  b)\a  -  b)^ 

.-.  I.  c.  m.  =:  3  X  4  X  5  (a  -  6)»  («  +  6)»  (a^  +  ai  +  6^)  (u^  +  62\ 

^  60(a»''  +  a?6  -  a"" b'^  -  2a^ b^ -  2a«6*+  2^*6"*  +  2a»6'  +  a%*  -  ab> 

-61") 

EXBRCISB   XXIV. 

a(a-b)  m(2a+mx-m*) 

1    — 2    — ^ 

■  a(x  +  y)  •        m{3a^  +  m) 

c(l  +a)  0*6(1  + 6  + «i) 

^-  n(l  +a)  "*•    x(l  +b  +  m) 

ate*  ojV 

6. 


b(a  +  c)  '  x^a^xm  +  ay  +  x^y-z^ 

7x'V(3-5x) 


7.  — 


a  -  TO 


14xV  (a-77i)(a  +  »i) 

(a  +  6)(a2  -  a6  +  6^)  (a  -  6)(o  -  6) 

9.  -^^ — r-^riX, iv—  10. 


(a  +  6)(a  -  6)  '"•  (a  -  6)(a»  +  a6  +  6») 


£x.  xxm,  ixiv] 


ALGEBRA.  19 


(a  +  L)(a^-ab  +  b^)  (a'  -  m^Xa*  +  a^m'  +  m*) 

(a--  m^)  (a^+  m")  1{x'  -  3j;  +  6) 

^'*-         a^(a^~m-)  ll(a;^  -  3x  +  5)' 

(X  -  7)  (X  -  4)  (2x  +  3)(2x  +  3) 

^^-  '(x'^Xx  +  3)  ^*^'  (2x  +  37(  X  -  4) 

x-(x  +  2y  +  Zf)  ^(a^  -  a/>  +  6^)(a  -  6) 

(a^  -  m^)  (a*  +  m^)  (a^  -  m2)(a^  +  ^2) 
■^^-  a^(a-^my-  m''*(a  -  ,7i)  ~  (0"-=  -  Hi'')(«   -  ;n~)" 

(ac  +  6c)  +  (ad  +  6(/)  c(a  +  i)  +  rf(a  +  6) 


20. 


(a/n  +  bin)  +  (2flp  +  2bp)       m(a  +  6)  +  2;j(a  +  b) 


(c  +  rf)(a  +  6)_  (X  +  «)(x  +  b) 

{nir+~2p){a  +  t")  ^  ^  ■  (X  +  c)  (X  +  6 ) 

(X  -  1 ) (2x2  +  3x  -  5)  ^„  +  ,/j)  (^a  +  2a/)i  +  m^  -  x^) 

^^-  (x-l)(7x-  5)  ^^'  {x-  -  (ri  -  /n)'}{(a  +  m)=^  -  x^) 

(a  +  ?/i)(a^  +  2am  +  m^  -  x^) 


(a^  +  2a/;i  +  ni^  -  x^)  (x^  -  a^  +  2a7«  -  m'^) 
(a*  +  X*)  (a»  -  a*x*  +  x*) 


24. 


(a*  +  x4)  (a' 6  _  a'-x*  +  a^x«  -a^x'*  +xi< 


Exercise  XXV. 

a*  +  a^  +  a  -  a^  -  rt  -  1  +  2        a^  +  1 
2_  _____  _     ^_  ^ 

3ffx  +  9a  -  7/x  -  37/  -  (3a2-  30)     3ax  +  9a  -  x//  -  3;/ -  3a^+  30 
^'  X  4-  3  "  ~  x  +  3 

3ax  -  3ay  +  xij  -  y^  ~  2a  -  xy       Sax  -  3ay  -  2a  -  y^ 

4. = 

X  —  y  X  —  y 

Sa'x  +  3ax'^  -  ay'  -  xy'  +  am  +  mx  -  3ax*  -  xi/* 

5 

a  +  X 

3a-x  -  ay'  -  2xy'  +  am  +  mx 

a  +  X 


20  KEY   TO  [Ex.  XXV,  xxvi. 

xyz  +  2mxy  +  mz^  +  27/1^2  +  xyz  -  z^m  -  2mH      2xyz  +  2mxy 
z  +  2//i  '  ~        a  +  2/ft 

_2xij{z-rm)  {a-^bf-{a-bf 


z  +  2m  •  rt  +  6 


a  +  b  u  +  b     ~       u  +  b 

a^+m--a^+m^       2tu^  a^  +  x^  -  a^  +  2ax  -  x'^       2ax 


EXEKCISB  XXVL 


2x^ 

2.  «  -  x)  rt>  +  x'  (a  +  X  + 

2  a  —  X 

a'  -  ax  "•    * 


flX  +  X* 
«x  -  X* 

2x^ 

y'  +  y* 


3.  X  +  y)  x2  +  2x(/  +  j/^  +  x8  -  y*  (x  +  J/  +  x^  -  xy  +  /  -    ^  _!_ 

X^  +  XJ/ 


a;*/ 

+  f 

^ 

+  y' 
x". 

-2/* 

»» 

-  xi'y  -  I/* 

-  x2y  -  xy' 

xf-y* 

xy^  +  2/' 

-J/*- J/* 

3 

4.  J?i  -  p)  5m'  -5p^  +  3  (5//t^  +  binp  +  5p'  +  ^_ 

5m'  -  biii^p 

biii^p  -  bp'  +  3 
bm^p  -  5mp' 

5inp  -  5jt'  +  3 

bmpi*  —  5p^ 


.X.  xxvii,  xxvtii.]  ALGEBRA.  21 

a-l  1 

5.  ab  -  b)  a^b  -  ab  -  J  +  I  (a  -  rz — — r:  =  <*  ~  "T" 

-a  +  1 

b(ia  +  1) 

6.  VI  +  ft)  7n  +  ub  +  5(/?n  (1  +  5a  ~ — r — 

TO  +  /^ 

G(/7«.  +  lib  —  b 
bum  +  5a6 


-  4a(^  -  6 


y&C 


Exercise  XXVII. 

2(x  +  y)^x     2{4x  +  y)     (x-y)(2x-3y) 
^-     2(x^-7/)    '  2(7-2^7^)=  2(x-'  -  7/^) 

«     4j.-    a--+l  3x  +  2       3(r(x2-l)    4x(a:2-l) 

^-  T'  T'  x^-Tl'  ''"'^     '3      '  "  "Six'-iy  '3(x*-l)' 

3(x^+  1)  (3x  +  2)  Cx''  -  1) 

®*  eo^a^  -  by  6a\u''  -  b')'  *°^  6a»(a2  _  52^ 


Exercise  XXVTII. 
4am  +  37?j  -  26c  y(r  +  3)  x  +  2  (ff  -  6) 


26/ft  "•  y^(x  +  3> 

(rt  -  6)^  -  (re  + />)2        -4o6  4a6 

315x-  18x  +  35x  +  63x* 


63 

x'  +  1/  (x  +  yY  -  xy  (x  +  y)       x^+x'y  +  2xy''  +  y^-x^y-xy' 
(^  +  yf  "  (i  +  2/)' 


c  (a  -  ft)  +  a  (6  -  c)  -  6  (a  -  c)         0 

6.  — ^ ^ ^^ — ' ^^ =  -;—  =  0. 

abc  abc 


22  KEY  TO  fEx.  xxvm. 

m  (m  —  p)  —  p  (771  4-  »)     m^  —  mp  —  mp  —  p' 

7    =  ,  Ac. 

(in  +  p)(m-p)  {m -i- p)im  -  p)   ' 

3  (2«  _  1)  _  4  (1  -  5a)  -  7  r2a  +  1)  ^   12g-14   _  14-12r! 

9.  Multiplying  both  num.  and  den.  of  1st  fract.  by  -  I  in 
order  to  change  the  signs  of  the  den.  we  get 
x(x  -  IG)  +  (X  +  2)C2x  +  3)  -  (2  -  3x)(2  -  i) 


4-x^ 
a:'-  16x  +  (2x'+  7x  +  6)  -  (4  -  8x  +  3i«) 

J~2 


=  &c. 


x+y     x+y     x+y     x-y     x+y     x-y     x+y-r+y 

10. +  — r —  —  +  -,     =  -r i— = ;;: =  *f' 

a  0  a  b  0  0  b 

(to  +  ]>)(in  -  p)  +  (p  +  x)(p  -x)  +  (m  +  x)(x  -  m) 
(p-x){x  -  mX"i-p) 
(m^  -  p^)  +  (p^  -  x^)  +  (x^  -  ;;|2)  0 


12 


(p  -  X)(X  -  7/l)(/ft  —p)  (p  -  x){X  -  Hl)(«l  -  p) 

{a  -  b)(b  +  c)  +  (b  -  f)((i  +  b)  2ab  -  2ac 


=  0. 


(a  +  6)(6  +  c)  ab  +  be  +  ac  +  be  -  be  +  b* 

lab  -  2bc  -  lib  +  2ac 

_     trf. 

ab  +  ac  +  be  +  b^ 

l+x-(l-x)       3(1  +  2x)-3(l  -  2x)  2x  12x 

^^-         r-^         "^  l-4x^  ^  iTJ?  "*"  r-^4? 

2x-8x»+ 12x- 12x»         14x-20i' 


(1 -x^)(l  -  4x^)  l-5x^f4i* 

14.  Multiplying  both  terms  of  each  of  the  last  two  frac.  bj 
—  1  we  get 


a(a-b){a-e)       6  (a  -  6)  (6  -  t- )       t(a-c)(c-6) 

bem  (6  -  c)(c  -  6)  -  acm  (a  -  c)(c  -  6)  -  oAro  (6  -  c)(a  -  b) 
abe  (a  -  6)(a^'c)(6^-/(c  -  6) 

fn(26V-6'c-6c»-flV+2a«/ic+«<:»-a6f«-a»6»+o6'-a6V)      m 
'  a6cC26 V'- 63c^6c'^ VT27='6r  +  a c » - ah^^'b'^Tab^'^^^ab^)  ^  (iiT- 


1 


£x.  xxvm,  xxii.)  ALGEBRA.  23 

OTHER\nSE    THUS 

Multiplying  both  terms  of  2nd  fractioa  once  hy  —  1,  and  of  3rd 
fraction  twice  hy  —  1,  we  get 


a{a  -  b)(a  -  c)       b{a-b){b  -  c)       c(a-c)(^b-c) 
whence  we  have  /.  c.  m.  of  the  deu.  =  ubc(^a  —  6) (6  —  c)(a  -  c) 

bna(b  —  c)  —  acm(a  —  c)  +  abm(a  -  6) 

.*.  the  given  fractions  =  — — t~: rr> \7I ; — 

^  abi:{a  -  b)(a  -  c){b  -  c) 

b'cm  —  br^/n  —  a^cm  +  ac^m  +  a^bm  —  ab^rn 

a^c(a  -  6)(a  -  c)(6  -  c) 

ac'm.  —  bc'^in  —  a^cm  ■\-b'cm  ■¥  a^bm  —  ab^m 

abc(^a  —  b){a  —  c){u  —  c) 

c^m{a  -h)  -  cni(a^  -  b-)  +  abm(a  -  b)       c'ln  -  cm(a  4-  6)  +  abm 

a6c(a  -  6)(a  -  f)(6  -  c)  ~       abc(^a  -  c){b  -  c) 

m(c^  —  ac  ~  be  +  ah)       m\(ab  —  be)  -  (ac  —  e^)\ 

abc{a  -  c){b  -  c)      ~         ahc{a  -  c)(6  -  c) 

m\b(^a- c)-c(a -e)\       m(^b  -  c)(a  -  c)        m 

abc(^a  -  c)(b  -  c)      ~  abe(a  -  c)(J}  -  c)  ~  abc 


ExEBCtSB  XXIX. 

2x  X  3x     3x'                             2m  x  x^xv* 
I     =  - — ■  2 2 

5  X  2a        5a  '  xyx  my  x  x~ 

2(a+b)       x(a~b)       2a  -  2b 

xy  3(a  +  6)  ~       3y 

3a       X  +  1       x-l       3(ar  +  l)(x-l)       3r«  -  3 

■     1  2a  a  +  b  ^  2(a  +  6)  ~  2a  +  26 

(a-x)((7  +  x)       (a  +  b)(a-b)  a  a(a  -  6) 

a  +  b  a  +  X  x(a  —  x)  ~         x 

a^-  ffi*        a*  +  m'        (a  +  m)(a^  +  m^)         u'  +  ma''  +  m'^a  +  m* 

6.    -     — X- =- =-  

my  a  -  m  my  my 

(a  -  x)  (a  +  x)         4ax*  4x(a  -  x)       4ax  -  4x^ 

7. ;. X 


3  IX  a  +  X 


24  KEY  'rO  [t^-  XXIX,  XXX. 

(x-7)Cx-6)     (x-5)(x-4)     (J-7)(3:-4)     x'- lli  +  28 
^-       x(x  -  5)      ^  ~(x  -  6)      ~  x^  ~  *=' 

ubcdm  am 

(tf-2)(ffl  +  2)     a'^-l      a-  2      (a  -  2)(a  -  2)     (a -2)' 


10. 


11 


(«-  -  1  j  2a        a  +  2  2a  2a 

(x  -  fl)(x -t- fr)  x(x  +  6)  +  c  (x +  6) 


x(x  +  6)  -  a(x  +  6)     x(x  +  c)  +  <i(x  +  c) 
(x-fl)(x  +  a)       (x  +  6)(x  +  c)       x+a 
(X  +  ii)(x-«)  ^  (jr+  c) (x  +  d)  ~  x  +  rf 

(x  +  4)(x-3)     (x-5)(T+7)     Cx-3)(X4-Y)     j^+4t-21 
^^*  (x^  8)(x  -  5)  ^  (X  +  4XX-11)  ~  (x^'8Xx-ll)  ~  x'-lSx+SS 

l-a  +  a2       g'  +  g+l       {(a»  +  1)  -  ff}{(a'' +  1)  +  g} 
1  u^ 


(2a  +  4m)(2a  -  4m)  5a  n  +  2/71 

^*"  o"-  2m.  ^  5(2a  ■)-  4m)(2a  +  4/tt>  *^       o 

2(a  -  2OT)(rt  +  2)n)        1 
2(a  -  2m)(a  +  2m)  =*  T  ' 


EzcaciSE  XXX. 


1. 

\         X   _    I         !/    _    !/ 
X    ^    y    ~   X        X    ~  X* 

a  +  X       a  ~  X       a+x           a          a+X 

a             a             a          a  —  x      a  —  x 

3. 

a+  b       (a  -  h)'       a  -  b 
a  -  6  ■■'  (a  +  6)*  ""  a  +  6 

4. 

(anx^)(a+xKa-x)     (y+2)(y-2)        3a        

yH-2              '^       a-x       '<a»+x«  =  3a(<'+x)(y-2} 

5. 

X-  3       (x-9)  rr-  P')       x-3 
x-9  '^  (X  -8)  (X-  7)  "'  *-  7 

tx.  XXX.  XXXI.]  aLgeera.  25 

fl»   +   b^  a-i   +   62  ^2   +   ^2  flZ  -   62  1 

(a'  -  x')  (a'  +  x^) 


^    a2  -  6'''  •   a^  -  62  ~  a^  _  6'=  ^  a^  +  6^  ^    1    ~  ^ 


7 

(or  -  x)^  '"  a  +  X  '"  a-*  +  ax  +  x^  ""  d^  —  ax  +  x^ 

(a  -  x)(a^  + /7x  +  x'-')(a  +  x)(a^  -  flx  +  x^)         1  a-x 


(u  —  x)  (a  -  x)  a  +  X      a^  +  ax  +  x' 


1  1 


a^  -  ax  +  x^        1 

3(a2-l)       2aCa  +  6)       3rt('7^-l)       3a^  -  3a 


9. 


2(a   +6)  x^-1  x^-  1  x^-  1 

(xy  +  y^)  +  y^  +  X  (x  +  y)       (2xy  +  2;/^)  +  x  (x  +  y)  -  xy 


xy  +  y^  xy  +  y" 

2y*  +  2xy  +  x*             a;y  +  y^ 
. ^ — ■ — . —  j^ ^  1 

xy  +  y'  2y^  +  2xy  +  x^ 

4oZ&2  4o6  4d^b^  a*  -  6^  ab 


a-6 
3  5(a-6) 

'  lOtt  +  96  ~  10a  +  96 
15 


1  ax 

3. 


a+2x  a  +  2x  "'     3x  -  1  10(3x-l) 


1 5  -  6x  +  C>a 


10  3  (15-6x4- 6a) 

IQq  +  lOx  -  6  '   "2(1  Oa  +  lOx  -  6) 
15 

fia  -  2a 

1  -  4a^  _  4a                       1-1* 

^    TTj^  "  iV4T»             ''•        2        *  ~  " 

1  -  4^  1  -  a' 


26  KEY  TO  [Ex.  XIII,  itxii, 

rt2  +  6^  _  flft 

b  fl2  -  b'        a(a^-  ab  +  6^         {a-  b){a  +  6) 


9. 


a-b  a^  +  6»    ~  a-b  (a  +  6)(a^-at +^^> 

xY  -  1  -  xY  -  1  -  1 

xy  xy  _  xy 


1-- ^ 1- — —       1- 


xy  xy  —  I  -xy 

~  xy  -  I               ~  xy  -  I  xy-l 
xy 

j-1                    -1                   j^l  -1 

xy                     xy                     xy  xy      _         1 

xy-l              1  -  xy        1  -  1  +  xy  xy              i^y* 

^~     -  1           ^~       1                     i  T 


i^^}- 


df+e  cf  bd/+be  +  c/  ad/+ae 

— >—  b  +  vv-~ 


/         _      ^  df+e      ^        df+  e        ^  bdf+bf^ 
adf  —  uc  W(//  —  ac  adf  —  ac  ad/  —  a 


bd/+be  +  c/      bd/  +  be  +  cf       bd/+be  +  c/       bd/+be  +  c/ 
adf+  ae       o(df+  e) 
adf  -  ac  ~  a{d/-  c) 
2  +  8m*  1  +  4m» 

Am  -  8//1*        III  -  2m^    _     -  1  -  4m*  1  +  4m* 

^^'  4m  +  8m*  "      1  +  2ffi     ~  m(l  -  4m*)  "  m(4m* -  I) 
-8/ft  ^~i 


Exercise  XXXII. 

1.  12x  +  4x  =  84  -  3x,  or  19x  =  84,  or  x  =  ^ 

2.  lOx  -  X  =  5x  +  20,  or  4r  =  20,  or  x  «  6 

3.  168x  -  28x  +  12x  =  63x  -  231  +  84x  +  756,  or  5x  ^  525, 
or  X  =  lOP 


Ex.  xxxu.]  ALGEBRA.  27 

4.  30x  -  105  +  9x  -  3  =  5x  +  40  -  30x,  or  64x  =  148,  or  x  =  2tV 

5.  56  -  4x  +  20  =  84  -  7x  +  49,  or  3x  =  57,  or  x  =  19 

6.  56x  -  8x  =  2Ix  +  7  +  14x  +  84,  or  13x  =  91,  or  x  =  7 

7.  8x  -  65  =  35  4-  2x,  or  6x  =  100,  or  x  =  16§ 

8.  15x4-45  -  12x -  48  -  960  =  -  20x  -  20,  or  23x  =  943,  or  x  =  41 

9.  80r  -  8x  -  7G  =  300  -  35x  -  55,  or  107x  =  321,  or  x  =  3 

10.  112X-I-480  =  3024-  39x  +  84,  or  151x  =  2628,  or  x  =  Hf^V 

11.  208x  -  442  +  308r  -I-  374  =  858x  -  4433  +  143x,  or  -  485x 
=  -  4365,  or  X  =  9 

12.  4x  +  4  -  3x  =  6  +  14  -  3x,  or  4x  =  16,  or  x  =  4 

13.  360x  -  160x  +  200  +  48x  =  2040  4-  60  -  180x  4-  45x  4-  15, 

or  383x  -  1915,  or  x  =  5 

40x  -  60   34x  -  108 

14.  Multiplying  by  12  we  get  x  4-  s -  g 

39x  4-12   36  -  23x 
=  12 ^—-—^-. 

lOOx  -  240        136X-432 
This  X  4  and  reduced  gives s ^ =  3x  -  36, 

or  800x  -  1200  -  952x  4-  3024  =   105x  -  1260,   or  257x  ^  3084, 
or  X  =  12. 

15.  60x  4-  30x  4-  15x  -  36x  4-  252  =  120x  -  156,   or  51x  =  408, 
or  X  =  8 

16.  336  -  lOx  4-  10  -  776  4-  56x  =  16x  -  3x  4-  11  -  144,  or  33x 
-  297,  or  X  rr  9 

17.  30x  4-  20x  4-  15x  4-  12x  4-  lOx  =  60x  4-  25x  4-  240,  or  2x 
-240,  or  X  =  120 

18.  12x  -  20  4-  X  4-  60  -  9x,  or  4x  =  -  40.  or  x  =  -  10 

125x  4-  500    125x  4-  500 

19.  36  4-  20x  -  20x  -  86  -  ^j.  _  ^^    ,  or  g^  _  ^g   =  50,  or 

125x  4-  500  =  450x  -  800,  or  325x  =  1300,  or  x  =  4 


28  KEY  TO  [Ex.  xxxtv 

120  +  TOx  ir.T-65       55a:  -  85 

20.  331  -  30j: +  9  +  5x  = r (i) 

=  the  given  equat  x  10 

960  +  5603; 
2720  -  200x  -  — — y =   15i  -  65  -  llOx  +  lYO  (ii)   -   .'t) 

reduced  and  x  8 

945a-  +  960  +  560.r  =  23535  (in)  =  ii  reduced  and  x  9 

laOSx  =  22575,  or  t  =  15 

144.C  -  432 

21.  9r  +  20  -^  ~  5x^34 —  "^  ^^      0^  =  given  equat.  x  36 

lOOx  -  80  =  144x  -  432  (ii)  =  i  reduced  and  x  (5x  -  4) 

44x  -  352,  or  X  =  8 

22.  30x  +  20x  +60  -  15x+60=  12r  +  CO  +  1900,  or  23x  =  1840, 
or  X  =  80 

23.  90x  -  35x  -  70  =  75  +  20x  +10-51  +  9x,  or  26x  :=   104, 

or  X  -  4 

12x2  +  36t+27 

24.  15x+  lOc^-  10x2+  18  ^  27  +  18x  -  - — — (i) 

3  +  4x  ^ 

=  given  equat.  x  (3  +  2x) 

12x^  +  3Gx  +  27 
3x  +  9  =  3+l^x ^"-^  ~  ^^^  transp.  and  collected 

9x  +  27  +  12r^  +  36x  =  12x2  ^  ^C^x  +  27  (in)  =  (ii)  x  (3  +  4x) 
.-.  9x  =  0,  or  X  =  0 

2'x  -  n9 

25.  6x  +  12  -    .        —  =  6x  +  7  (i)  =  given  equat.  x  9 

5  +  lOx  =  21X-39  (ii)  =(i)red.andx(l+2x);  llx  =  44.-.x  =  4 

c-b 

26.  ax  =  c  -  t,  or  X  = 

,       .  '^  ,  3b'+3bc 

27.  9ax-  3J^=  3bc  -  2jx,  or  llax  ^  36c  +  3ft^  or  x  =  — r^ 

28.  86x  -  Cr  =  a  -  6»  +  3ax,  or  (86  -  6  -  3a)x  =  a  -  h', 

a-b' 
°''  ^  ^  86  -  6  -  3a 


£x.  xxxii.J  ALGEBRA,  29 

20.  4.t^i)r  -  Ca^-h  2ax  -  2abx  -  abx  +  b^x  (i)  =  given  equa.  x  2ab 
('ia%  +  2a-  ah  -  b'^)x  =  6ir  (ii)  -  (i)  Ir.ansp.  and  bracketed 

•'•  ^  ""  U^b  +  2a  -  ab  -  b^ 

30.  I5j,bc  -  10?x  -  5ac  ~  20ab  -  Ibbx  -  abcx  +  bh  (i)  -  given 
eqiiat.  x  56c 

(156  +  abc-  10c)x  =  20ab  +  b\  ■(-  bac  -  15abc  (ii)  -  (0  trans- 

I'oi-ed  and  bracketed 

20r/6  +  6''c  +  5^tr  -  \5abc 

^  ~  156  +  ubc-iUc 

31.  hdx  +  ndx  +  bcx  -  bilf,  or  (hd  +  a<i  +  bc)x  ~  bdf,  &c. 

32.  abx  +  4a2  -  4a^  +  126j;  -  4rt6j:  =  'ka^b'^  -  lOa^  +  126x  +  ^a?x, 
by  multiplying  the  given  equation  by  4a;  and  this  reduced  and 
T  by  a  gives  Six  +  4'«  -  10a  -  406^,  or  (36  +  4a)x  =  10a  -  4a6'' 

10a -4a6'' 
•■•  "^  =  "   36  +  4a 

33.  abx  —  a^x  —  b'^c  +  a6c  =  6^x,  or  (a6  —  a^  —  b^)x  ~  b'c  —  abc, 

bc(b  -  a) 

or  X  =   -r ^ rr, 

ab  -  a''  -  b^ 

34.  lla^  -   3«x  -    lla6  +  36x  -  (tiu^  4-   Cr.l-  -  5ax   -    56x) 

=  (a  +  6)^  +  2x  (i)  =  given  cquat.  x  (a^  -  b^) 

2ax  +  86x  -  2x  =  b^  +  lOab  -  4a^      (ii)  -  (i)  reduced  and  tran.sp. 

b'+  VJab  -4a^ 
(2a  +  86  -  2)x  =  6^  +  19a6  -  4a^  or  x  =  —7. — — .r — r,— 

za  +  00  —  ^ 

a 

35.  a^  +  2ax  +  x'^  -  4a6x  =  x^,  or  (46  -  2)x  =  a,  or  x  =  --. — 

3abc      hx  2nh  +  62  \  0^6^ 

^^-  a  +  6  ~  u'  ^ ^  "  ^+2^16^^  +  (^aTJ?  "  ^'■'^  ^') 
=  given  equa.  with  num.  and  dea.  of  1st  term  x  3,  and  2nd  tmd 
5th  terms  factored 
3rt6c      hx  (      a'     1  a'b' 

a+b-'a  \(^^T6?;  ■*■  (^T67  =  ^^^^  <")  =  (0  ^ith  2nd  term  red 


30  KEY   TO  [Ex.  xxjLi:. 

3abc  abx  a^b' 

TTi'b  '^^'  +  0^T6?j  =  ^{3c  +  (^^^i^.j  (IV)  =  (in)  with  1st  and 
3rd,  and  2nd  and  4th  terms  factored 

ab  ab      ) 

.-.  X  =  ——7   (v)  ^  (iv)  ^  }3c  +  ; T—,\ 

37.  3000  +  1720X  -  2210j:  ^  203j--  (i)  =  given  equa.  x  1000 

693x  ^  3000,  or  X  =  4/3^ 

'   3x  23x 

38.  y  +  6x  -  ax  ^  3a  -  -— -,  or  33x  +  594x  -  09nx  =   297(7 

-  23x;  or  650x  -  99(/x  =  297a,  or  (650  -  99a)x  =  297«, 
297a 


"'■  "^  ~  650 -99a 

39.  42(x  -  i)  +  35(1  -  X  -  I)  -  30(x  -  1  _  -|  j  =  I05x  +  30x, 

by  multiplying  the  given  equation  by  105;  and  removing  the 
brackets  from  this  we  get  42x  -  14  +  35  -  35x  -  14  -  30x  +  30 
+  lO.r  =  135x ;  or  148x  -  37  .-.  x  =  i 

40.  V2ax-96-  756=  180 -456  -  35c, or  72  jx  =  180 +  396 -35c 

180  +  396 -35c 
72a 

41.  a'''6'  +  a^x  -  6^x  -  x^  -  3a6  +  3a6x  =  ex  -  ac  +  ax  ~  ir' 

a'x  —  b^x  +  3a6x  —  ex  —  ax  =  3cb  —  ac  —  a^6' 
(a?  -  6^  +  3a6  -  c  -  a)x  =  3(z6  -  oc  -  oV 

3a6  —  ac  —  a^b' 
•'•  '  "  o*+'3a6  -b'-c-a 


t. 


Ex.  ixxui]  ALGEBRA.  81 

EXKKCISB  XXXIII. 

1.  Let  X  =  greater,  then  47  -  x  -  the  less,  and  x  -  (47  -  x) 

=  13,  or  2x-47=  13 

2x+  21 

2.  Let  X  =  the  less,  then  x  +  21  -  the  greater  ;  — - —   =  3, 

or  2x  +  21  =  3x 

2x       2x 

3.  Let   X    =    money;    "^  +  T"  -  1'"^*'   P''''^   away;    then   x 

2x       2x 
=  y  +  Y  +  ^^2-50 

X  -  21 

4.  Let  X  =  the  number;  then  — r-; —  =  5 

5.  Let  X  =  the  quotient,  then  2x  +  3x  +  4x  =  54 

2x  3x 

6.  Letx  =  debts  ;  then  —  =  1st  payment,  and  --  =  remainder; 

3        3x      9x  2x      9x 

.'.   -^-  of  —  =  —  =  2nd  payment ;  then  —  +  „^  +  192  -  x 

7.  Let  X  :=  the  number  of  cattle  in  the  drove, 

XXX 

then  v  +  "r  +  T'  +  9  =  ^ 
3        b        5 

8.  Let  X  =  the  number  of  sheep  in  each  flock  ;  x  -  19  is  twice 
as  great  as  x  -  91,  that  is  x  -  19  =  2x  -  182 

X  T 

9.  Let  X  =  the  number ;  then  —  — Y  ~  ^ 

3         X  3x 

10.  Letx  =  the  number;  then2x-  -^-  of —  =  25, or  2x-  —  -  25 

X 

11.  Let  X  X  the  number;  then  x  +  --  =  39 

/  X        x\ 

12.  Letx  =  the  number;  thenx- f  —  +  „- j  =  17,  ( 

=  17 

2x  -  15  3x 

13.  Let  X  -  the  number ;  then  — ^ +  7  =  —  +  3 

5(x  +  ll) 

14.  Let  X  -  the  number  ;  then r =  85 


82  KEY   TO  lEi.  Aixiii 

X      2x     3x       llx 

15.  Let  X  -  tie  number ;   then  ":,r  +  ^  +  ~r  ~  ""a""  "*■  ^' 

36 

16.  Let  X  =  price  per  barrel;    then  —  =  number  of  barrels; 

36  /36 

and 5  -  number  of  barrels  sold  the  second  load  ;    /  —  -  5)x 

=  21,  or  36  -  5x  =  21 

17.  Let  X  =  distance  in  miles,  then  ix  =  half  distance  ;  Jx  -=•  ^ 

X  X 

=  -;r  =  times  in  hours  ^'J  travels ;  ^x  -r  4  =  —  "  times  in  hours  B 

28i 
travels;  then  kx  -  ix  =    „.  -  =  H,  or  x  =  26? 

18.  Let  X  =  the  time  in  hours,  and  since  the  three  runs  cf 
stones  severally  require  72,  84  and  90  hours  to  empty  th » 
granary,  they  will  in  1  hour  empty  respectively  -^g,  -j^^-  and  /i 

XX  X 

of  it,  and  in  x  hours  they  will  empty  v;:,  —  and    .;  similar!; 

X  X 

the   teams  will  respectively  fill  in  x  hours  t^    and   —  ;   thej 

X  X  X  X  X 

7 2  "^  8 4  '*'  90  ~  GO  ~  7l  "" 

19.  Let  X  =  date  of  abolition  of  slavery  in  Canada;    then 

3(x  -  1780)  +  1620  =  year  of  massacre  of  Lacfaine.     Therefore 

x  +  3(x-  17S0)  +  1620 

^^ —- +  116  =  1862 

20.  Let  X  -■  A's  share,  then  x  -  120  =  B'a,  and  x  -  106  =  Cs. 

Therefore  x  +  x  -  120  +  x  -  106  =  7400 

24x  -  300 

21.  Let  X  -  price  in  cents  of  a  music  lesson,  then  r^ — 

/24x-300\ 
=  price  of  a  drawing  lessson ;  therefore  32x  =  24  (  ^ —  1  +  1000 

23.  Let  X  =  the  number  of  volumes  on  science  ;  then  3x 
=  number  on  travels,  3x  =  number  on  biography;  4jx  =  number 
on  history,  and  9x  -  number  on  general  literature.  Therefore 
X  +  3x  +  3x  +  'Ix  +  9x  =  14.^5;  whence  x  =  70 


Ex.  SXXtll.] 


ALGEBRA.  B3 


23.  Let  X  =  length  of  Niagara  river,  wherefore  ix  -  6  =  leugth 
of  Rideau  canal ;  then  2(5x  -  6)  -  100  =  230 

24.  Let  X  =  days  required  to  finish  the  work.  Then  since  ^ 
does  -iV,  B,  -jJj,  and  C,  -,V  of  the  work  in  1  day,  A  and  B  work- 
ing 1  day,  and  B  and  C  working  2  days  will  finish  -iV  +  -jV  +  Va 
=  iVo  of  it,  and  the  part  remaining  to  be  done  =  [2*0  i    i°  ^  '^'^js 

c4  does  75<As. ;  B,  Tzths.  and  C,  T^ths.   of  the  work,   therefore 

^  +  ^  +  ^  =  iSB»  or  15x  +  12x  +  lOx  =  109 

25.  Let  X  =  greater  part ;  then  n  -  x  the  less ;  and  x-(a-  x) 
=  a-  c 

26.  (i)  Let  X  =  minute  divisions  the  hour  hand  passes  over; 
then  since  the  minute  hand  travels  12  times  as  fast  as  the  hour 
hand  it  will  pass  over  12x;  but  the  minute  hand  also  passes 
completely  round  the  circle  (60  minutes),  and  then  in  addition 
over  the  x  minutes.  Therefore  60  +  x  is  also  equal  to  the 
number  of  minute  divisions  passed  over  by  the  minute  hand  ; 
then  12x  =  60  +  x,  or  llx  =  60,  or  x  =  S-^^i]  hence  the  hour 
=  5-iV  X  12  =  1  h.  5-,V  m. 

(ii)  To  be  opposite  the  hands  must  be  30  minutes  apart ;  then 
letting  X  =  space  in  minutes  passed  over  by  hour  hand,  and 
remembering  that  the  minute  hand  travels  12  times  as  fast,  and 
also  goes  over  30  +  x  minutes,  we  h*«e  12x  =  30  +  x,  or  llx 
=  30,  or  X  =  2^\  and  2^\-  x  12  =  32//  past  12 

(ill)  By  similar  reasoning  to  the  above  12x  =  15 +  x,  or  llx 
=  15,  or  X  =  1-ji-,  and  1/f  m,  x  12  =  16-i*(-  m. 

27.  Let  X  =  price  in  dollars  of  first  field  ;  then  x  +  90  -  25 
=  price  of  second  field,  wherefore  (x  +  90  -  25)  +  90  =  2x 

28.  Let  X  =  days  required  by  ^  and  C  to  finish  the  remainder  ; 
then  Tj'j  -  (/,)-  +  -g',-)  =  T^^5  =  part  C  does  in  1  day,  .-.  in  11  days 

C  does  -jVu l" I  ^'^'^  B  and  C  together  in  5  days  do  ^'^^  +  j_"j  =  ^^ 

G 


34:  KEY   TO  [Ex.  xxxni. 

=  ^'j.     Ilcnce   part   remaiuing  to  be  done  =  1  -  (faVrr  +  /o) 

X         19x 
=  iVsrj  -  -iW^  =  U} ;  then  ia  x  aayg  j1  and  C  will  do  —  +  - -- 

X         19x 
•■•    50  "^  1300  ""  3ia;  or  26x  +  19z  =  896,  or  x  =  igjj  daj8 

3i 

29.  Let  X  =  Cs  snare  in  cents  ,then  -y  -  2540  =  Ds  share; 

■^(-^  -  2540)  4-  4000  =  y  +  2984  =  Fs share,  and  (x  +  y  -  2540 

4x  3x        4r 

+  Y  +  2984)  =  2x  +  444  =  ^'s.  Then  x  +  y  -  2540  +  y  +  2984 

+  2x  +  444  =  718900,  or  4x  +  888  =  718900,  or  4x  =  718012  .-.  x 

3x        538509 
=  $1795-03  =  C's  share  ;  y  -  2540  = = 2540  =  $743-895, 

&c. 

30.  Let  X  =  the  number  of  days  required  ;  then  since  4  men 
can  do  it  in  9  days,  1  man  can  do  -^^g  of  it  in  1  day;  similarly  a 
woman  can  do  V,,,  and  a  child  jj^  of  it  in   1   day.     Hence 

^  +  ;r;^  +    ^^TT    =  l,  or  35x  +  54x  +  42x  =  1260,  or  131x  =  1260 
Ob        70  oO 

31.  Let  X  =  right  hand  digit,  then  14  -  x  =  tiie  left  hand  digit 
and  10(14  -  x)  +  x  =  the  number.     Hence  i'7(140  -  9x)  =  ^x 

32.  Let  X  =  value  of  the  property;  then  8600  -  x  »  gain  had 
the  note  been  good,  and  x  -  (8G00  -  640)  =  x  -  7960  =  loss 
when  note  proved  worthless.     Hence  x  -  7960  =  j(8600  -  i) 

33.  Let  X  =  weight  of  head;  then  x  -f  9  =  weight  of  body. 
Hence  x  =  9  +  J(x  +  9),  or  2x  =  x  +  27,  or  x  =  27  =  weight  of 
head ;  and  body  =x  +  9  =  27-f9  =  36.  Hence  fish  weighs  9  +  27 
+  36  =  72  lbs. 

34.  Let  X  =  his  capital ;  x  +  Jx  -  1000  =  capital  at  end  of  1st 

year;  Ji  -  1000  +  l(Jx  -  1000)  -  1000  =  capital  at  end  of  2nd 

16x  -  21000      16x  -  21000       1  /lex  -  2inoo\ 
year  = ;    ^ 


1  /16x  -  2inoo\ 
+  y( i j-1000 


Ex.  sxxni.]  ALGEBKA.  35 

64x  -  84000 
=  — •„- -  1000  =  capital  at  end  of  3rd  year.     Hence 

C4x-  -  84000 

^„ 1000  -  2x 

X 

35.  Let  X  =  the  distance  in  feet,  then  —  =  number  of  revol- 


utions of  the  fore-wheel,  and  -j-  =  revolutions  of  the  hind-wheel. 

X 

T 


X  X 

Hence  —  =  -t"  +  w  .•.  ox  =  a.r  +  abn^   whence  bx  —  ax  =  abn   .•. 


abn 


36.  Let  X  =  number  of  minute  divisions  the  hour  hand  passes 
over  before  the  minute  hand  overtakes  it;  then  the  minute  hand 
must  pass  from  XII  to  XII,  i.e.  60  minutes  plus  x  minutes  in  order 
to  overtake  the  hour  hand,  that  is  while  the  hour  hand  passes 
over  X  minute  divisions  the  minute  hand  passes  over  60  +  a: 
minute  divisions,  but  the  minute  hand  moves  through  twelve 
times  the  space  the  hour  hand  travels  in  a  given  time.  Hence 
12x  =  the  space  travelled  over  by  the  minute  hand,  while  the 
hour  hand  goes  over  x  minutes.  Hence  12x  =  60  +a;  .-.  llx  =  60 
and  consequently  x  =  S-^i ;  that  is  the  hands  will  be  together 
for  the  first  time  after  XII  when  the  hour  hand  has  passed  over 
5-j5,  of  the  minute  divisions,  i.  e.  in  5-^\-  x  12  =  1  h.  5,5f  m.,  and 
similarly  they  will  be  together  again  1  h.  S/f  m.  afterwards,  and 
so  on.  Hence  they  will  be  together  at  1  h.  5/,-  m.,  2  h.  10}^?  m., 
3  h.  16-1*1- m.,  4  h.  21-1^,- m.,  &c.,  and  they  will  be  together  a^ 
often  as  1  h.  5^^^  m.  is  contained  times  in  12  h.,  i.e.  11  times. 

37.  Let  X  =  the  greater  part,  then  96  -  x  =  the  less.     Hence 

X 

~=-  +  3(90  -  x)  =  30;  clearing  of  fractions  we  have  x-\-  2016  -21x 

=  210,  whence  x  =  90-,'y-  =  the  greater,  and  96  -  x  =  96  -  90-f  j 
=  5i"y  =  the  less. 


86  KEY  TO  fEx.  xsxiu. 

3x  3x     9x 

38.  Let  X  =  B'3  share,  then  —  =  ^a  share,  |  of  —  =  --  =  C's 

3x      9x 
share,  consequently  X +  —  + —  =  25C0,  whence  by  clearing  of 

fractions  4x  +  Cx  +  9x  =  10240  ;  that  is  19x  =  10240,  whence  x 
=  $538-94i^  =  B's  share,  .-.  ^'s  share  =  ^  of  B'a  =  $808-42-2^-,  and 
C's  share  =  |  of  .4's  -^  $1212-63-,V 

39.  Let  X  =  rate  down  .-.  28x  =  distance,  and  x  -  5  =  rate  up 
the  river,  and  x  -  3  =  rate  up  the  lake  ;  length  of  river  =  ^  of 
28x  =  12x,  .-.  length  of  lake  =  16x 

12x         16x    _  19/28x\  3  4  19 

^^^"^    x-5  "^  x^3  ^  21  \x^/   •■•  1^5  "^  7^  ~  3(x-5) 


10 


(^r3)  =  3(^^)  •••   12X-60  =  lOx-30,  x=15,  x-5=10, 
X  -  3  =  12,  and  28x  =  420 

40.  Let  X  =  the  whole  property,  then  $1800  +  ^(x-  1800) 

=  $1800  +  y  -  $300  =  y  +  $1500  =  share  of  the  eldest ;    also 

X  -  f -T-  +  $1500)=  —  -  $1500  =  part  remaining,  and  $3000 

(5x  5x  5y 

Y  -  $1500  -  $3600)  =  $3600  +  ^  -  $850  =  3 g  +  $2750 

=  share  of  the  second,  but  these  shares  are  equal.     Therefore 

X  5x  X 

— +  $1500  =  ^-  +  $2750,  whence  x  =  $45000  and  —  +  $1500 
b  36  6 

=  $9000  =  share  of  each ;    also  $45000  -=■  $9000  =  5  =  number  of 

children. 

41.  Let  X  =  the  left  hand  digit,  then  x  +  7  =  the  right  hand 

Higit ;  also  lOx  +  x  +  7  =  the  number,  and  x  +x+7  =  2i  +  7 

lOx  +  x+7  7 

c  sum  of  the  digits.     Then  — ^        » —  =  2  +  ox  +  1  ^^*^°^^^ 

llx  +  7  =  4x  +  14  +  7,  and  .•.  x  =  2,  x  +  7  =  9,  consequently  the 
number  ia  29. 


Ex.  XXXIII.]  ALGEBRA.  37 

42.  Let  X  =  5's  share,  z  -  20  =  C's,  and  f  (2i  -  20)  +  80  =  .Ta. 

2(21-20) 
Then  x  +  x  -  20  +  — ^ r +  80  =  2100  ;  whence  lOx  -  100 

+  4r  -  40  +  400  =  10500  .-.  14x  =  10240,  and  x  =  $731-425  =  B'a 
share  ;  also  $731-42'f  -  $20  =  $711-42f  =  C's  share,  and 
^($731-42^  +  $7ll-42'J)  +  $80  =  $657-14f  =  ^'s  share. 

43.  Let  X  =  the  number  of  rows,  then  x^  +  75  =  number  of  trees 

also  X  +  6  rows  each  containing  x  —  5  trees  =  x^  +  x-30  +  5  =  the 

number  of  trees.     Then  x^ +x- 30  +  5  -  x^+ 75;  whence  x=  100 

.-.  x^  +  75  =  10000  +  75  =  10075  =  number  of  trees. 

n 

44.  Let  r  =  one  part,  then  a  -  x  -  the  other ;  and  x  =  — (a  -  x) 

na 
.-.  mx  =  na  —  nx,  or  mx  +  nx  =  tm  .•.  x  -  — ; — ;    also  a  —  x 

m  +  n' 
na  ma  +  na  —  na  ma 


m  +  n  7/1  +  n  m  +  n 

45.  Let  X  and  60  -  x  =  the  two  parts,  x  being  the  less  ;  then 
x(60  -  x)  =  Sx^  .-.  eo  -  X  =  3x,  and  x  =  15  =  the  less  ;  whence 
60  —  X  =  45  =  the  greater. 

46.  Let  X  =  the  growth  in  acres  of  one  acre  of  grass  for  one 

week.     Then  the  growth  of  Si  acres  for  4  weeks  =  x  x  ^^  x  4: 

40x 
=   -r-  ;  and  the  growth  of  10  acres  for  9  weeks  =  .r  x  10  x  9  =  90x. 

Therefore  the  whole  quantity  of  grass  eaten  in  the  first  case 

40x               40x  +  10 
=  -  _-  +  3i  = ^ acres,  and  the  quantity  eaten  in   the 

second  case  =  90x  +  10 

Hence  in  the  first  case  the  quantity  of  grass  eaten  by  one  ox 

40x  +  10  20x  +  5 

= -„- X  i  X  -jij-  =  — i^^ — ,  and  in  the  second  case  thequan- 

90x  +  10 
tity  of  grass  eaten  by  one  ox  =  (90x  +  10)  x  |  x  ^'j-  =  — r-- — 

But  by  the  question  an  ox  in  the  first  case  eats  as  much  as  an 
ox  in  the  second  case 


i>8  KEY   TO  [Ex.  xxxiii. 

20X+5     90x  +  10 
Therefore  — W7^ —  =  — r-r — ;  whence  x  =  -^  of  aa  acre 

„  20x  +  5     2.0  4.  5  80 

Hence  — ^j—  =      ^^    ■  =  ^^  ^  ^^  =  /^^  =  fractional  part  of  au 

acre  of  grass  eaten  by  one  ox  in  one  week,  .•.  one  ox  in  18  weeks 
will  eat  -^^  X  18  =  f  acres. 

Now  since  each  acre  increases  at  the  rate  of-,^j-  of  an  acre  per 
week  24  acres  will  increase  2  acres  per  week,  and  in  18  weeks 
the  24  acres  increase  by  36  acres,  and  therefore  become  equiv- 
alent to  60  acres 

Then  60  acres  ■^  §  acres  =  36  oxen. 

47.  Let  X  =  the  first,  then  nx  =  the  second,  and  mx  =  the  third. 

a 

Therefore  x  +  iix  +  mx  =  a,  whence  x  =  — — - —  =  first;  second 

na  ma 

=  nx  =  r- ; —  ;  and  third  =  mx  =  ;— 

1  +  /n  +  »i '  1  +  m  +  n 

mx  px 

48.  Letx  =  the  first,  then =  the  second,  and  —  =  the  third. 

'  n  ? 

mx      px 
Therefore  x  +  — — -  +  —  =  a,  whence  nqx  +  mqx  +  npx  =  nqa,  and 


Q 
anq 


=  the  first  part 


Second  part  = 


nq  +  ?ftg  +  np 

mx         m  anq  amq 


nq  +  mq  +  np         nq  +  mq  +  np 


px        p  anq  anp 

Third  part  =  —  =  —  x  ; ; =  ; ; 

'■  q  q         nq  ■{■  mq  +  np         nq  +  mq  +  np 

49.  Let  X  =  the  number  thrown  by  the  first  aftef  the  second 

commences ;  then  x  +  36  =  whole  number  thrown  by  the  first ; 

1.V 

•  =  the  number  thrown  by  the  second.     But  every  4  charges 
8 

of  the  first  consume  as  much  powder  as  everj^  3  charges  of  the 

second,  and  they  are  to  consume  equal  amounts   of  powder, 

X  +  36       tx 
.•.  — z —  =  — ;  whence  x  =  216  =  balls  thrown  by  the  first 


Ex.  XXXIV.]  ALGEBRA. 


39 


after  the  second  commences.     Therefore  |  of  216  =  189  =  balls 
thrown  by  second. 


EXKKCISE   XXXIV. 

ft 

10.     4x  +  6y  =  2a  1  11.  12x  +  4:uy  =  Am 

I5x-6y=3b)  12x  +  3by  =  3n 


19x  =  2a  +  36  4ay  -  3by  =  An  -  3n 

2a  +  3b  (4a  -  3b')y  =  Am  -  3)i 


^^       19 

3y  =  a-  2x  '^  ~  4a  -3b 


Am  —  3/) 

y  = 


Aa  +  66  Aam  -  3'^n 

3x  =  m-  ay  -  m  -  — ;; rr— 

•'  4a  -  36 

4affi  —  3bm  —  Aam  +  '^un 
3y  = T^ 3x  =  — 


a  — 

19 

19a 

-4a- 

66 

19 

15a 

-  66 

19 

5a- 

26 

^y-         lo  ^^^      4a -36 

an  —  bm 


Aa-  36 
3an  —  367/1 


y=       ia  «  = 


19  4a  -  36 


12.  2a6x  -  Aaby  =  26^  13.  x  -y 


2abx  -  aby  =  ac  x^  -  y 

3aby  =  ac  -  26^  i^-y)(^  +  y) 


ac  -  262 
^"^"306" 


:} 

)(x  +  2/)  =  6-l 
a(x  +  y)-bj 


ax  +  ay  =  b 


2ac  -  46^  ax  -  ay  =  a" 

axr-b  +  2ay  =  b  + ^r — -  ~ ^— 

^^  2ax  =  a^+  b 

362  ^  2ac  -  Ab^  a^  4-  6 


ax  = 


36 

2ac  - 

62 

36 

2ac  - 

6» 

2ay  ~  b  -  a? 
b-a^ 


dab 


40  KEY   TO  [Ex.  2XIIV. 


14.  ex  —  ay  =  acm 

(m  -  c)x  +  (m  +  c)y  =  acm 


] 


c(m  -dfit  -  a(m  -  c)y  =  acm(m  -  c) 

c(rty^c)x  +  r(m  +  c)y  =  ac^m. 

mcy  +  c-y  +  amy  -  acy  -  ac^m.  -  acm~  +  achu 

acm(2c  -  m) 

"      7)ic  +  cr  +  am  -  ac 

la'-c^m  -  a^cm^ 

ex  -  acm  +  ay  =  acm  +  r, ■ 

7ftc  +  c'  +  am  —  ac 

am(mc  +  c^  +  am  -  ac)  +  2d?cm  -  ahn^ 

X  =  5 

mc  +  c'  +  am  -  ac 

am^c  +  amc^  +  a'^m'^  -  ahm  +  2ahm  -  a^m^ 

mc  +  c^  +  am  -  ac 

am^c  +  amc'  +  d'cm. 


15. 


mc  +  c^  +  am  -  ac 

bm 

bn 

X 

+  —  =  ab 

y 

bm 

mq 

X 

-—y'  =  bm 

bn 
J"- 

mq 
— —  =  ab  -  bm 

16.  (x-y)(x  +  y)  =  55| 

X  +  y=  11     J 

ll(x-y)  =:  55 

X  -y  =  5 

x  +  y=  11 

2x  =  16 

(ab  -  bm)y  =  bn  +  mq  x  =  8 

bn  +  mq  2y  =  6 

•'  ~  ab  —  bm  y  =  3 

n  n 

=  a-  J  =  <'-  b7+l^  17.  459x  -  463y  =  -  495-» 

ab  -  bm  -  5x  +    19y  =  131    J 

abn  -  hmn  2295x  -  2315y  = -l4"75 


bn  +  mq  -  2295x  +  8721y  =  60129 

abn  -i-  amq  -  abn  +  bmn  6406y  =  57654 

bn  +  mq  y  =  9 

aq+bn  5x  =  I9i/  -  131  =  171  -  131 

mq  +bn  qj.  -  40 

mq  +  bn  3.  =  g 

aq  +  bn 


Ex.  XXXIV.]  ALGEBRA.  41 

18.  ex  —  ay  =  acp 
ax  ~  cy  -  a?  +  c^ 
acx  —  d^y  =  d^cp 
acx  -  c*y  =  a'^c  ■\-  (* 

c^y  —  d"y  =  a^cp  -  a?c  —  c" ' 


v?cp  -  al^c  -  c" 


c'  -  a" 


a?c'''p  —  flV  —  <:* 


+  2666J 
=    1261 

=  89i7J 


/x  =  a"  +  c^  +  CM  =  «  +  <-  + a iS 

^  c  —  a 

c*  -  a*  +  a^c^p  -  o?c^  —  c*  ac^;j  —  a'  —  ac' 

c^  -  a^  c^  -  a' 

(19) 

35i-  -  126  +  2Qxy  +  49!/  =  2Sxy  +  76j/ 

72x2-  goxy  -  Vo'^  +  42)/  -  f  i  =  72x2-  go^jy  +  930y-  1002x 

35x-      27y=    126 

3293X  -  2960y 

115255X-    889Ily  =  414918 

115255X  -  lOSeOOi/ =  312095 

146897"=! 02823  •'•  2/  =  " 

35x  -  126  +  27y  =  126  +  189  =  315  /.  X  =  9 

(20) 
i(u'^  -  b^)x  +  5(a2  _  62)y  ^  8^26  _  2a62 

Z(d^  -  ¥)x  +  3(a  +  6  +  c)by  =  3a''6  +  Qab^  + — 7— 

"'^^  Zab^c 

5(a^  -  b^)y  -  3{ab  +  b'^  +  bc)y  =  8d^b  -  2ab'^  -  3d'b  -  6ab^  -  — —r- 

5a^b  -  3a-b^  -  8ab^  -  3a6% 
(5a-  -  8b'  -  3ab  ~  3bc)y  = — r 

(5a2  -  86^  _  3ab  -  3bc)ab 
(5a^-  862-  3ab  ~  3bc)y  =  ^ ^^—^^ 


y 


ab 
a+b 


8a^b-2ab^  8a^b-2ab^      5ab      8a^b  -  2ab^  -  5a^b  +  5ab' 

3a'^b  +  Sflft^       3ab(a  +  6)  3ab  ab 

^^"  "      u^  -  b'      "  ~a'^  -  b'     '    ^^  ""  S^  •■•  ^  ""  ^Tfr 


42 

KEY  TO 

[Ex.  xxxv. 

ElKEClSB  XXXV. 

1 

—  + 

X 

(6) 
1        5 

7  ~  "6" 

(7) 

X  +    y  =    xy 

X   +      2    =    2X2 

X 

1 

7  ~ 

1         3 

T  ~  T 

15          3          1 

T  ~  T  ~    4    ^12 

2j/  +  22  =  Zyz\ 
1         1 

1         1 

1 

—  + 

1               t 

2                       "12 

3 

=  4 

-7  +  ^  =  2 

y 

1      1 
T-7  =  ' 

2 

8 
=  —  ;   81/  =  24;  y  = 

1          7 

+  T  =  r2 

7         1         1 

^  12  ~  T  "^  T  •■•  ^ 

y 
1 

¥ 
1 

2         2 
— =  2 

2         y 

2         2 

T  +  7-^ 

'*' 

1 

1 

1         5 
5        11 

2 

-  =  5  ...  z  =  - 
4 

7  =  1  ■••  V  =  4 
1       1 

1      1 

4 
3 

(8) 

X  +  3)/+  22  = 

6 

(10) 
Add  all  four  equations  to- 

3x +  5i/  -  22  = 

7;i   • 

gether  and  then  - 

-  3 

8x  -  21/  +  22  = 

2n] 

t;  +  x  +  2/  +  2=23 

4x  +  8y  =  6    + 

7ft  1 

V 

+  x  +  y        =""13 

12x  +  24y  =  36  +  3;ft 
88x  +  24y  =  8»i  +  16ft 


irex 


=  5?ft  +  16ft-36 
5?ft  +  16«-3ft 


5m  +  16ft  -  36 
W 


76 

+  8i/  =  6  +  7ft 


2=   10 

»  +  X 

+  2  =    17 

y        =6 

p  +  y 

H-  2  =  18 

X 

=    5 

X  +  y  +  2  =  2  i 
=    •> 


Ex.  XXXV. J 

8y  = 

ALGEBRA. 

5m  +  16«-  36 
b  +  m-            ^g 

48 

8y  = 

22b+147n-l6n           lib  +  1m 
19              '  ^  ~             76 

-  8n 

s  = 

5m  +  16«  - 
7i-4tx  +  y;  z-n-             ^^ 

-  36 

116+  1m- 
+             76 

8n 

z  = 

z  = 

I6n  -  20m  -  64re  +  126  +  lift  +  ' 

Im  -  8n 

76 

236  +  4:71 -I3m 

76 

(9) 

abx  +  b^y  = 

be 

- 

abx  4-  acz  = 

■  a' 

bhj- 

■  acz  ■■ 

=  be  -  a' 

c^y  +  acz  ■■ 

=  be                     26c  -  a* 

(IS'  +  c'')y 

=  26c  -  a^ '  ^  ~   62  +  c2 

az 

26c^  -  a'^c 
^b-cy-b        ^2  ^.  c2    i  «2 

63  -  6c2  +  a^c 

6«  +  6c2  -  26c2  +  a'c 

62+c2 

z 

ax  = 

X  = 

a6'^  +  ac' 

2h\  -  6a2 
^  c  -  6i/  -  c  -     ^.,  ^  ^2    J  ax 

C.8  _  62c  +  a^b 

ah"^  +  ac^ 

cb' 

2  +  c3  -  262c  + 

d'b 

62+c2 

( 

11) 

ax  +  ay  -^  c 

iz  =  a:'  +  ab  +  ac 

hx  Jr  cy  +  az^  a 

2  +  62  +  c2 

(a  -  b)x  +  (a  -  c)i/  =  ab  +  ae  -b'  —  (^ 

6x  +  6y  +  6s  =  a6  +  62  +  6c 

(u-  +  ay  +  6s  =  a2  +  62  +  c2 

(6  -  c)x  +  (6  -  a)J/  =  a6  +  6c  -  a2  _  c* 

(a  -  6)a;  +  (a  -  c)y  =  ab  +  ac  -b^  - 

(6  -  c)x  +  (6  -  a)y  =  ab  +  be  -  a^  -  c' 

{2ab-b'-a^)x  +  (a-c)(b~a)y=  2ab'^  +  aJbe-b^-b(^-a^b-a^c  +  a<^ 

{ab  -  ac  -be  +  c^)x  +  (a  -  c)(6  -  a)y  =  a-b  -  a^  -  ac^  -  6c2  -}  a'c  +  c' 


44  KEY   TO  [Ex.  XXXV. 

(b'  +  a^  +  c^-ab-ac-bc)x^b^  +  c'-a^-2ab'''-abc+2a^b  +  2a}c-2ac' 

b^+c^-a^-  2aP  -  2ac2  +  2(1^6  +  2a'c  -  abc 
^  "  b'^  +  a^  +  c^-ab-ac-bc  =b  +  c-a 

(a  -  c)y  =  ah  +  ac  -  b'  -  c^  -  (a  •-  b)  (b  +  c  -  a) 

(a  -  c)y  -  a^  -  ab  -  c^  +  be  .•.  y  =  a  +  c  -  b 

x  +  y  +  2  =  a  +  b  +  c.-.2  =  a  +  b  +  c-(b  +  c-a')-(a-\-c-b) 

z  =  a  +  b  +  c - b - c  +  a  -  a  - c  +  b  ;   z  =  a  +  b^c 

(12) 
nx  ^  ii'^y  +  a'z  =  ami 
ax  +    y  +  az  =  n     r 
(fx  +  ay  +    z  ~  p 
ah/  -  y  +  (^z  -  az  =  am  -  ri] 
ay  -y+     z  -  az  =  p 
(a^  -  l)y  +  (a^  -  a)z  =  am-n] 
(o  -1)2/-  (a  -  l)z=p 

'"] 

(a^  -  a)y  -  (a^  -  a)z  =  ap   -  an  j 

{2a' -a-l)y  =  ain  -  n  +  ap  -  an  .-.  y 2o'  -  o  -  I 

(a  -  l)z  =  (a  -  \)y  -  p  +  71 


(a^-  l)y  +  («^-  a)z  =  am 

-  an  \ 

am-n  +  ap-an 


("-1)^=  2^1 -P^" 

am  -  71  +  ap  -  an  -  2ap  +  2an  —p-¥n 


(a.  -  1)2  = 


2a  +  1 


am  -ap  +  an  -  p  am  -  ap  +  an  - p 

^"' "  ^^^  "  2^r+l  •■•  *  ""        2o2  -  a  -  1 


/am  —  n  +  ap  —  an  +  am  —  ap  +  an  —  p\ 
X  =  vi  -  a(y  +  z)=  m  -  a  (  2a^  -  a  -  1 ) 

/2am  -  n  -  p 

X  =  m  -  al  -T—i ~ 

{20*  -  a  -  I 


p\  2d^m  -  an-  ap 


2o*  -  o  -  1 

2o*//i  —  am  —  m  —  2a^m  +  an  +  ap  ap  —  am  +  an  —  m 

^  ^  2a* -a- 1  •■•  *  ^       2o»  -  o  -  1 


Ex.  XXXVI.  ALGEBRA.  45 

EXBEOISH  XXXVI. 

1.  7(x  +  y)  +  4y  =  50,  and  2(x  -  y)  +  3x  =  16 

2.  X  +  y  =  a,  and  bx  -  cy  =  0 

3.  Let  a;  =  price  in  cents  of  hay  per  ton,  and  y  -  price  of 

oats  per  bushel :   2x  +  35y  =  4400,  also  —  =  reduced  price  of 

4x  8x 

oats  and  —  =  increased  price  of  hay  ;    then  —  +  28y  =  5120. 

8x 
Hence  the  equations  are  2x  +  35y  =  4400,  and  —  +  28y  -  5)20 

4.  Let  X  =  length,  and  y  =  breadth ;  then  xy  =  area.  Then 
(x  +  20)(i/  +  24)  =  xy  4-  4180,  and  (x  +  2i)(y  +  20)  =  xy -i-  3860, 
which  two  equations  when  reduced  give  Gx  +  5y  -  925,  and  5x 
+  Gy  -  845 

5.  ix  +  iy=  11;   ix-1  =  iy 

6.  X  +  y  =  144,  and  fx  -  ^y  =  IJ 

X        4x 

7.  X  +  y  =  48,  and  —  =  — ,  or  xy  =  16x,  or  dividing  by  x  we 

get  y  =  16 

8.x  =  ^(y  +  z);  y  =  J(x  +  s)  +  6  ;  z  =  ^x  +  y)  ~3  ;  whence 
by  reduction  we  get  4x  -  5y  -  5-  =  0  ;  -  x  +  2y  -  c  =  12,  and 
-x-y  +  3z  =  -9 

9.  Let  X  =  sulphur,  y  -  saltpetre,  and  z  =  charcoal 
X  +  y  +  z  =  4000  ;  y  +  r  -  x  =  3240 ;  x  +  y  -  2  =  2760 

10.  X  +  y  +  2  ^  72  ;  ix  =  iy  ;  Jx  =  is 

11.  Let  X  =  space  occupied  by  one  shilling,  and  y  =  space 
filled  by  a  ten  cent  piece ;  then  16x  +  27y  =  1,  and  llx  +  13y 
=  -,5g^,  whence  x  =  7^14,  and  y  =  -/i,-;  wherefore  the  purse  would 
hold  ^^l^  =  40'jJ  shillings,  or  ^/  ?  44J  ten  cent  pieces. 


46  KEY  TO  [Ex.  xxiTi 

12.  Let  X  =  number  of  lines,  and  y  -  number  of  letters  in  a 
line;  then  xi/  =  number  of  letters  on  a  page.  Then  (x +  3)(y +  4) 
=  xy  +  224,  and  (x  -  2)(y  -  3)  =  xy  -  145 ;  or  4x  +  Sy  =  212,  and 
3x  +  21/  =  151 

13.  Let  X  =  left  hand  digit,  and  y  =  right  hand  one  ;  then  the 

lOx  +  y 
number  will  be  represented  by  lOx  +  y.    "Whence  9     ,  9..  _  ^ 

lOx  +  y 
=  3,  and     _       ^  =  13 ;  or  4x  -  5]/  =  -  12,  and  23x  -  My  =  65 


5y 

twenty-five  cent  pieces;    then  lOx  +  25y  =  8160,  and  —  -  6x 


14.  Let  X  =  number  of  ten  cent  pieces,  and  y  =  number  of 

reuty-five  cent  pieces;    then  lOx  +  25i 
=  4 ;  or  5!/  -  12x  =  8,  and  5i/  +  2x  =  1632 

X        x(a  -  1) 

15.  Let  X  =  rate  before  the  accident  .*.  x  —  —  =  

a  a 

=  rate  after  the  accident 

Let  n  =  the  number  of  miles  from  Kingston  at  which  the  ac- 
cident occurred. 

n  n  an  n 

•■•  x(«  -  1)    =  -^  +  ^  •••  x(a  -  1)  =  T  +  *  ('^ 

a 

n  —  c  n  —c  a{n  —  c)        n  —c 

And      ,„       . ,    =  +  d  .-.  —. rr  =  +  d  (11) 

x(a  -  1)  X  x(o  -  I)  X  ^   ' 

a 

an  n  n 

From  (i)  —T ^  -  —  =  — — — TT  =  6  (in) 

^  ^  x(a  -  1)        X       x(fl  -  1)  ^    ' 

a{n  —  r)        n  —  c  n  —  c 

From  (11)  -, ^.^  -  =  -7- TT  =  d  (it) 

^   '  x{n  -  1)         X         x(a  -  1)  ^    ' 

n  c  n 

,\  From  (iv)  —, rr  -  — ; TT  =  d.    But  (ill)  —. rr  =  6 

^    ^  x(a  -  1)       x(a  -  1)  ^     '  x{a  -  1) 

c  c 

••■  *  -  ^irri)  =  '^i  «^'  ''^  -  r::!  =  '^ 

whence  x(6  -d)  =  ^^.:x  =  ^j^—^^^^^ 


l-lx.  xxxYl.]  ALGEBRA.  47 

16.  Let  X  =  the  number  of  inside  passengers,  and  y  =  the  fare 
in  dollars  of  each 

Then  x  +  4  =  number  of  outside  passengers,  and  |(4y  -  h) 

-  fare  of  each 

Then  xy  +  (x  +  4)  x  i(4y  -  J)  =  45  ;  or  22x2/  +  32j/  -  x  =  634 
Also  i  y  =  fare  of  inside  passengers  for  half  way,  and  Y^iiy  -  i) 

-  fare  of  outside  passengers  for  do.,  and  the  whole  fare  was  in- 
creased by  -iV  of  $45  =  6- 

Then  iy  +  -3^-(4y  -  D  =  6 ;  or  38;/  =  171 ;  or  y  =  $4i,  and  this 
substituted  in  the  first  equation  for  y  gives  us  22  x  |x  +  32  x  S 

-  X  =  634,  or  98x  -  490  ;  whence  x  =  5  =  inside  passengers 

17.  Let  X  and  y  =  the  digits  ;  then  the  number  will  be  lOx  +  y 
Then  lOx  +  y  =  2xy  (i)  and  lOx  +  y  =  4x  +  4]/ ;  or  6x  -  Sy  =  0  ; 

or  2x  -  3/  =  0  (n) 

Adding  equations  (i)  and  (ii)  we  have  12x  -  2xy  =  0,  and 
dividing  this  by  2x,  and  transposing,  we  have  y  =  6 ;  whence 
X  =  3,  and  the  number  =  36 

18.  Let  X,  y  and  z  be  the  digits,  then  the  number  will  be 

100x4-  lOy +  c 

lOOx  +  lOy  +  z.     Then  y  =  Ux  ■{■  z)  ; — =  48,  and 

X  +  y  +  z 

lOOx  +  lOy  +  2  -  198  =  100-  -}-  lOy  +  x.     These  reduced  give  the 

equations  -  x  •)-  2y  -  s  =  0,  52x  -  38y  -  47=  =  0,  and  x  -  2  =  2,  <fec. 

19.  Let  X  =  the  oz.  of  j1,  and  y  =  oz.  of  5 ;   then  x  +  y  =  p  (1) 

b 

since  p  oz.  of  A  lose  b  oz.  in  water,  1  oz.  will  lose  — ,  and  .•.  x 
J-  p 

bx  cy  bx      cy 

oz.  lose  —  ;  similarly  y  oz.  of  B  lose  —  oz.  in  water  .•.  —  +  — 

J)  '  .''  '  p  P       P 

=  a  (II) 

From  (11)  bx  +  cy  =  ap  (ui)  and  multiplying  (i)  by  b  we  get 

6x  +  6y  =  hp  (iv);  then  (ui) - (iv)  gives  us  cy-by  =  ap- bp 

(«-^)P      .    .,     ,  (g  -  «)P 

.-.  y  =     ^_i,     ;  similarly  x  =     ^_^ 


48 


g^EY  TO  t^^-  ^^^"^^ 


-a 

fe^        M        Tf        00        o  I 


+  1  <B  (0  ¥^ 

i:-  H  O  II  II 

1  •  -§  5  ^ 

H  3  m  .'  •: 

+  ,  cs  ^     ^     ^        •  ,^ 

'^  »  .r,  O    O    O 


o    f:;     5 


3 

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H  bo  bfi  U)  bo  ^ 

-^j  '?  ^3  ,a  J 


Ex.  xxxYiit.]  ALGEBRA.  49 

BSBRCISB  XXXVIIL 

I.  (2a  +  3)S  =  (2a)6+6(2a)«3  +  15(2a)''3^+20(2a)33'+15(2a)23* 
+  6(2c)3«  +  36 

8.  (3  -  2m)«  =  35  -  5  X  3*(2»t)  -!-  10  x  3'(27«)^  -  10  x  3\2my 
+  5  X  3(2?ft)*  -  (2my 

9.  (3a-2yy  =  (Say  -  5(3«.)*(2(/)  +  l0(3a)\2yy-  I0(3a)2(2.v)« 
+  5(3a)(2y)*  -  (2,jy 

10.  (26  -  5c)'  =  (26)3  _  3(26)2(5c)  +  3(26)(5c)''  -  (5c)' 

II.  (3x-4ijy=  (3.r)*-  4(3x-)^(4y)  +  G(3a;)2(4y)^  -  4(3x-)(4i/)^ 
+  (4!/)* 

12.  (ab  +  3cy  =  (aby  +  5(aby(3c)  +  10(a6)^(3c)^  +  10(a6)-^(3c)3 
+  5(a6)(3c)*  +  (3c) « 
.     13.  (2ac  -  xijzf  =  (2acy  -  3(2ac)^(x)/2)  +  3(2ac)(xj/z)2  -  (xyzf 

14.  {(a  +  6)  -  cf  ^  (a  +  by  -  3(a  +  6)2c  +  3(a  +  6)c2  _  c^*  =  a» 
+  3a^6  +  3a62  4-  6*  -  3c(a^  +  2a6  4-  6^)  +  3c^(a  +  6)  -  c* 

15.  {2a  -  (6  +  c)}*  =    (2ay  -  4(2a)»(6  +  c)  +  6(2a)'(6  +  cf 

-  4(2a)(6  +  cy  +  (6  +  c)*  =  16a*  -  32a^(6  +  c)  +  24a'-'(6^  4-  26c  +  c^) 

-  8a(6^  +  36^c  +  36c2  +  c')  4-  6*  +  463c  4-  66  V  4-  46c3  4-  c* 

16.  {2(a  4-  6)  -  3c}'  =  2«  x  (a  4-  6)^  -  5  x  2-^(a  +  6)*(3c)  4-  10 
X  2\a  4-  6)*(3c)2  -  10  x  2^(a  4-  6)^(30)'  4-  5  x  2(a  4-  6)(3c)4  -  (3c)5 
=  32(a»  4-  5a46  4-  10a36^4-  10a^63  4-  5a6*  4-  6^)  -  240c(a*4-  4«*6  4-  6a262 
4-  4a6'  4-  6*)  4-  720c2(a*  4-  3a^6  4-  306^  4-  6^)  -  1080r'(a^  4-  2a6  4-  6^) 
4-810c4(a4-6)  -243c« 

17.  {(]  +x)-  x^}^  =  (1  +  xy  ~  4(1  4-  xy(x')  4-  6(1  4-  xy^x'^)'^ 

-  4(1  4- a;)(a;2)3  4- (x^)*  =  1  4- 4x  4- 6x24- 4.r'' 4- X*- 4x2(1 4- 3x  4- 3x^  4- a;3) 
4-  6x*(l  4-  2x  4-  x2)  -  4x^(1  4-  x)  4-  x« 

18.  {(a  -  6)  4-  2c}s  =  (a  -  by  4-  5(a  -  6)'*(2c)  4-  10(a  -  by(2cy 
+  10(a  -  6)2(2c)3  4-  5(a  -  6)(2c)*4-  (2c)5  =  a=  -  5a46  4-  lOa'^Z-  lOa^fi* 
4-  5a6*  -  6«  4-  10c(a*  -  40^6  4-  Ga'b''  -  4ab^  4-  6*)  4-  40c\a^  -  3a^b 
+  3aP  -  63)  4-  80c3(a2  -  7.ab  4-  6^)  4-  80c*(a  -  6)  4-  32c5 


50  KEY   TO  [Ex-  XXXIX. 

EXBBCISB  XXXIX. 

(1)  '  (2) 

4  +  2j;-12x*  x^  +  2x^  -  2x*- 

ix^  -  Sx*  X*  -  2x* 
+  9x*  -i-x^ 

(3)  (4) 

4x^- 12x''-2x*  l-a  +  4a^-2a' 

9x*  +  3x»  ia'-2'J'-fa* 

+  ix6  4a*  -  Aa" 


(5) 

1  +  2x  -  x2  -  a;S  V  2x4 

x^  -  x'  -    X*  +  2x* 

ix4+  Jx"  -X« 

ixs-x'' 

+  X8 

(6)  (7) 

4a^  -  4fi2x  +  Sa'x^  1  +  2bx  -  2cx' 

aV  -  ia^x'  6^x2  -  26cx» 

+  ia'^x*  + 


(8) 
a?  -  2ubx  -  2acx^  4-  2a<ix' 

b^x^  +  26cx»  -  2bdx* 

c^x*  -  2cdx^ 

(9) 

1  -  2a  +  262x2  -  2c'x«  +  2d*x* 

a'  -  iab'^x^  +  2ac'x«  -  2a<i*x* 

b*x*  -  26Vx»  +  26»(i*J« 

cBi6  -  2(^d*x'' 

+  <i'x» 


Ex.  XXXIX.]  ALGEBRA.  51 

(10) 

{(a  +  iff  =  (a^  +  3a?b  +  Saft^  +  fca)^  = 

+66 

(11) 

\(a  -  c)2f  =  (a2  _  2ac  4-  c^)* 
=  {(a?  -  2ac  +  c^Yf  =  (a*  -  4a'c  +  6a^c^  -  4ac^  +  c*)^  = 
a«  -  8a^c  4-  Ua^c'  -    Sa^c*  +    2a*c* 

lea^cz  _  48a«c'  +  32a*c*  -    8a^c^ 

36a*c*-48a%«  +  12a^c6 

+  r« 

(12) 

(aV  -  4cx  +  4)2  =  a*x*  -  8aV  +  Su^x^ 

IGa^x^  -  32ax 
+_16 

(13) 
4-12r+16x2-    2x3+ljx* 

9x2  _  24x3  ^  3_j.4  _  2x5 

16x*  -  4x«  +  §x6 

ix6  -  Jx^ 

+ix» 

(14) 
1  -  4x  -  2x2  +  43.8  _  2x* 

4x2+4x3-8x*  +  4x'' 

X*  -  4x«  +  2x6 

4x8  -  4x^ 

+  X8 


i>'2  KEY  TO  [Ex.  XLi. 

Exercise  XLI. 

(3) 

4j;*  +  12x'  +  5x2  -  6x  +  1  ^2^2  +  3x  -  1 
4x* 
4x-  +  3x\  12x*+  5x2 
12x3+  9^2 


4x"  +  6x-l\      -4x2-6x  +  l 
-  4x2  -  6x  +  1 

(4) 

This  may  be  worked  by  the  rule  or  it  may  be  bracketed  so  aa 
to  show  the  sq.  root :  thus  x*  -  2x2(i/2  +  1)  +  (yZ  +  1)2 

(6) 
9a4  +  12a'  +  34a2  +  20a  +  25(3a2  +  2a  +  5 
9a* 
C'i2  +  2a\     r2a3+ 34a* 

12a»4-    4a2 


Ga2  +  4a  +  5\  30a«  +  20a  +  25 

SOa^  +  20a  +  25 


(12) 
(X  -  yy  -  2(x2  +  i/^)(x  -  y)2  +  2(x*  +  y*)  =  x*  -  4jr^y  +  6x2yi 
-  4x(/*  +  2/*  -  (2x*  -  4x3|/  +  4x"y^  -  4x/  +  2(/*)  +  2x*  +  2y* 
^  X*  +  2x^y"  +  y*  ;  and  ^Jx*  +  2x'^y'^  +  y*  =  x^  +  y^ 

(13)  (a2  -  b'  -I-  c*  -  d«) 

a*  -  2a«  6«  +  i*  +  2a2c2  _  26''c2  +  c*-  2a2<i2  +  ib-d"^  -  2c«d3  +<i* 

«4 


2a2-i2\  _2a262  +  6* 
-2a2  62+j4 


2u2-26-+c2\  2a-c2-262c;+c* 
2a*c«-262c2+c* 


2a2  -  26*  +  2c*  -d2\-2o'd*  +  26«<iJ-2c*d*+</* 
-2a«rf«  +  26«rf»-2c*rf»+<i4 


Ex.  ^Li,  xLir.j 


ALGEBRA. 


(14) 

1 


—  Ix  + ^x* 


2  -  jx  +  xn   2x'''  -  ^x*  +  ^x* 


2x2  _  42-3  ^  2-4 


2  -  jx  +  2x2  _  la-sx  _  1^3  ^  1^.4  _  ja;6  4.  ^gj.6 
-  ix*  +  -^x*  -  Jx'  +  ^IjXS 


(15) 

x'     x*  y2  X       y 

ix*  +  —  +  —  -  xy  -  2  +  —  (Jx*  +  —  _  — 
*  y      y^  X-  ^  y       X 

J. -r* 


X2    + 


:)" 


.             x^' 

x' 

»       7 

+ 

r 

3.3 

x2 

y 

+ 

r 

2x       V  \  '/^ 

x2  +  — -—  I  _xy-2+^ 

y        X  /         ^  X- 


-  ^-y  -  2  +  -I 


Exercise  XLII. 

(2) 
a6  +  6^5  _  40^3  +  96a  -  64(a«  +  2a  -  4 


3a  ^ 


+  6a3 


+  4a2 


3a*  +  6a5  +  4a'' 


6a*  -  40a3 

6a5  +  12a4  +  8a» 


Sa*  +  12a*  +  12a2 

-  12a2  -  24a 


+  16 


3a*  +  12a3 


-24a+  16 


-  12a'*  -48a3  +  9Ga  -  64 


-12a'*  -48a3  +  96a -64 


54 


KEY  TO 


[Ex.   XLII. 


(4) 

aS  -  6a«  +  15a*  -  200^  +  ISa^  _  6a  +  l(a«  -  2a  +  1 


3a* 

-  6a«  +  15a*  -  20a» 

-6a^ 

+  4«2 

-6as  +  12a*-    8a» 

.3a*  -  6a^  +  4^-= 

3d*-  I2a»+  12a2 

3a*  -  12a»  +  150^  - 

6a  +  1 

3a^  -  Ga 

+  1 

3a*-  120^+  15a2- 

3a*-  12a='+  loa^  -  6a  ■{-  1 

6a  +  1 

(6) 


12x2- Sax +  4a» 


8xe-36ax'  +  102a2j:*-l7laV+204a*x'''-144a5x  +  64a6 
8x6 


12x* 

-36ax*  +  102a'''x*-  I7la»x» 

-18ax» 

+  9a^x2 

12x*- 1  Sax*  +  90^x2 

-  36ax«  +  54a2x*  -  2  7a»x» 

12x*-36ax»+27a2x^ 

48o^x*-l  44a'x'+204a*x*-l  440"  x 

24aV-3Ga«x 

+  64a6 

+  16a* 

12x*-36ax»+51aV-«-36a*x+ 16u* 

48a-x*-l  44a»x'-204a*x--l  44a'x 

+  64a6 

(8)  {a+b  +  c  +  d  +  e 

03  +  3a«6  +  3a6«  +  b^  +  3(a  +  i)*c  +  3(a  +  b)c^  +  &c. 


'"'  +  3a6^j. 

3a*b  +  3a6«  +  6» 

3a*  +  3a6  +  6« 

3a«6  +  3a6«  +  6» 

3(«+*)%3(a 

+  i)c^,. 

3(a  +  bye  +  3(a  +  b)c*  +  c' 

3(a  +  i)i  +  3(a  +  b)c  +  c* 

3(a  +  by-c  +  3(a  +  A)c'  +  c* 

3("  +  i  -^  0*  ^.  3(,  ^  , 

+  c)d  ^  ^, 

3(a  +  6  +  c)*d  +  Ac. 

3(a  +  -!'  +  cy  ■ 

1-  3(a  +  6 

+  C)(f  +  d* 

3(a  +  i  +  c)«cf  +  &c. 

£x.  XLm'j  ALGEBRA.  65 

Exercise  XLIII. 

/      _4  _S\-3        /      _3_S\-3        /      _ia\-3  i9.       , 

6.  (a     5xa     5)      =(^     s      s)      =  (a     '^)      =a^^=V«'' 
8.    {  (a='6^)Ha56V)^  xa'Z-v({a56}JcO'xa-ii~^c~'^-'^} '^^ 


p  \r»t 


-j,4^a 


( '-     :     ^\*^  •  1/4"^    /  -  +  jL  +  n-"*   x'»"^ 

Vy'  X  i/«  X  y    '  \v*      «        / 

/     n.l  +  .nrt+yr.Nrat 

Ix         "'         / 


y  X  y*  X  y   '  \y' 

yitri  2:^3t  +  mrt  +  prj  j.4»rg 


_  ^.nst  +  mrt  *  pn-  mergy4»ra  -  4ng  -  4r»  -  7»a 
_   -gmtr,'!  - 4)  +  rnt  +  prjtyina -ij- a.'4n ♦  7«; 

(10)  (11) 

J  -  3ab^  +  3ah  -  b^  (a' +  x^)  -  (ax)* 

a^  -  3a«6*  +  3ab  -  a*i^  (a^  +  x'  )  "^  -  (ax)  ^ 

-    a^b^  +3ab-3ah''  +  b^  =  a^  +  2a^x^  +  x^  -  a'x^ 

a'^-Aah^ +Gab-Aah^ +  b^  »  a*  +     Jx^  +  x^ 


66  KEY  TO  [Ex.  iim. 

(12) 

4x~  2x^  y~  ^  +  2xK^  -y-  ''■  +  y"  K^ 

2x'ty     ^-2* 

8x"  -  4xy  ~  "^  +  4x=*  -  2x^2/-i  +  2x^y  ~  ^.i* 

Axy     ^-2x='j/-i  +  2x'i/     ="2*-^     ^  +  3/-!-* 

-  4xz*  +  2x^1/ ~  ^z^  -  2x^s^  +  2/-!;*  -  2/  ~  ^c^ 
8x^  - 4x^y  ~  ^  +  6x^y  "■  ^z*  +  2y  - iz*  - y  ~  ^  -  2x^c^  -  y  ~  ^2* 

(13) 

-3x-*i/-2x-'\9x-^j/-  4x-''y-^(-  Sx'^  +  2x-*y-i 
gx-i'y  +  6x-* 

-6x-»  -Ax-^y-^ 
~6x-»-4x-'^y-^ 

(14) 

7     Z^^         5     Z~2         A    Z"i  2     _  4         i     ITS 

-a'i     '-a-'i     ^ -a*b     ^-a'b     "-a"*     »  -  J-i 

1     —i  5     _1  »     _5  i     _1         3     _5.         2     _8 

-a'i     '-a"*     *-a^6     ' -a^b     ' -a^b     "-a'b     ' 

£._9         5        a        i        4.        2_fi        s_&        I       ? 

^     _  ^  A  I  ^i^     _  5  2     1_  (i  '      _  " 

-a'-i     ^-a'-i'^-a'i     "-a'i     »-a'6     '-6-i 
-ah^^~ah~^-ah~^-a^b~^-ah~^-b-^ 


£x.  xuii.j  ALGEBRA.  67 

(15) 

x~^  +  1  +  x^)x-^  +  x~^  -I  +  x^  +x{x~^  -x'  ■'  +  l-z^+x' 

_   o  _    I 

x-^+x     ^+x     ' 


—  x' 
—x' 

-§ 

- 1  + 

-  X 

x*  4-x 

^-1 

X 

x~ 

^  +  x^  +x 

*  +  1  +  X* 

-  1  +a; 

-  1  -  x^  -  x^ 

^      i 

x^  +  x^  +x 
X*  +  x'  +  X 


(16) 

=  a^  -  2a^  +  2a^  +  2a^  _  2a  -  2a}  +  2 


a«  -  2a^  -  2a  +  2a^  +  2  -  2a~  ' 

l_2a~^-2a-i  +  2a~2 


a  +  2a^  -2-2a"*  +  2a-i 


a-^+2a     *-2a-« 
a-2-2ff.~2 


-  a^-2a^+3a^-3a+2J  +3-6a~^  +«-»  +  4a~* -a"* -.2a  ^ 

(17) 

a?  4-  2a*-l-2a"*  +  a~^(a*4-l-a""* 

o^^ 

2a* +  l)         2a*  -  1 

2a* +  1 

2a''  +  2-a~*)  -2-  2a "'  +  a~* 
-2  -  2a~*  +  a~^ 


58 


KEY  TO" 


(Ex.  XLIII, 


(18) 


(x*  -  2z^  +  3  -  2x~  W 


-i  .  .-§ 


X*  -4x+10x*-  16r^+19-16x~*+  10x~'-4z-i  +  x~^ 


2r    -  2x')-4a:+ lOx* 


—  4x  +    4x 


2x^  -  4x^  +  3)      6x^-16x*  +  19 


6x'-12i*+    9 


2x^  -  4x^  +  6  -  2x     ^  )  -  4x*  +  10  -  16x  ~  *  +  lOx  ~  * 
-4x*+    8-I2x~*+   4x~' 
2x^-4x*  +  G-4x'"^+x~^)      2-    4x~  * +  6x"  ^ -4x-i  +  x 

2-    4x~^+6x~'-4x-^  +  T 
(19) 
2x%j^  a-l-xV*")^  V-3^~    2/  +3x'y-i-xy-'(x~'y-  x^ij-'^ 


_  « 

_  i 


x-iy" 


-  3x     *y  +  3x'y-i- xy"" 

-  3x~*y  +  3x*y"*  -  xy~' 


(20) 


|x^  -  2x^y»  +  3y^ 
X»_6ii»y6  +  21x*y'-44ry'+63x'y*-64a:'y*+27y 


3x3 


-6xy« 


+  4x'y* 


A  +  2    1 

3x»-6xyo  +4x'y' 


-6x3yi  +  2Lr^y*-44»y^ 
-6x*y6  +  12x^y'-   8xy' 


Sx''-  12xy^  +  12x^y' 


9x^y*-18x*y^ 


+  9y' 


3xi  -  12xy6  +  21x'y*  -  18x*y* 


+  9y 


9x3y*-  36xy*-H)3x'y^-54x*y^+27y 


9x^y*-  36xy*+63x^y'-54i^y''+27j 


Ex.  xLiv.]  ALGEBKA,  69 

Exercise  XLIV. 

1.  2^  =  (22)i  =  4^;  7^  =  (75)^  =  343^;  2^  =  (2*)^  =165';  (i)^ 

^{(T)T';{a)T=a)-(^)* 

2.  a  =  a^  -  (a^y  ;  3^  =  9^  ;    (|)^  =  (\^)^  ;    (2a)^  =  (40^)^  ; 
(3(1^6)^=  (9a*b^)^  ■    (4a;Y)'  =  (16x*i/6)i 

=  (TiB)-*;  {(2a)-}-^-  (i^)~*;{(3a^*)T*=(2w)"'^; 

{(4xY)-}-*=(^)"* 

a-a^=(a*)^;  3*  =  (81)*  ;  (#  =  (^g-^)*  r  (2a)^  =  (16a*)*  ; 
(Sa-^i)^  =  (81a864)*  ;  (4xY)*^  (256x^y^)^ 

3.  «===  a^  (a-*)-^  -  (^^)"  ^  3^  =  (S'l)"^  =  (i)"^  5 

{(2a^63)-f  i=  (^^)"S  {(ac==)    T  ^  =  (-^)"^  (a^)"^-^ 
=  (-.^/4-)"^  3-^  =  ^  =  {(i)-^}-i  =  81-^j  (I)-'=(^)»=3^4V 

4.  a)"^x(5)U(A^x5)U  (5)i;  1(^^)-^  =  1^^)5  . 

2  /3a6\i  a/6\J      a 

5-y(t;     =5-iV3a6  =  W3a6;y(^-^j     =^^6; 

§(D^  =  K\*)^  =  §  X  IV14  =  i^Yi  ;  ^(^i^)i  =  |(.2^^.)J  =  -^4.»/20-; 
3a/6\-5      3a/2\3      3a/4\i      3a/46\i      3a.  _ 

t(,tj    =t(,tj  =^\j^)  =t(p;  =r6^^* 


60  KEY  TO  [Ei-  XLiv. 


25  \i 
90^ 


/3a-\-|  /9a<\-i  /25\i 

6-2«(-5-)     '=^2a(^^j        -2«(^^j     =(8^. 

/200\J  /9a\-i 

5  (^~4    y      ~    5  (^  16  y      ^    (^3125  '^    16  j     ~  ^3125^ 

2      /am  -  pg\  i  am-  pq"]  i 

(am  +  pq)^  x  (^^^^;^^r^^ J     =  {(am  -^  pqXam  +  /J?)  x  ^^j;^^^  j 

=  (aV-p^y,  or  („2^/_^25.j 

7.  (135)^  =  (27x5)*  =  3-^5 ;  ^/T62  =  V81  x  2  =  9V2  ; 
V80  =  Vl6x5  =  2^5 ;    T^324  =  7  ^27x12  =  7  x  3^12  =  21^12  ; 

(--)^  (-11^)  =  (yo4T)  n^)    "^^^  "^ 

8-|60iT^;    =1    3G(aH-x)^     (-   =1 36(a  +  xr^6a(a+^)  j.* 


=  6(^T^)V6«(«  +  ^); 
c     /  fc'^m'^       1  \       a      cm    I  \        cm    I  n      cm 

. f(«r- 22)29(6 +2)1 

J3'(a™  *  "x)  =  a^a^x  ■  ^V' ^ 1  =  (az  -  z^»  x 

=  ^-^^^{(6  +  2)(c  +  c)«-i} 

3V2  and  3^3,  or  18^  and  8l\  or  island  81?;  or  (5832^6 
and  (6561)^ 

or(^i)*j  (44)^  ^nd  (5103)*;  or  <ai!^j<i^)^, (85184)*  and  (5103)% 
or  (3528^'^)^,  (85184)^  and  (5103)" 


Es.  xLiv.]  ALGEBRA. 


61 


10.  1-V2  +  llv'2-'s/2  -  8V2  +  35/2  =  59V2  -  9v^  =  50V2  ; 
8V(0  +  4-/r5  -  VV15  +  Vlf  =  4V3  +  WIS  -  VVT5  +  ^VT5 
=  4V3  +  4VT5  -  2VT5  =  4V3  +  2Vl5 

11.  2V7  +  eV"?  +  3^3  -  4-^3  =  8V7  -  -^3  ; 

—  —      c*   —       /  <^*\    — 

3aU\lac  +  2a-Vac  -  -^V««  =  (  ^ai'-"  +  2a^  "  "b  1  V«c 

r2«Pi»i^^5+3a^"6;7^5-c^7^"=(2aP6"  +  3a3-»6-c2)^a365 
13.  GV200'=  6V100  X  2  =  60V2  ;    35V60  =  35V4x"i5  =  1Q\J\5 ; 

3  2  11 

(3  x  G«)  X  (4  X  606)  =  (3  ^  4)  x  (2166  ^  36006 
=  12^64  X  12150  =  24Vr2T50 


11.  16^'  X  86  =  yia  X  16x8x8x8=  V^^  x  32  =  4V32  ; 
28tt'6  X  ttff  =  28a6  =  28aV« ;  2  x  3^  x  72^  =  2  x  (2'7  x  72)6 
=  2V1944;  7V24X  J^'5  =  14^3  x  ^^5  =  "^^ 
ax     by      c'^d  i  i  'I 

1^-  67  ^  ^  '^  T  ""  ^^""-"^  ""  ^^^^   ""  ^'^^  ^ 

-  :r)/[(aa:yi'^  X  (iy)"i*^' x  (cc)'i'=^}  =  xy(a^b^<?x^ifz^y^ 
(X  -  ^/^j  -h  2/)(Vx  +  V!/)  =  ^-^  +  y^-     [See  Alg.  Art.  179.] 

16.  (2V3  +  SVi)(3A/2]  -  4V3)  =  (2V3  +  Ts'/^Xl^  -  ^3) 
=  3V30  +  yi^O  -  8V9  -  A\/45  =  3V30  +  VS  -  24  -  fV5 

"  17.  iVI  =  see)'  =  iV6 ; KD^  =  KW'  =  iWi4; 
2(9  -  f )^  =  2V^  =  Wio ;  m  -  \^)^  =  Wil  =  w\n  =  -/^Vlso 

.4x(^)*  =  W^7 

3/iy4\       3/VTiN       3/16Y^      3/16xl25\i  — - 

T  V^T/  "  3  V  VV  ~  3  U  /     "3  V243;    -  3  V729;       sV^'' 
-^ii^J  ^  ""3-Vl^3J  -  3  [a^.0    -  3  U6x-6>l    -  3aa;V« 


62  KEY   TO  [Ex.  xLiv. 

4V'3     5^4     6V7_4V9      5'V256      6^/49 
2V3  ~  2iy3  "^  2V3  "  2V3  ~  2'-^27    "^  2'V27 

=  2V3  -  K¥7^)^  +  3(1?)"^'^'=  2^3  -  —^^5038848  +  1^964467; 


c 


20.  Multiplying  by  V'^  -  6  we  hare  (V?)^  -  6^  =  7  -  36  =  -  29 
Multiplying  by  V3  +  V2  we  have  (V3)"  -  (V2)^  =3-2=1 
Multiplying  by  4V3  +  eV^Twe  have  (4V3)^-  (6V2l)2=  48  -  90  =  -  42 
Multiplying  by  \^\  -  ^2  we  have  (WO'-  (iV2)='=  ff-^=  2^1; 
Multiplying  by  5Vi  +  SVe  we  have  (WO'-  (3V.^)'=  -^-^'2  =  -tV4TJ 
2(V3-2V5)  2V3-4V5        2V3-4v5     4V5-2V3 

^^*  (V3  +  2V5)(V3-2V5)  ~  (V3)^-(2Vo)^~  ~3  -  20     "         17 

(V2   +   V3)(2V5    +    3V6)  2VTO    +    2\/T5    +   SyH  +    3VT8 

(2V5   -   3V6)(2V5   +    3V6)  ^  20  -  54 

2VIO  +  2-vn3  +  6V3  +  9V2 
~  -34 

(2V3  +  Vn)(7V8  +  8V7)  _   14V24  +   7^85  4-   16V21   +  8/7? 
(7V8  -  8V7)(7V8  +  8V7)  ~  392  -  448 

28V6  +  14/22  4-  16/21  +  8/T7 

=  =T6 =  ^'^^ 

3(V3  +  V^)  3V3  +  Z-Jx     {a'Jm-m-Ja){a'Jm-m^a) 


22. 


(■\/3-'Jx)(^j3  +  'Jx)~      3-x     '  (^a^Jln-m'Ja){a^Jm+m^a) 
(a^Jm  -  ■ni'Jay       a^m  -  2am^nia  +  Tn^a       a  -  2-Jma  +  m 
a^m.  —  fii^a      ~  dhii  —  in?a  ~  a  —  m 

(2  +  3V§)(WJ  +  ?V.?)       §VJ  +  W2  +  Vi  +  2V? 


iV2+-,\Vl5  +  iV3  +  ?/lO 


-ii 


&c 


L .    :.i-iv,  sxv.]  ALGEBRA  63 

(Vx-  +  X  +  1  -  ^Jx^-x-  l)(Vx^  +  x+  1  -  Vx^-x-  1) 

3    —  — — — — —  — 

(Vx^  +  X  +  1  +  V^''  -  ^  -  1)(V^^  +  X  +  1  -  V-=^  -  X  -  1) 


(Vx"  +  X  +  1    -    a/x^  -  X  -  1)^ 
(Vx^  +  X  +  D*  -  (Vx-  -  X  -  1  )2 


x^  +  a;  +  1  -  2(Vx^  +  x  +  l)(Vx^  -x-l)+x^-x-l 

(X2  +  X  +  1)  -  (X2  -  X  -  1) 

2x^  -  2Vx*  -  x^  -  2x  -  1       x2  -  Vx^  -  x^  -  2x  -  1 


24. 


2x  +  2  X  +  1 

V3-V2-V5 


(V3  -  V2  +  V5)(V3  -  V2  -  V5) 

V3  -  V2  -  V5 _    V3  -  V2  -  V5 

{(V3  -  V2)  +  V5H(V3  -  a/2)  -  V5}  ■  (V3  -  V2)='  -  5 
V3  -  V2  -  V5         V3  -  V2  -  V5        (V3-- V2  -  V^)  x  V6 


""   3  -  2V'6  +  2-5  "  -  2V6  -  2V6  x  V6 

VT8  -  a/12  -  V30       2V3  +  ^30  -  3V2 
"  ^12  ~  12 

(1  -3V2)(1  +  3V2+V3)  V3-3V6  -  1'^  _  V3  -  3V6  -  17 

(r+TV2  -  V3)(  r+3V2  +  V3)  "  (l  +  3V2)^  -  3  "        16  +  6V2~~ 

(V3  -  3V6  -  l'r)(16  -  6V2)       26V3  -  2YV6  +  5lV2  -  136 
"       (16  +  6V2)(16-6V2)       "  92 

(2  +  3V3)(1  +  2V3  +  V2)  20  +  ^3  +  2V2  +  3^6 


(1 +2V3-V2)(1  +  2V3  + V2)  11  +  4V3 

(20  +  7V3  +  2V2  +  3V6)(11  -  4V3)  _  136  -  3V3  -  14V2  +  25V6 

'  (11  +  4V3)(11  -  4V3)  ~  73 


Exercise  XLV. 


6.  Let  V42  +  3V174|  =  V^  +  V 
Then  V42  -  3Vr74|  =  Vx  -  Vy 
Or  V1764-  1568  =  Vl96  =  14  =  x  -  y 
Also  42  +  syi  74g  =  X  +  2Vxy  +  y  or  42  =  a;  +  y 
.-.  2x  =  56  or  X  =  28 ;  2i/  =  28  or  y  =  14 

.•.V42  +  3Vr74|  -  V28  +  V"!^  =  2V7  +  ^\l 


64 


KEY    TO 


[Ex.  XLV. 


9.  Let  Va  -  2V«  -  1  =  V^  -  V2/  (0 
then  Va  +  2^a  -  i  =  V^  +  Vl/  (")  J 
or  Vtt^  -4a  +  4  =  a;-y 
or  a  -  2  =  X  -  1/ 
Squaring-  (i)  we  get  a  =  x  +  y  .-.  2a  ~  2  =  2i,  4o. 


10.  Let  ^/2a  +  2^/(1^  -  6^  ==  V-»^  +  Vj/ 


V2a  -  2Va^  -  6^  =  V^  -  Vv 
V4a^  -  4a2  +  46^  =  -^46^  =  2&  =  x  -  y 
And  2a  =  X  +  y 


2x  -  26  +  2a ;    or  x  -  a  +  6 
2y  =  2a  -  26  ;    or  y  =  a  -  6 


11.  Let  Vs  +  V39  =  V^  +  Vl/ 
Then  Vs  -  V39  =  -Jx-^/y 


Or  V64  -  39  =  V25  =  5  =  X  -  Jf 
Also  8  =  X  +  1/ 
.-.  2x  =  13  or  X  =  -i;,',  and  2y  =  3  .-.y^ 


■.  ^J8  +  39  =  VV+  VI  =  iy/^  +  iV3 


12.  Let  J—  +  ih'Ja:^  -  b^  --  -Jx  -i- ^y 


VX  -  i  Va^  -b^  =  -^x-^/y 


\a*  \a*      a-b*     i*     a«     6« 


a*     6*  a«     t« 

Then  2x  =  y  -  Y  ;  or  X  =  -  ^— 


And 


x  +  y 


6* 


or  J/  =  -7-  .••  &o. 


Ex-  XXVI,  xLviiJ  ALGEBRA. 

EXEKCISE  XL VI. 


1.  V32  -  V24  =  V8(2  -  V3)  ••-  VV32  -  V^4  =  VV^C^  "  V^) 

=  VT8  -  V2  

2.  3V5  +  V40  =  V5(3  4-  2V2)  .-.  VsVo  +  V^O  =  VV&(3  +  2V^) 

=  V5V3T272  =  V5(V2  +  1)  =  V5(V^  +  1)  =  V20  +  V5 


3.  3VC  +  2VT2  =  V6(3  +  2V2)  .-.  Vs  V^  +  2Vr2"=  VV6(3  +  2V2) 
=  V6V3T2V2  =  V6(V2  +  1)  =  V^CV*  +  1)  -^  V^  +  V6 

4.  VTS    -   4   =   V^  -   Vl^  =  V2(V9   -  V8)   =  V2(3  -  V^) 
.'.  VvTs  -  4  =  VV2(3  -  V8)  =^  VWC^  -  V8)  =  V2(V2  -  1^ 


Exercise  XL VII. 

1.  4'f^^  -  2ypn^  =  12V"^  -  4-/"^  =  s/ITs 
(a  +  V"^)  +  (a  +  V"-^)  =  2a  +  (V^  +  'Jc)sJ~^ 

2.  V'=^  +  V"^^  +  V^^^Tl  =  (V5  +  V7  +  Vii)V^^ 

3.  V?  +  6V"-~2  =  V^  +  Vy ;  Vt  -  6V^  =  V^  -  V.y 

V49  +  72  =  11  =  x-y 

And  7  =  X  +  y 

.'.  a:  =  9,  and  y  =  -  2 

4.  (4V*=^  +  7V*^)(4V'^  -  V^^)  =  (4V^^)'  -  O^T^if 
=  (16  X  -  3)  ~  (49  X  -  2)  =  -  48  -  (  -  98)  =  -  48  +  98  =  50 

5.  ('vrr2-.3V^^)^  =  -2-6V6(V^^^+(9x-3)  =  -29  +  6v'6 

V2-V"^ V2-V^       V2  -  V^~5 

■  (V2+V^)(V2-V^)   ~    2 -(-5)    ~  ' 

7.  ai=3  X  -  V^n:  =  -  dy^yj^n. ;  +  l ;  V"^ ;  -  1    [See  Algebra 
Art.  193  (III)] 

8.  (a  -  V^^  =  0?-  2ffV"^^  +  (  -  a)  =  a'  -  2aV'^  -  <» 

B 


66 


KEY  TO 


(Ex.  xLVn. 


9.  (V2  -  V"^*=  (V2)'-  3(V2)==(V"^  +  3V2(a/^)^  -  (V"^)' 
=  V8  -  6^/A^/^n.  +  3V2(  -  4)  -  (V4V"=~i)^  =  2V2  -  12a^=T-  12V2 
+  S-f^  =  -  4/"^  -  10V2 

10.  Let  V-2- 2V^^5  =  V^^-Vy;  then  V-2  +  2V-  15 
=  V^  +  Vy  •■•  V*-  4(  -  15)  =  v^  =  8  =  a;  -  1/,  and  x  4-  y  =  -  2 

.-.  2x  =  6  ;   X  =  3  J   2y  =  -  10  ;   and  y  =  -  5.    Hence  V*  -  V? 

11.  We  are  to  find  the  square  root  of  0  ±  y  -  1 

Let  Vo  +  V~  =  V^  i  Vy 

then  Vo  T  V"-^  =  V^  T  Vy 

or  V»J  -  (  -  1>"=  Vl  =  1  =  ar  -y,  and  J?  +  j  =  0. 

.-.  X  =  I,  and  y  =  -  J 

Hence  Vo  +  V  -^  ;  that  ig  of  ±V~  =  Vi  iV"-! 
-  JV2  +  JV"^^ ;  or  JV2  -J-/^^ 

12.  Let\/3H-42V^  =  V^  +  Vy;  thenV31-42V-2=?V^-Vy 
.  .  V9tJl  -  l764(-2)  =  V9t>l  +  3528  =  V4489  =  67  -  x  -  y 

and  31  =  X  +  y  .-.  2x  =  98,  and  x  =  49  ;  2y  =  -  36,  or  y  =  -  18 

Hence  V^  +  Vy  =  V49  +  V  -  18  =  '  +  3V~^ 

4  + V"^^  _  (4  +  \r^)(2  +  V"^)  _  8  +  4V"=T+2V"^-2 
2-V^2~(2-V"^)(2+V^2^)'"  ^  -  (  ^  2) 

6  +  6V^^  

.  ^ =   1  +  V  -  2 

14.  7  -  /ri\  14  -  VT5  -  7V^^  -  2V^^(2  -  \'^ 

14  -  2V^ 

-  7V^-^  -  Vl5 

-  7V'^  -  Vl5* 

15,  (o  +  6V":ri)(a_J/ri>  =  aS-(JVTT)2=.a»-(t»r-l> 
x;  a»  -  (  -  fc'')  =  a^  +  6« 


13. 


*Thu9  -V-5x-V-3  =  -  V^V  -  1  X  -  V3V  -  1  =  V  16(V  -  1)' 
e  V^  X  -  1  =  ~  ^/TZ 


£x.  xLVtr  '  ALGEBRA.  67 


EXERCISH  XLYIII. 

1.  12  +  a;  -  4  +  4Vx  +  x ;  -or  4Vx  =  8 ;  or  V*  =  2  v  or  «  =  4 

2.  3z  -  6  =  2x;  or  x  =  6 

3.  X  -  24  =  X  -  4a/x  +  4  ;  or  4Vx  =  28  ;  or  Vx  =  7 ;  or  x  =  49 

4.x-  2Vxy  -+ai*'-  =  a  +  x,    or—  2»Ja  +  ax  "  ^  =  a ; 
-2   +  -=V«;or-=V'^+2;or-^^  =  ;^;^;^r-2TvS 


VVVVxi:T23^4^-5  +  ^  =  -3;orVvv^:irT23-^4^5->-6  =  9; 

or  V'VV^TTH^  -^^  =  ^  5  °^  VV^^Tl23  +  4  +  5  =  9;  or 


VVx  +  l23  +  4  =  4;   or  Vx  +  123 +4^  16  ;   or  Vx  + 123  =  12 ',    or 

x  +  123  =  144;  orx  =  21 

Va 

6.  ^ax  -  Vx  =  V'^ ;  or  V^(V^  -  1)  =  V'^i  or  V^  ~'ja  —  1 ' 

a 
or  X  =  7-7- — TTg 
(Va  -  1)" 


7.  2x+Vx*-x^  =  x*  +  2a:  +  l;  orVx*  -  x*=  x*4- 1;    orar*-ar* 
=  x*  +  2x2+l;  or3x2  =  -l;  orx2  =  -J;  orx  =  +  V'^  =  ±W^ 

8.  X  4-  34Vx  +  168  -  152  +  42Vx  +  r^  or-8Va>=  16  ;  or  Vx  =  2  . 
or  X  =  4 

9.  Vx  +  Vx  +  2  =  —         :  or  VxVx  +  2  +  x  +  2  =  4 ;  or^xJxT^ 

Vx  +  2 

s:  2  -  X ;  or  x(x  +  2)  =  4  -  4x  +  x^ ;   or  x^  +  2x  =  4  -  4x  +  jr^  ;  or 
6x  =  4  ;  or  X  =  J 

10.  c  +  X  +  2Va^  -  x^  +  a  -  x^ --ax  ;  or  2Va^  -  x*  =  ax  -  2a ;  01 
4a2  -  4x*  =  a^x^  -  4a2x  +  40=*;   or  aV  +  40?**  Aa'x;   or  a^x  +  4a 

4a» 


68 


KEY  TO 


[Ex.  XLVm. 


11.  a^+  2ax  +  x^=  a^  X'jW+l?;   or  2ax  +  i^  =  x^]l>^  +  x' ; 

or  2a  +  x  =  ^WTx^-,  cr  4d^+ 4ax +  x^-b^  +  x^;  ot4ax=b'-4a^; 

b^  -  4a2 

or  X  =  — 

4a 


12.  '^b'^  +  ax  ■[•  x'^  =  a  - b  - x]   or  b^  +  ax  t  x'^  =  a*-  2ab  -  2ax 
+  b^+2bx  +  x^;  or  3ax  -  2bx  -  a' -  2ab ;  or  x(3a~2b)  =  d'-2ab; 

a^-2ab 

13.  X  +   2a^x  +   Sb^/x   +  Gab  =   Aab  +  b^x  +  Aa>Jx  -r  x; 

ab  a^b^ 

or  2b'\/x  -  2a'\Jx  =  -  2ab]   or  V^  ~"^T^  ;  or  x  =  ,    _  ^.^ 

5(1  

14.  ^x  +  V4a  +  X  =  — ;  or  V^  +  ^^  +  V^a  +  4ax  +  x  +  x-* 

Vl  +  ^ 

=  2a ;  or  V4a  +  4ax  +  x  +  x^  =  2a-'Jx-\-x^)  or  4a  +  4ax  +  x  +  x* 

=  4a''^  -  4a  Vx  +  x^  +  X  4-  x^ ;  or  4a  +  4ax  -  40^^  =  -  4ffl  V*  +  ^'  i 

or  1  +  X  -  a  =  -  V^Tx ;  or  1  +  2x  -  2a  +  r^  -  lax  +  a*  =  x*  +  x  ; 

2a  -  a"  -  1 
or  X  -  2ax  =  2a  -  a*  -  1  ;  or  x  =  — \~2a — 

15.  X  -  32  =  256  -  32  'Jx  +  x ;  or  32  V^  -  288  ;  or  V^  ~  9  I 
or  X  =  81. 

b  /'    be    \{        c         /    4bc    \i  b  c 

16.  +2-1  -i 2       + =  I  ~2 i"       ;  or  ---—  + 

a  +  X         \a^  -  x''J        a-x      \  o'  —  x'  J    '        o  +  x     a-x 

=  0  ;  or  c6  —  6x  +  ao  +  ex  =  0 }  or  6x  -  ex  ■=  ab  -t-  ac ;    or 

a  (ft  +  <•) 


ft-c 


3Vx 


17.  X  +  V^  —  V^''  -  X  =  -«—   =   given  equation  multiplied  by 

Vx  +  's/x  ;  or  V-J^  +  1  -  V-^  -  1  =  I  (dividing  by  V^)  ;  or  2-/^'  +  2 
=  2'<^x  -  1  +  3  ;  or  2Va;  -1=2  '^x  -  1 ;  or  4x  -  A^/x  +  1  =  4x  -  4  ; 
or  4  V^  -  5 ;  or  16x  =  25  ;  or  X  =  f  g. 

18.  X  +  o  =  c*  -  2c  'Jx+'b  +  x  +  b;  or- 2c '/x  +  b=a-b-c'; 
or  4c*  (X  +  ft)  =  (a  -  b  -  c=0*;    or  4c-x  =  {a-c^-b^-  Ac^b  ] 

(^a-c*-  by-4c^b      /a-c='-ft\' 


4c» 


=(^^y- 


Ex.  XLViii,  xLii-.]  ALGEBRA.  69 


1        2        1        1  I    4  9  1        2         |4 


14449  149 

x"      ax     a^      a^ 
4x  =  8a;  or  x  =  2a 


-^  +  "+-2=^7  +  — 5  \  or  —  +  —  =  —  :    ora+-4a;=   9a  :    or 
x^      ax     a^      a^      x^  ^         x       a       x   ' 


or 


Jx  +  a  +  Jx  -  a     m        2  ^x  +  a     m  +  1 

20.    -^iz= ^-=.  =  -r   or     ^ = -. 

Vx-  +  a  -  V^  -  a      ^         2  V-r  -  tt     ™  -  1 

^Jx  +  a     m  +  1  X  +  a     7n^  4-  277i  +  1 

x/j;  _  (I  ~ m  -  i  '       X  -  a  ~  m^  -  2/?i  +  1  ' 


2x     2(m2+l)  X      m'^+l  a  {m^  +  I) 


EXSBCISB  XLIX. 

1.  x*=  9;  or  X  =  ±  3 

2.  18  -  18x+  18  +  18x  ^  100  -  lOOx^  ;  or  lOOx*  r;  64  ;  or  .E^  z:  .^^ ; 
or  X  -  i  I 

3.  4x2  :^  Sx^  +  9  ;  or  x^  =  9  ;  or  x  =  f  3 

4.  4x2  -  8  =  1  ;  or  4x2  =  9 ;  ^j,  ^2  _  2  j  or  x  -  +  g 

5.  x^  -  6x  +  9  =  13  -  6x  ;  or  x^  =  4 ;  or  x  =  i  2 

6.  3  (x2+  lOx  +  25)  -  1x  =  23x:  or  Sx^^  -  75;  orx2=-25;  or 

X  =  4  5  V^^l 

216x2  +  36        216x2  +  36 

7.  10x2+l7-10x2+8=-j-^^;gj^^;  or25.-y^^^^:^^;  or 

275x2-  200  =  216x2  +  36;  or  59x2=  236;  or  x2  =  4 ;  or  x  =  +  2' 


8.  V9  +  2x2  -  9  .  or  9  +  2x2  =  81 ;  qj.  3^2  =  72  ;  or  x*  ^  36  ;  or 

X  =  +  t) 


9.  ^/{x  -  3)(x  +  3)  =  3u ;  or  x2  -  9  =  9a2 ;   or  x'-*  =  9a2  +  9  ;   oy 
X  =  i  3  Vtt^+  J 


70  KEY   TO  [Ex.  xLii. 


10.  ■\/a'  -  X*         X  a  a^  -  x^     X*      2a     a* 

X         ^   T   ~  1'  °^      x^       ""  P"  ~  T  "^  ^  ' 
a«  X*      2a     a*  a-*      2a 

°'  ^-^=6^-T"^^'  °'6^  =  T-^i   orx2=  2a6-i^  or 


X  --  +  V^aA  -  6' 


a«  la*  a*  2a«  /a* 

4a-*      4a'62  4a*  4a^62 

or  —7-  -  —  ii"  +  ''*  =  — r  -  4i* ;  or  —  ,-  =  56* : 

X*  x^  X*  x^  ' 

4ttZ  4a«  2a  2a 

or  ^  =  56=^;    or  x''  =  gp-  or  x  =  +  -yVi  =  1  ^V^ 

12.  3ax2  -  cx^  =  d  -  1  -  6  ;    or  x^(3a  -c)  =  d-l-6;    or  x* 
d-l-b 


3a-  c    ' 


^-iV(^i^) 


1.3.  V«^  -x^  =  aVl  -  a:^  -  xVa*  -  1 ; 
a^-x^=  a*-  a*x^ -  2a^X'^{l  -  x'0(a=^ -  1)  +  a'l' -  a^' ; 


or  2xV(l  -  x'OCa^'-  1)  =  a^  -  a^x^-  1  +  x^;  or  2xV(l  -  x*)(a*-  1) 
=  (a2-l)(l-x=i);  or2x  =  V(o^-l)(l-a;^)i  or4x='=  (a^- 1)(1 -x^) 

a»-l 


14.  xV6*  +  a;H62+x»=c6^;  xV6*+i*=<:6^-6^-x*;  or  i^xHi* 

^  c^6*-  2c6*-  2c6-x2+  fc*+  26^x2+  x*:  or  2c6*x2-  6V=  tr^i*-  2c6* 

+  6*;  or  2cx^-x2=  c-6=*-2c62+i^=  (c2-2c+ 1)6*;  or  (2c  -  l)x' 

(c  -  1)262  (c  -  1)6 

=  (c-l)^6^  orx-^=     .^^_^     ;  or  x  =  i -=. 

V2x  , /2x    

— V3  +  ix+  jx=  ix-3;  or-2Y  -VS  +  i-c 

2^  /2x   X  2x  x' 

=  -6-y;   or  Y -^VS  +  Ix  =  3  + -y,  or  j(3  +  ix)  =  9  +  2x+ - 

X«  X*  X* 

or  -g-  =  9  +  y  ;  or  -  g  =  9 ;  or  x^  =  81  x  2  ;  or  i  =  i  9^3 


Ex.  XLXK,  L.]  ALGEBRA.  71 


11  1  » 

16.  a  +  X  +  3((i  +  xy(a  -  xy  +  3(a  +  x)'(a  -  xy  +  a-  x  =  b^; 

or  2a  +  3(a^  -  a:')*(^a4-x  +  ^a-x)  =  b^;   and  by  given  equation 

^a  +  X  +  ^a  -  X  =  b  ;   substituting'  this  in  the  last  equation  we 

oave  2a  +  3b^a^  -  x'^  =  b'' ;  or  3b^a^  -  x'^  =  6*  -  2a! ;  or  ^d^  -  x'^ 

6»-2a  /63_2fl\»  /6*-2tt\3 


36 


;  or  a^  -  x2  =  (^-^^""j  5  or  a-=  a^^-  i^-^T'J 


V/fi"-  2a\ 


Exercise  L 

1.  2x2  +  8x  =  90 ;  or  x^  +  4x  =  45  ;  or  x*  +  4x  +  4  -  49  ;  or 
r  +  2  =  ±'7;  x  =  5;  or -9. 

2.  x^  -  19  =  8x  -  10  ;  or  x^  -  8x  =  9  ;  or  x^  -  8x  +  16  =  25  ;  or 
«trx-4  =  i5;  x  =  9;  or-1 

3.  x^  -  8x  =  20  ;  or  x^  -  8x  +  16  =  36  ;  or  x  -  4  =  i  6  ;  x  =  10  ; 
or  -  2 

4.  X*  +  12x  =  45 ;  or  x^  +  12x  +  36  =  81 ;  or  x  4-  6  =  +  9 ; 
«  =  3;  or-  15 

5.  3x2+2x  =  85;  orx2+§x=\5;  orx''+ Ix  +  ^  =  2^5  +  |  =  ^^^  ; 
or  X  4-  i  -  i  -I3& ;  z  =  5  ;  or  -  5| 

6.  Sx^  -  14x  =  -  15 ;  or  x^  -  -i3^x  =  -  5 ;  or  x^  -  i<x  +  </ 
=  -6  +  ^^=^;  orx-5=±§;  orx  =  3;  orlj 

7.  5x2-236x  =  -4'7;  or  x^-sssj;  =  _  11  .  or x2-igax  + HV* 
=  -^^  +  -'W*  =  H¥'i  orx-H^-iHi;  orx-sgsori;  that 
is  X  =  47  or  ^ 

8.  4x2  -  8x  =  320  ;  or  x"  -  2x  =  80  ;  x'-*  -  2x  +  1  =  81 ;  x  -  1 
=  i9;  x  =  10or-8 

9.  x^  -  2x  =  -  a"  J .  or  x^  -  2x  +  1  =  1  -  a^ ;  or  x  -  1  =  ^/l-d^; 
ov  X  -  I  i  '^l  -  a^ 


72  KEY  TO  [Ex.  L. 

10.  5x'^  +  4x  ^  273  ;  a;^  +  |x  =  ^.p  j  x^  +  ^x  +  ^V  =  J-ifi  +  jV 
-J-^?^;  ^  +  |  =  ±='s^;  x=7or-7f 

11.  1x^  -  20x  =  32  ;  x^  -  s.Qx  =  3^2  ;  x^*  -  ^^'^x  +  i^<\,a  =  '^V  +  Va*^ 
=  W  ;  ^  -  't^  =  i  -7*  ;  X  =  4  or  -  If 

12.  x2  _  7x  =  -  12  ;  x2  _  7x  +  *4^  =  -  V  +  \^  =  i  i  -r  -  ^  =  i  i; 
X  =  4  or  3 

13.  Sx'-i  -  llx  =  -  6 ;  or  X*-  Jjix  =  -  2  ;  or  x^  -  -^-x  +  ^-^ 
-  _  2  4-  '  ^'  =  A?  •  X  —  J  I  =  +  '^  ■  X  =  .^  or  J 


14.  acx^  +  bcx  ~  adx  =  bd  ;   or  x^  + 


be  —  ad         bd 


X  = 


ac  ac 

be  —  ad  /be  —  ad\  ^    bd       {be  —  ady 

2ac     I  ~  ac  4a^c'' 


or  x^  +  —  X  +  /      ...      1  =  —  + .-272 


be  -ad     (be  -  ad\  ^   Aabcd  +  6 V  _  2abcd  +  a^i' 


^^•^  +      «c      ^\    2ac   y  -  4a^c^ 

6V   +    2abcd   +   a^rf^      /be  +  ad\^  be  -  ad        be  +  ad 

be  +  ad     be  -  ad  2ad      d  2bc  b 

"•^•'"i"!^^ — "uT'  ^  =  '2^  =  7;  ^"•"  =  -2it?  =  -T 

1  X  —  V-c 

15.  ,—  = -. —   by   dividing   the   given   equation    by 

X  +  V-^  ■)    or  4  =  x^  -  2xVx  +  x  ;    or  4  =  x(x   -   2^x   +    1) ; 

4  2 

or  —  -  (V--  -  1)' ;    or  71-  =  ±  (V^  -  1)  (i)  i  or  2  =  X  -  Vj;  ; 

or  X  -  V-c  +  i  =  I  +  i  -  I ;    or  V-»--  -  i  =  i  ^  ;    or  V-r  =  2  or  -  1 

.-.  X  :»  4  or  1 

2 
Taking   the   minus  sign   in  (i)  -n-e    have   -.^  =  -  V^-"  +   1 

or  2  =  -  X-  +  V-^  i  or  X  -  V-»:  =  -  2  ;  or  x  -  V-^  +  i  =  -  5  +  ♦  -  ~  4 
0,  ^x  -  I  =  ±  V^ ;   or  V-^-  =  i  ±  i  V^^  =  Kl  +  V^^^ 
or  X  -:  i  (1  1  2  V^^  -  7)  =  i  (-  6  i  2  V^^"?)  =  i  (-  3  i  V  -  • ) 
The  rejected  factor  x  +  V-^"  =  0  gi^es  us  x  =  0  or  1 

16.  x^  -  X  =  210  ;   x'^  -  X  +  i  =  *Ji  +  i  =  ^\^  ]    x  -  J  :-•  ±  =,«  , 
X  =  15  or  -  14. 


/' 


Ex.  L.]  ALGEBKA.  73 

17.  4x2  +    36   =    3x2   +    48   _   Hx  ;     or    x^  +    \\x    -    12  ; 
•  x2  +  1 1  X  +  -ifJ-  =  A^  +  -i-sJ-  =  i-Aa ;  X  +  V  =  i  V  ;  X  -  1  or  -  12 

x-2     x  +  2     x-3     x  +  4      x2-4x  +  4  +  x2  +  4x  +  4 

18.  — -;,  + „  =  -—„  + 


x  +  2     x-2     X  +  3X-4'  x''-4 

x2- 7x  +  12+x2+ 7x+ 12  2x2+8       2x^+24 

or 


x^-x-12  '         x2-4       x^-x-12' 

+  4         x^  +  12  2  x^       2x2-x  X      2x  -  1 


x^-4     x^-x-12'  8         X  +  24 

or  x^  +  24  X  =  8x  -  4  ;    or  x^  +  16x  =  -  4  ;    or  x^  +  16x  +  64  =  60  ; 
or  X  +  8  =  V60  =  1  2  Vl5~;  or  x  =  +  2  Vl5  -  8 

The  rejected  factor  x  =  0  gives  us  the  other  value. 

19.  49x='  +  42x  +  9  =  10(2x2  +  4x  -  6  -  2x2  +  3x  +  9) 
=  10(7x  +  3)  ;  or  49x2  +  42x  +  9  =  70x  +  30  ;  or  7x2  -  4x  ^  3  ; 
x2-^;  x=^x2_^x  +  ^ia-=  ?i-+^V=4y  ;  J:  -  t  =  ±  7  i  J^-lor-? 

20.  ax2  -/x2  -hx  -  ex  -  -h  -  c\  orfx'^  -  ax''  +  bx  +  ex  -  b  -i-  c 

/6  +  c\        b  +  c 
(/-  a)x2  +  (6  +  c)x  =  6  +  c  ;  x2  +  fyT^ j-^'  ^/TT^! 


G^:)^ 


6  +  c\2     b  +  c      /  b  ■>r  c 


X  +       „^       „.         =  7^-  + 


2/-  2a j      f-a      U/-  2a 


▼   f  -  a 


6+c\2        6+c 


f-a      \jLf-  -laj       2/-  2a 

1  111  1  mx  —  ax  +  am, 

'  a  —  m  +  x  ~   a       m       x  '  a  -  m  +  x  ~  ainx 

amx  =  Zamx  —  a^x  +  a'ln  —  m'x  —  ani'  +  7?ix2  -  ax^;    ax^  -  nx^ 

+  a^x  —  2amx  +  m}x  =  <^m.  —  am'  =  am,{a  —  ?n) ;    (a  —  m)x^ 

+   (a*  _    2am  +  7?i2)x    =   am(^a  —   m)  ;    x^  +   (a  -   m)x    =   am ; 

(a  —  m)2                  (a  -  m)^       4a)n  +  (a  _  77;)- 
x2  +  (a  -  7n)x  + =  am  ■{■  j =  ^ 

a*  +   2a'»  +   OT^         (a  +  m)2  a  -   m  a   +   /n 

=     : ;     i-   +    — 7i —  =   ±   — ^ —  ; 


4 
a  +   m        a   -  m 


=   ffi   or  —  a 


74  KEY  TO  [Ex.  L,  LI. 

_  2(a  +  i)         <a-  by 

22.  abx^  -  2x(a  +  6)Va6  =  (a  -  6)^ ;   z*  -        _      x  =  ^ — r— 

2(a  +  6)        (a  +  by  _  (a  -  by       (a  +  6)^  _  2 (a'  +  6=') 
~       Jab  ab        ~         ab  ah        ~  ab 


a  +  b         V-(«^  +  **)  a  +  b  ±'j2(a^+  b-) 

3;  -   -  ^  =  ±  -= j  X  =  _ 

'^ab  '^ab  'Jab 


ExilBCISB  Lt. 

1.  3x'  +  2r=85;   361^  +  24x  =  1020  ;   36j:*+ 24x  +  4=  1024  ; 
6x  +  2  =  i  32  ;  6x  =  30  or  -  34 ;  x  =  5  or  -  5} 

2.  4x2 -4x  ^840;  4x''-  4x  +  1  =  841 ;  2x  -  1  =  i  29  ;  2x=  30 
or-  28;  x  =  15  or  -  14 

3.  64x2  _  48J.  ^  1350  ;    54^2  _  48x  +  9  =  1369  -,    8x  -  3  =  ±  37  ; 
X  -  5  or  -  4i 

4.  X*  -  26x  =  -  25 ;   4x*  -  104x  +  676  =  -  100  +  676  =  576  ; 
2x  -  26  =  i  24 ;  2x  =  50  or  2 ;  x  =  25  or  1 

5.  5x»+4x  =  273;  100x^+80x=  5460;  100x2+ 80x+ 16  =  5475  ; 
lOx  +  4  =  i  74  ;  lOx  =  70  or  -  78  ;  X  =  7  or  -  7J 

6.  4x''+  8x  =  21  ;    x'+  2x=  3^1-;    4x*  +  8x  +  4  =  21  +  4  -  25, 
2x  +  2  =  i  6  ;  2x  -  3  or  -  7 ;  X  =  IJ  or  -  bj 

7.  llx«+  7x-4  =  14x;  llx»-  7x=  4;  484x2-  308X  =  176; 

484X'' -  303x  +  49  =   176  +  49  =  225;    22x  -  7  =  1   15  ;    x  =   I 

or  -  A 

be 

8.  a''x2+(a6- ac)x  =  6c;  ax^+(6-c)x  =  — ;  4u^x2+ 4a(6-c)x 

+  (6  _  cY  -  46c  +  (6  -  c)'*  =  (6  +  c)2 ;    2ax  +  6  -  x  =  ±  (6  +  c)  j  j 

e  b 

lax  =  2c  or  -  26  ;  x  -  —  or 

a  9 


Ex.  LI.]  ALGEBRA.  75 

9.  12x2  ^  120  =  16x  +  135  ;  12x2  _  iq^.  ^  ^5  .  3^^  _  4x  =  ^  ; 
36x2- 48x  =  45.  36x2- 48x  +  16  =  45  +  16  =  61 ;  6x-4  =  iV(JT; 
6x  =  4  i  V61 ,  x  =  l(4±  V6T) 

10.  Yx2  -  4xV3  +  (2  -  V3)x  =  2  ;  (Y  -  4V3)x2+  (2  -  V3)x  ^  2  ; 

-"'  +  2^rV3  =  7~r^  C^^°^«  7  -  4V3  =  (2  -  n/3)2] 
4  18  19 


•2  -  ^3"  ^  7  -  4V3  ~  7  -  4V3  "^  7  -  4V3  ~  7  -  4V3 
13  4  2 


^•^  +  9  _  .,/:-i  -  ±  o  _  ,/Q  ;  2x  -  - 


2  -  V3  ~  -  2  -  V3  '  2  -  V3        2  -  V3 

1  2 


~  2  -  V3  2  -  V3 

11.  x2  +  6ax  =  62.    a;2  +  6ax  +  9a2  =  6^^  +  9a' ;    x  +  3a 


==  ±  V9a*  +  6'' ;  X  =  i  V9a2  +  jz  _  3a 

45 -9x  63  +  36X     19(45  -  9x) 

12.  -^:^-3x.x ^^— ;   _^_— -=40x-€3j 

855  -  I71x  =  40x2  +  5Yjc  _  139  ;   I0x2  +  57x  =  261 ; 
400x2  4.  2280X  +  (57)2  =  10440  +  3249  =  13689 ; 
20x  +  57  =  i  117;  20x  =  60  or-  174;  x  =  3  or  -  8-/^ 

13.  xa  -  5x  =  -  m"^ ;    4x2  _  20x  +  25  =  -  4^2  +  25  ;    2x  -  6 


-  +  V25  -  4wi« ;  2x  =  5  ±  V25  -  iin^  ;  x  =  i(5  f  V25  -  4w*) 
14.  771x2  _  „_j;2  -  ^mx'Jn  =  -  mn;  (in  -  7i)x2  -  (2m'\}n)x  =  -.  mn 
2m\ln  mn  ^m.\Jn  4mn 

771  -  W  771  -  71  '  m  -  71     ~        771  -  71 

Sm^/n  Ain^n  4m,^n  4mn  4mn^ 

4x2  _  — !_   + 


771  -  71  (/7l  -  7i)2  (HI  _  7i)2  771  _  ;j  („i  _  „)  2 

277iVft         27iV''i  2n\Jm  +  2mV«        2m'Jn  -  2n'Jin 

2x =  + ;  2x  = —  or — 

771-71         m  —  71  in  -  n  m  —  n 

'Jinn('Jn  +  ^m)       »Jmn(A/m  -  t/n)               *J'fiin              Jmn 
t  = —  or ;  X  =  -7 or  — 


76  KEY   TO  [Ex-  LI.  Lii 


15.  1+: 


/a  +  1\  /I  +  X  +  x^  -  x(l  -  x)(l  +  X  +  x^)\ 
~  (a  -  y  (^  1  +  X  +  x^  ) 

1  +  X  +  x^       a  +  1     2  +  2x2       2o     1    +  x^ 

.l^::r+-x'  =  o^i  -2x—  "  T'  "^~  =  "!  1  +  x^  =  ax  ; 
x^  -  ax  =  -  1 ;  4x^  -  4ax  +  a*  =  o'  -  4 ,  2x  -  a  =  f  V"*  -  ■*  i 
X  =  1  (a  +  Va^~4) 

16.  X*  +  3x»  +  6  -  X*  +  3x»  +  13x2  +  ijx  -  60  ;  ISx*  +  7x  =  66  ; 
676x^  +  364x  =  3432  ;  676x='+  364x  +  49  =  3481 ;  26x  +  7  =  i  59  ; 
26x  =  52  or  -  66  ;  x  =  2  or  -  2/j 


EXBKCISE   LII. 

1.  x  +  2  =  0,  and  x  +  7  =  0  .-.  (x  +  2)(x  x  7)  =  x'  +  9x+  U^O 

2.  (x-4)(x  +  2)(x-  l)x  =  0;  or  x*  -3x*-6x»+  8x  =  0 

3.  (X  -  2)(x  +  2)(x  -  3)(x  +  3)x  =  0  ;  or  (x*  -  4)(x»  -  9)x  =  0  ; 
or  x»  -  13x»  +  36x  =  0 

4.  (x-5)(x  +  5)(x-2)(x  +  2)(x-3  -  V2)(x-3  +  V^)  =  0 
or(x2-  25)(x^- 4){(x - 3/-  i^2f\  =  (x»-  25)(x-'- 4)(x^- 6x  +  7)  -  Oj 
or  x6  -  6x'  -  22x*  +  l74x'  -  103x'*  -  600x  +  700  =  0 

5.  (X  -  l)(x  -  2)(x  -  3)(x  -  4)(x  -  5  -  ^J6)(x  -  b  +  ^6)  ^  0\ 
ov  (X-  -  3x  +  2)(x2  -  7x  +  12)(x^  -  lOx  +  19)  =  0  ; 
x«  -  20x''  +  154x*-590x»+  1189x'-  1190x  +  456  =  0 

6.  (x-5)(x-4)(x-  l)x  (x-2-V^^)(-E-2+V"^3)  =  0| 
or  (x^  -  9x  +  20)(x^  -  x)(x2  -  4x  +  7)  -  0  ; 
J.6  _  i4j,.^  +  7^j;4  _  206x'  +  283x^  -  140x  =  0 


£x.  Lii-l  ALGEBRA.  1 1 

I.  (x  -  5)(x  +  2)  =>  0  ;  or  J''  -  3x  -  10  =  0,  aad  (r*  -  6x'  +•  5x^ 
+  \2x  -  60)  -f  (x'^-  3x  -  10)  gives  us  i^  -  3x  +  6  =  0  ;  then 
a:2  _  3x  =  _  6  ;  x^  -  3x  +  |  =  9  -  2  A  =  -  J^?  ;  x  -  J  =  ±  ^  V"  15  i 
x=  i(3  +  V-  15) 

8.  (x  -  1  -  V'-^)(^  -  1  +  V^)  =  0  ;  or  x2  _  2x  +  7  ^  0 ;  and 
therefore  (x*  -  4x'  +  Sx^  -  8x  -  21)  ■=•  (x^  -  2x  +  7)  gives  us 
a;«  _  2x  -  3  =  0.  Then  x^  -  2x  =  3  ;  x^  -  2x  +  1  =  4  ;  x  -  1  =  +  2  ; 
X  =  3  or  -  1 

9.  (x^  +  6x2  _  3920)  -=•  (X  -  14)  gives  us  x^  +  20x  +  280  ^  0  ; 
X*  +  20x  =  -  280  ;  x^  +  20x  +  100  =  -  180  ;  x  +  10  =  +  6  V"^  i 
x  =  -  10  f  6V^ 

10.  X  =  0  is  evidently  another  root,  then(x*-  6x'+  13x^-  lOx) 
■f  (x^  -  2x),  gives  us  x'''  -  4x  +  5  =  0  ;  x*  -  4x  -  -  5  ;  x^  -  4x  +  4 
=  -1;  x-2  =  ±V~l;  ^=2  ±  V-^ 

II.  Cx-3)(x+4)x  =  0;  therefore(x''-2x*-25x='+26x2-t-120x) 
f  (x*+  x'-*  -  12x)  gives  us  x^  -  3x  -  10  =  0  ;  x^  -  3x  -  10  ; 
4x2  -  12x  +  9  =  49  ;    2x  -  3  =  i  7  ;  2x  =  10  or  -  4 ;  x  =  5  or  -  2 

12.  X  =  0  is  obviously  another  root.  Then  (x- V- 2)(x  +  '/- 2) 
=  x*  +  2  =  0 

.-.  (x*-x«-  2x-4)  -f  (x2+  2)  =  x2-x-  2  =  0;  that  is  x^-x  =  2, 
whence  x  :=  2  or  -  1 

13.  Alg.  Art.  20G,  when  the  roots  are  equal  4*  =  4  x  2  x  r; 

cr  16  =  8c;  or  c  =  2 

b  c        $i-y        b 

14.  Alg.Art.  208(Cor.),  )3  +  7  = and/Sv^—  .-.  — -^  = 

0  6        116  c     I       a 

•r  —  = .-.  —  -! =  -  — ;  Also  since  /3-y  =  — i  TT  ="  — 

a  c        ^       y  c  '^'      a'  ey      c 

b  a 

Heuce  —  =  the  sum  of  the  roots  and  —  =  their  product  and  the 
c  c  '^ 

b  a 

equation  is  x*  +      x  n =  0  that  is  ex*  +  6x  +  a  =  0 

1  c  c 


^8 


KEY  TO 


(Ex.  Ln,  Lilt 


15.  Alg.  Art.  208   $  +  y  =  -p  and  Sy  =  q 

/B"  +  7«  =  /3«  +  72  +  2j37-  2i9>=(/3*  +2;3-y  +  7«)-2/3-y-  (P  +  y)- 
-  2;87  =  p«  -  2<7  (I) 

(/3  -  7)^  =  /3*  -  2/87  +  y''  =  (3^  +  7^  -  2a37)  =  p«  -  2?  -  29from 
(I)  .-.  =p«  -4?  (II) 

P^  -y^  =  iB-i-y)  (&-y)~-  p  (f  V  i**  -  ^g),  aiace  from  (11) 


$-y  =  i^/p-^  -  Aq) 
1         1   _  j3  4-7 

"is"  "^7"  ^-y 


33_  ^3  ,  (32  +  g-y  +  7?)  (3  -  ^)  ^  03?  +  7«  +  jS7)<i3  -  7) 


EXERCISB  Lin. 

1.  X  -  6V^  +  9  =  25;  V^  -  3  =  +  5  ;  V^  =  8  or  -  2  .-.  ar  =  64  or  4 

2.  V^  -  4V^  +  4=1;  V^  -  2  =  ±  1  ;  ;^x  =  3  or  1,  .-.  i  =  81  Or  1  • 

3.  X*  -  14a;2  =  _  40  ;  i*  _  Hj^  +  49  =  9  ;  x^  -  7  =  +  3  ;  x^  =  10 
or  4,  .-.  X  =  +  2  or  +  -JTo 

4.  x'+  14Vx^=  1107;  x^+  14Vx^  +  49  =1156;  x-  +  7  -  +34; 
x«  =  27  or  -  41 ;  x*  =  3  or  ^-41;  .-.  x  =  9  or  ^1681 

5.  x-2Vx+6  =  2;  (x  +  6)-2Vx  +  6  =  8;  (x  +  6)-2Vx+6  +  l  =  9  ; 
^x  +  6  -  1  =  i  3  ;  V^  +  6  =  4  or  -  2  ;  x  +  6  =  16  or  4,  .-.  x  =  10 
or  -  2 

6.  x*-Jx2=248;  x« -  Jx2  + A' =  248  +  i^  =='?§' i  ^-i  =  ±^4'; 
x»  =  16  or  -  fi^S  .-.  X-  i  4  or  +  JV  -  «'2 

7.  x6  -  8x»  =  513  ;  x^  -  Sx^  +  16  =  529  ;  x»  -  4  =  f  23  ;  x»  =  27 
or  -  19,  .-.  X  =  3  or  ^-19 

8.  (X  +  5)  -  Vx  +  5  -  6  ;  (x  +  5)  -  V^^  +  5  +  i  =  6  +  i  =  »/  ; 
^x  +  5  -  i  =  1  S  ;  V^  +  5  =  3  or  -  2  ;  x  +  5  =  9  or  4,  .-.  x  =  4  or  -  I 


« 


iix.  Mit.i  AtSiiBiiA.  ?l) 

9.  *Jx(x'^  +  X  -  6)  =  0,  .'.  V^  =  0  and  .-.  x  =  0 

Also  x^  +  X  -  6  =  0,  .-.  x^  +  X  =  6  ;   x^  +  x  +  i  =  ^4^  ;   a:  +  ^  =  +  -J, 
.-.  X  =  2  or  -  3 

10.  Clearing  of  fractions  2x  +  2^Jx  =  16  -  x  ;  3x  -  2Vx  -  16  ; 
36x  -  24Vx  +  4  =  192  +  4  =  190  ;  Q'Jx  -  2  =  +  14  ;  6Vx  -  -  12 
or  16,  .-.  V^  =  -  2  or  f,  .-.  X  =  4  or  7^ 

11.  '^xTTi  +  Ifjx  +  21  =  12  ;  '^IcTTl  +  Vx  +  21  +  I  =  ^^  \ 
yx  +  21  +  i  =  i  i ;  V-^  +  21  =  3  or  -  4 ;  x  +  21  =  81  or  256  ; 
r  =  60  or  235 

12.  -Jxix  -  2  -  V^;)  =  0,  •••  V^  =  0)  •■•  x  =  Q.     Also  x  -  2  -  Vx  =  0, 
X  -  V^  =  2  ;   X  -  Vx  +  i  =  f ;  V^  -  i  =  +  i ;    V^  =  2  or  -  l, 

.*.  X  =  4  or  1 

x"  +  X*  +  2       x'  +  X*  -  2 
^^-     x"  -X*      ""     a;^_a:'^    •     Then  (by  Art.  106  -  vii) 

2x«  +  2       2x''  -  2     x«  +  1       x'  -  1 

2J-4+2  ^  2x^-2  '  x*+l  "^  i^^ ;  3:^  -  x5  +  x^  -  1  =  X'  -  X* 

+  x'  -    1,   or   X*  -  x^  =   X*  -   X*  ;    x"  -  x*  +  x*  -  x*  =  0  ; 
:*(x-  l)  +  a:^(-r-l)  =  0;  (x- l)(x*  +  x2)  =  0, .-.  x- 1  =  0  or  x  =  1. 
Also  X*  +  x^  =  0,  .-.  X*  +  x"  +  J  =  1 ;   x*  +  J  =  +  |  ;   x»  =  0  or  -  1, 
,  X  =  0  or  +  V  -  1 

9(6 -V^;)  _        7x^  -  3x  +  4  23(x-2Vx) 

^^-  ^^TTV^  -  (6  +  Vx)(x  +  2Vx)  ^  ~6+Vx    .-'multiplying 
by  the  denominator  of  the  2nd  term  we  get 
9(36  -  x)  =  7x2  -  3x  +  4  +  ^Z^x^  -  4x) 

or  324  -  9x  =  Ix"^  -  3x  +  4  +  23x2  _  92^. .   39x2  _  86x  =  320 ; 
or  x2  -  ^Ix  =  AjJ ;   x2  -   igx  +  (Jg)2  =  Vo^  +  5fiaci  =  i.^Ai2  . 

15.  x»  -  3x2  +  3z  -   9   =    0;    3.2^^   -  3)  +  3(x  -  3)   =  0 ; 
(X  -  3)(x2  +  3)  =  0,  .-.  X  -  3  =  0  or  X  =  3 ;    also  x'^  +  3  =  0, 
x*  =  -  3  ;  or  X  =  ±  ^~^ 


80  KEY  fO  t^-  ^-^ 

16.  ix-3)(x-A)  =  2-  2V2V(^-l)(^-"2)  +  (^  "  ^^^  "  2) 
or  x^  -  7i  +  12  =  2  -  2V2Vj:'  -  3x  +  2  +  x'  -  3j  +  2 ; 

or  -  4z  +  8  =  -  2V2V^'  -  3x  +  2  ;  2i  -  4  :-  VW-^  -  '^^  +  •^ '» 
Ax^-  16x  +  16  =  2(a;2-  3x  +  2)  =  2x^-  6x  +  4 ;  or  2x^-  lOx  =  -  12, 
.-.  x^-  5x  +  25^  =  2;^.  _  Y  =  I.     Hence  x  -  |  =  +  i,  .-.  x  =  3  or 

17.  x'  -  1  -  3x  +  2  +  1  =  0;  or  x^  -  1  -  3x  +  3  =  0 ; 
orx^-l-3(x-l)  =  0;  or  (x  -  l)(x^  +  x  +  1)  -  3(x  -  1)  =  0  ; 
or(x-l)(x-^+x-2)=0,.-.x-l  =  0orx=l.  Al8OxHx-2  =  0, 
.•.x^  +  x  =  2;  orx^  +  x  +  i  =  |;  orx  +  l=  +  3,  .•.x=lor-2 

18.  Since  (^x^+ax  +  b  +  ^x^-ax  +  bX^x^^^''  '  s/^'-o^^  ^') 
=   2ax,  dividing  these  equals  by  the  given  equation  we  have 

^r^T^TTb  -  V^^^^^^T6  =  ^,  and  adding  the  given  equation 

to    this  we    get  2Vi^  +  «x  +  6  =  ^  H-  r;    or,  by  squaring, 

4x^  +  4ax  +  4b  =  -^  +  4ax  +  c^  .-.  4x'-      ^      -  c«      41., 

c     /c'-  46 
or  xH4c''  -  4a^)  =  0^(0=  -  46),  .-.  x  =  i  y  V  ^-^"^ 

19.  Reducing  the  terms  of  the  first  member  to  a  common  dene 

xVx-xVn-x+xVx4-xVu-.r     _^  ^ 

minator  and  adding  we  get  x  -  (a  -  x)  '  V^  ' 

aft 

_^V^ ^ .    or    2x2  =    2bx   -   ab,    or   x-  -   6x    =>    -   y  { 

°^  2x  -  a  "    V-^"  ' 

,     r,.  4.  L  -  _  _  —  --  -— : ;  whence  x  -  —  =  i  W*'-  2">S 

i»-6x  +  — -   ^        2  4        '  2 

Orx  =  iChiV'^^^^^^^) 

20.  Clearing  effractions  We  get 
(Vrr^  +  V:i^^9)'=  2V:^+60x^+9xr5rO4-89;    that  is 
x+60  +  2V^?T60i'^^'^f^O  +  ^'+^  =  2VxH60z'-h9x  +  540  +  89: 
orx*+x=20;  x'+ x  +  1  =  20  +  i  =  V,  ^  +  i  =  i  |  .•- x  =  4  or-( 


Ex.  LIII] 


ALGEBRA.  81 


21.  a:i2_i  =  o;  or  (x6+ l)(r6  -  1)  =  0  ;  or  (x^  +  l)(x3+ l)(x^- 1) 
=  0  ;  or  (x2+  l)(x*-  x2+  l)(x  +  l)(x^-  r  +  l)(x  -  l)(x^  +  x  +  1)  -  0 
.-.  we  have,  separately,  x^  +  1  =  0 ;  or  x^  =  -  1,  .-.  x  =  +  V  -  1  i 
Also  X*  -  x2  +  1  =  0  ;  X*  -  x2  +  i  =  -  3,  .-.  x^~  i  =  ±  y^^, 
.-.  X  ^  i  VUl  i  V"-^3)  ;  Also  x  +  1  =  0,  .-.  x  =  -  1  ;  Also 
x2-x  +  l  =  0;  orx2-x-- 1,  .-.  x  =  Kl  iV^);  Also  x  -  1 
=  0,.-.x=l;  Alsox2  +  x  +  l  =  0;  orx^-f-x^- 1,  .•.x=  i(- 1  i  V-3) 

22.  x'  -  6x*  +  llx  -6  =  0;  multiplying  by  x  we  get 
x^  -  6x3  +  iij;2  _  6x  =  0 ;  or  X*  -  6x3  +  gj-z  4.  2.c2  -  6x  =  0  ; 
or  (x2  -  3x)2  +  2(x2  -  3x)  =  0,  .-.  (x^  -  3x)^  +  2(x2  -  3x)  +  1  =  1  ; 
or  x2-  3x  +  1  =  i  1,  .-.  x2-  3x  =  0  or  -  2;  x(x  -  3)*  =  0, 
.-.  r  -  3  =  0,  or  X  =  3.     Also  x^  -  3x  =  -  2  ;  whence  x  =  2  or  1 

23.  x»  -  4x2  +  X  =  0  ;  or  x{x'^  -  4x  +  1)  =  0,  .-.  x  :^  0.  Also 
x^  -  4x  +  1  =  0  ;  whence  x  =  2  f  V3 

24.  x'  -  Sx^  +  llx  +  20  =  0;  multiplying  by  x  we  get 
T*  -  8x3  +  11x2  +  20x  =  0  .  oj.  J.*  -  8x'  +  16x2  -  bx^  +  20x  =  0  ; 
or  (x2  -  4x)2  -  5(x2  -  4x)  =  0  ;  (x2  -  4x)2  -  b{x^  -  4x)  +  25  =  £/, 
.-.  x^  -  4x  -  f  =  +  ^  ;  or  x2  -  4x  =  5  or  0  ;  x(x  -  4)*  =  0,  .-.  x  =  4. 
Also  x2  -  4x  =  5,  .-.  X  =  5  or  -  1 

X  +  a  -  6  +  6        /2x  +  rt  +  c  +  6-6\2  x  +  b  +  a  -  b 

25. —7^: =      o^  ,  >.  .  .. ;    or 


x  +  6 

a 

-b       N 

\' 

2x 
1 

+ 

b  +  c^ 
2 

)    ' 

x  +  6  \^        2x  +  Z)  +  c 

/2x  +  6  +  c  +  a-6\^  a-  b        ( 

=  \ — 2X+6  +  C — ;  ;  «■•  1  +  ^T"6  =  l^^  + 

a-b  2(a  -  6)  (a  -  bf 

'"■  ^  "^  xVft  "^  -^  "*■  2x'+T+l: "''  (2i~+  6  +  cj' '  °^  ^+"6  "  l^bVc 

a-b  2x  +  6  +  c-2x-26  a-b 

~  (2x~Tb  +  c)2  5    '^'^  "(^  +  6)(2x  +  6  +  c)    ~  (2x  -Tb  +  c)^' 

c  -  b  a  —  b 

or      ,— I  =  ;; — r»;^'~  ■)  or  2cx  -  26x  -  b^  +  c^  =  ax  -  bx  +  ab  -  b^: 
x+b2x  +  b  +  c'  ' 

€■'  -  ab 

or  (n  +  b-  2c)x  =  c^  -  ab  ;  or  x  =  „  ^  fj  _  2c 

P 


82  KEY  TO  [Ex.  LIU. 

26.  3x^-  14x2+  21a;  -  10  ==  0 ;  multiplying  by  3x  we  have 
9x*-  42x3+ 63x2- 30x=  Q;  or  9x*- 42x3+ 49x2+  14x2-30x  =  0; 
or  adding  x^  -  5x  to  each  side  (9.c*-  42x3+  49^;'-')  +  (isx^-  35x) 
=  x2-  5x;  or  (Sx^-  7x)2  +  5(3x2-  7^.)  ^  j^s.  =  x2  -  5x  +  ^^5, 
.-.  3x2  -  7x  +  f  =  +  (X  -  I) 


Then  3x2  _  7x  +  ^  ==  x  -  ^- 
that  is  3x2  _  gx  =  -  5 
whence  x  =  I3  or  1 


Or  3x2  -  7x  +  I  =:  ^ 
that  is  3x2  _  gx  =  0 
whence  x  =  2  or  0* 


27.  Assume  -^x  +  ^a  -  ^n,  then  cubing  each  side  we  have 
X  +  3x'a'  +  3x'tt'  +  a  =  n;  or  x  +  a  +  S^uxC-^x  +  ^a)  =  n; 
or  X  +  a  +  'i^Janx  =  n  since  ^x  +  ^u  -  ^Jn.  But  comparing  this 
with  the  given  equation  we  see  that  n  -  b,  .-.  ^n  =  ^b, 
.:  ^x  +  ^a  =  yb;  or  ^x  =  ^b  -  ^a,  .:  x  -  (^6  -  »3 

28.  (4x2-  9_p)  _  (4x2-  9x  +  11)'  =  -  5,  or  adding  11  to  each 
side  we  have  (4x2  _  gx  +  II)  -  (4x2  -  9x  +  11)»  =  6  ;  or  com- 
pleting the  square  (4x2  -  gx  +  11)  -  (4x2  -  9x  +  H)'  +  |  _  i^i  j 


.-.  (4x2 -9x+  ii)i  -i  = +  5;   or  (4x2-  9^  +  ^^i  =  3  01  -  2, 
.-.  4x2  _  9x  +  11  =  9  or  4,  Then  4x2-  gx  =;  _  2,  whence  x  =  2  or  1 ; 

1 - 

Also  4x2  _  9x  -;  -  7,  whence  x  =  --(9  +  V  -  31) 

o 

29.  Completing  the  square  we  have 

(x  +  6)2  +  2x'(x  +  6)  +  X  =  138  +  X  +  X*,  and  taking  the  square 

root,  X  +  6  +  V-r  =  iV(^ 38  +  x*  +  x);  or  (x  +  -Jx)  +  6  =  ±  Vx  +  xi  +  138  ; 

squaring,  we  have  (x  +  V^)^  +  12(x  +  ^/x)  +  36  =  (x  +  x')  +  138  ; 

or  (x  +  'Jxy+  ll(r  +  V-i)  =  102;  or  (x  +  Vx)2+ ll(x  +  Vx)  +  J-^i 

=  102  +  J-a^  =  ij9,  .-.  X  +  V-c  +  V  =  i  V»  •••  X  +  V-r  =  6,  whence 

X  =  4  or  9,  or  X  +  V-c  =  -  17,  whence  V-'^  =  i(~liV-  tJ7),  and 

.-.  x=  j(-33q:V-67) 

•We  throw  away  tho  root  x  =  0  because  it  arises  from  the  x  by  which 
we  multiplied  each  side  of  the  equation  in  the  solution,  and  is  couse* 
quently  net  a  root  of  the  given  equation. 


Ex.  Liii.]  ALGEBRA  83 

30.  X*  -  4x^  +  Qx^  -  4j:  4-  1  -  6,  or  extracting  the  square  root 
x"~  2x  +  1  =  +  ■\/G  ;  and  again  taking  the  sq.  root  x  -  1  =  +  V  i  V^, 
wiieoce  X  =  1  ±  '^f'  +  -^6 

31.  Squaring  we  have  4x^  -  4x'^=:  u^  -  2a^x*  +  a'-x^,  and  divid- 

4        4x2       I  4        4J.2 

iug  by  «"x*  we  get,  -r^.  -  — .,-  =  — r  +  2  +  x*  ;  or  -  ,— ,  —  -v-  -  4 

.ij-2+x-or(^x*-2  +  l)  +  (^,--^,)=-4j 

/           1\*4/           1\        4        4              4- 4a*       4 
1        2  2 1  2  


Let  -    _.(!  +  V"'  -  "')  be  I'epresenled  by  2b-j  then  we  have 
x^-—  =  26^;    or  x*  -  26V  =  1;    or  x*  -  2i-x2+  b*  ^  1  +  b*: 


or  x2-  62..  i  VI  +  6^  .-.  x2^  6n  Vl  +  ^*     (A) 


fiut   26^  -  -  -,(1   +  Vl  -  «*),   •••  ^'^  -  -  ^(1   +  VI  -  a*) 

r      1 V      1 

•'•  ^*  -  I  -  a^(l  +  VI  -  «')|   -  ^-4(1  T  2Vl  -  a^  +  1  -  a*) 

=  -(2  +  2V1  -  <^'  -  «")  .-.  6^  +  1  =  1  +  ^,(2  +  2Vr^r^  -  a*) 
=   1   +  ;^(2  T  2Vl  -  «*)  -  ^  =  -,(2  +  2Vr^^^)  .-.   1   +  6* 


~  ~i(^  +  V^  ~  >^'')     Substituting  these  values  for  6-  and  1+6* 
in  equation  A,  and  ihen  extracting  the  square  root  we  have 


VI  ,_    12  ^^I^" 

-  ^(1  T  Vl  -  a*)  +  Va^G  +  VI  -  a*)  or  using  ooly 

the  upper  signs 

a  ±  —  {  ■y'  -  1  +  Vl  ^~a*  +  V2(l  +  Vl  ^^*)  1 


84  KEY    TO  [Ei-  LIU 


or  X 


=  +  A|  _  1  +  ^1  _  „4  +  V(l  +  a^)  -  2V1  -  a*  +  (1  -  a»)| 
.-.  a;  -  +  -^  {  -  1  +  Vl  -  «*  +  V(Vl  +  a«  -  Vl  -  "'/l 
=    -.    — $  _  1  +  Vl  -  «*   +  V 1   +   «^  -  Vl  -   "T 

=  +    — 5Vl  -  a*  -   Vl   -   "'  +  Vl   +   «'^   -    IJ 

-  i  -^l(V'i~+^'  -  i)(/i  -  ^'  +  1))^ 

32.  {(.r   -    2f   -   xf   -   {{X   -   2y   -    r}    =    90 

.-.    {(X   -    2f  -  xf  -   {(X    -    2)2   -   x}    +    i    =    3$- ; 

or  {(X  _  2)2  -  x{  -  i  =  +  ^2%  .-.  (X  -  2)2  -  X  =  10  or  -  9 

that  is  x2  -  4x  +  4  -  X  =  10,  whence  z  =  G  or  -  1  ; 

or  x^  -  4x  +  4  -  X  =  -  9,  whence  x  =  i(5  i  3V  -  3) 

h        a 

33.  Dividing  through  by  x^  we  have  ax^ +  bx  A-c  +  —  + -^  -  0; 


or    ax'+ 


Let  X  +  —  =  !/ ;   then  x"  +  —  =  r  -  2,  and  substituting  these 
Tallies  for  x  we  have  a(y'  -  2)  +  by  ^  -  c  ;   or  ay^  +  by  ^  2a  -  c, 

-  b  ±V8a2  +  fc2-4ac  J_  _  -  />  4  V^^n^  +  ^a  _  4,,^^ 

whence  1/  =  go  .-.  x  +  ^  -  j.i. 


that    is    2rtx2  +    (6    T    V8«^  +    6*  -   4oc)x   =   -   2a,   whenci 


4  VSa^  +  6^  -  4ac  -  &  j  V  -  8a'  +  2&'  -  4gc  f  2b'/8a^  +  b^-  4ac^ 

X  -  ■  4^r 

Note.— An  equation  sucli  as  the  above,  in  which  the  coefficients  follow- 
ing  the  middle  term  are  the  same  as  those  prccediuR  it  but  rover?ed  in 
order,  is  called  arecurring  equation.  The  above  solution  aff<  r<is  .i  r.-.^pral 
method  lor  solving  sncb  recurrinr;  biquadratic  equations. 


f-x.  Liii.]  ALGEBRA.  85 

34.    \   (^^ ;  )  - —  -  ~  V  ("^~  '^''  squaring  both  sides 

we    have    .---+--    -- ^-^  -    ^,j       -    "^  "    ^  5 
2a:-/  .        a-iNi 


1x                1      X'*  —  a* 
or  x'^  -      (x*  -  a*)'  + ^ — ■  -  0  ;   or  taking  the  square  root  v\  e 

V(^-*-"*)  ,    x*-a*. 

^et,  x =  0  ;  or  transposing  and  squaring,  x'=  ~ a^~  ' 

«*     5a  ♦ 

Df  a^x^  =  X*  -  «* ;    or  x*  -  «^x^  =  a* ;    or  x*  -  a^x^  +  —  = 

'  '  4         4 


.".  s*  -  --  (1  +  V^))  whence  x  =  +  ^VK^  i  V^) 

2f(2x  +  4)  -  4(2  -  x){ 

35.  'Jlx  +  4  -  2V2  -  X  -  --1-^ :    or  factoring 

the    second    member,    we    hare   ^1x  +  4   -   2V2    -    x 
_  2{(V2x  +  4   -    2V2    -    r)(V2x  +  4    +    2V2    -    x)\ 

Tlien  dividing  each  side  by  '^Ix  +  4   —  1^1    —    x  we  have 

2('Jlx  +  4  +  2V2  -  x)  _ , , 

1  =  -^ .    ,  ■ ;  or  V9.r^+  16  =  2{v'2x  +  4  +  2V2  -x 

V9x^+16  ^ 

Now  squaring  each  side,  we  get  9x^  +  16  =  48  -  8x  +  16VS  -  2x- 
.-.  x^  +  Sx  =  32  -  Sx^^^-t-  IGVS- 2i- ;  or  x^  +  8x  =  4(8  -  2x^)  +  lCV3~2x^ 
or  x'-^  +  8  r  +  1 6  =  4(8  -  2x0  + 1  G'/s'^x^  +16.-.x  +  4  =  +  (  2  V8~2x2  +  4^ 
.-.  X  =  2V8  -  2x^,  or  x^  =  4(8  -  2x0  ;  whence  x  =  +  ^^2 

Also  X  =  -  2V8  -  2x^  -  8,  or  x^  +  16x  +  64  =  4(8  -  2.r-')  : 
or  9x^  +  16x  =  -  32,  whence  x  =  -  |  f  JV"!*  =  -  t(.^  T  V  -  14) 

And  by  equating  the  rejected  factor  ^2x  +  4  -  2-;/2  -  x  =  0 

2 

we  obtain  the  remaining  root  x  =  -- 


86 


KET-*J:o  fp'^  ^"^ 


2x^4-l+W4x^±_3  ^  ^^  ^^^^^^  ^,g^^,^^   Article   IOC 
Ix^  +  3  +  xV4j;''*  +3        ^ 


2  1-a 


2X''  +  3  +  X'^^x'-  +  3 
2 


—  =  1-a,  .-.  2x2  +  3  +  xV4j:^  +  3-  !_„ 


37.  {(x-l)(x-4)}{(x-2)(x-3)}  =  8;  {x2-5x  +  4l{x^-5x  +  6}  =  8 
.•.}(x^-5x)  +  4H(x^-5x)  +  6},thati3(x^-5x)^+10(x'=-5x)  +  24  =  S 
or  (x^  -  5x)^  +  10(x^-  5x)  +  25  =  9,_.-.  x^  -  5x  +  5  =  ±  3 
...  a-2  _  5x  =  -  2,  whence  x  =  i(5  i  V^^)-     ^'so  x'-"  -  5r  =  -  8, 

•whence  x  =  i(5  +  V-  '^) 

38.  {(X  -  l)(x  -  8)|{(x  -  2)(x  -  7)}{(x  -  3)(x  -  6)}{(x  -  4)(x  -  5)} 

=  {(x^  _  9x)  +  8H(x2  -  9x)  +  14}l(x^  -  9x)  +  \mf  '  9^)  *  20} 
-  (x^  -  9x)(l7x'^  -  153X  +  230)  +  401.  For  x*  -  9x  write  y,  v^cn 
we  have  (y  +  8)(v  +  14)(y  +  18)(y  +  20)  =  I7y'  +  230y  +  401; 
that  is  t  +  60y^  +  1308y^  +  12l76y  +  40320  =  l7y»  +  230y  +  401, 
subtracting  from  each  side  8y^  +  l7Gy  +  320  we  hare 
y*  +  60y»  +  1300y^  +  12000y  +  40000  =  9^*  +  54y  +  81,  or  taking 
the  square  root  of  each  side  y^  +  30y  +_200  =  i  (3y  +  " 
.-.  y'-'  +  27y  --  191,  whence  y  =  ^(  -  27  f  V  -  35) 

Also  f  +  33y  =  -  209,  whence  y  =  J(  -  33  ±  V253) 

But  y  =  x-^^9£,^^x-^  -  9x  =   K  -  27  ±  V  -  35),  whenW 

x=  J(9±V27  i  V"^3     __  __. 

Also  x«-  9x  =  J(;  -  33  ±  V253),  whence  x  =  K^  ±  ^15  f  2  V253; 


Ex.  Liti.]  AtGEBRA.  87 

39.  Multiplying  as   indicated  we   have  x^  -  Qx^  +  llx  -  G 
=  a-^  +  6^2  +  llz  +  6,  whence  Ux^  +  12  =  0,  .-.  x  =  ±  V^ 

40.  Reducing  as  indicated  by  the  question  we  have 

a:  +  1  -  S^Jx  +  1  +  6  +  5Vl  +  x  -  6^x  +  1  +  'Jx  +  I  -1  =  0; 

or  (x  -  5^/x  +  1)  +  sVx  -  5i/-c  +  1  =  -  7  ;  or  completing  the  square 

(r  -  5Vx  +  1)  +  5(x  -  5^/x  +  ly  +  2/  =  _  a^  whence  (r  -  5^x  +  1)^ 

-  5  +  V^  

= ^ ,  .".  X  -  5^-^  +  1  =  i(ll  ±  5-/  -  3)  =  a,  suppose  : 

Then X- 5^x4- 1  =  a;  orx-a  =  5Yx  +  1 ;  orx^-2ax  +  a'^=25x  +  25; 
or  x'^-  (2a  +  25)x  =  25  -  a^,  .-.  x  =  {(2a  +  25  +  5V4a  +  2"9) 
Bui  a  =  ^(11  +  5V  -  3)  by  supposition 

.-.  X  =  i{)l  +  5V  -  3  +  25  4  5V22  +  lOV  -  3  +  "•^f'j 
=    18    ±   i(V  -   3   +   V51    ±   10V~^3) 

41.  Arranging  the  given  quantities,  we  have 


(4x*-  8x3-  4^2^  ?x  -  1)  -  2(2x2-  2x  +  l)V4x*-  8x'-  4x2+  Sx  _  ^ 
+  (4x*  -  8x'  +  8x2  _  41  4-  1)  =  0,  and  taking  the  square  root 
V4x*  -  8x*  -  4x2  +  3x  _  1  -  (2x2  _  2x  +  1)  =  0  ;  or  transposing 
and  squaring  4x*  -  Bx^  -  4x2  +  3^  _  1  =  4x*  -  Sx"  +  Sx'  -  4x  +  1 
.-.  12x2  _  7j.  =  _  2j  whence  x  =  ^^4(7  i  V^^) 

42.  Multiplying  through  by  ax  to  clear  effractions 
c''bx''^  +  2a*c*^x2-  2ab  +  2a^c'^x^-  2ix  =  <tc'^x(x^-  w'^brx  +  «') 
multiplying  now  by  c  we  have 

a'^brx''^  +  2ft''x2  -  2abc  +  2o2x''  -  2ifx  =  ax*  -  a~'^bcx^  +  a*x ; 
or  transposing  and  changing  signs 

ax*  -  2aV  -  2aV  +  a*x  -  a-^6rx2  +  2&fx  +  2ahr  -  a^brx-'^  ~  0 
dividing  through  by  ax  we  now  have 

x^  -  2(7x2-  2d'x  +  a'-  bc(a-^x  -  2a''^  -  2x-i  +  hx-2)  =  0; 
orf/^x2(a'2x-2a"^-2x"i  +  ax"2)_;,r(a-2x_2a-i-3x"i  +  r/x'2)  =  n 
.-.  (rt2a:2_  ir)(a-2j  _  2a-i  -  2X-1  +  CX-2)  =  0,  or  factoring  the 
first  member  (a^x"^  -  bc)(a~'^x''^  +  a"ix"2)(x2  -  3ax  ^-  a'^)  =  Q 
,'.  a^x^  -  be  -  0,  whence  x  =  ±  a''^^Jbc 


88 


KEV  TO 


[Ex.   LIII,  LtV. 


Also  a"^x"^  +  a'^x"^=0;  or  — ;—  =  -  — 7 ;    or  ax'^  -  -  d'x, 
'        a-x  ax*  '  ' 

whence  x  =  —  a 

a 
Alsox^-  Sax  +  d^=  0;  ov  x^-  Zax  -  -  a'-,  whence  x  =  —(3  +  v'i>) 

43.  AdJ  x*  to  each  side,  lUea 

a-4  +  80.-^  +  22j^  +  24x"  +  9  =  x*,  and  taking  the  square  root 
X-+  4x  +  3  =  +  x^,  .-.  4x  =  -3,  whence  X  =  -  4  j  also  2x^+4x--3, 
whence  x  =  -J(2  f'\/-2) 

44.  Changing  signs,  adding  (4  -  2x*)  tu  each  side,  and 
arranging  wo  have  x*  -  4x''  +  4  =  4x*  -  4x^  +  I3x''  -  6x  +  9,  and 
now  extracting  the  square  root  +  (x^  -  2)  =  2x-  -  x  +  3, 
■.  2X--X  +  3  =  x'^-  2  ;  wuence  x''-  x  =  -  5,  and  .".  x  =  ^(1  +  -y'  ~  I'''.' 
Also  3x*-  X  =  -  1,  whence  x  =  J(l  +  y  -  11) 

x'^  +  2x(V3  -  V5)  +  (8  -  2 V fo)     x''-2x(V3-V5)  +  (8-2Vl5) 
X  -  ■v'3  t  V^  ~  X  +  -^3  -  ^5 

=  x-'  -  8  -  V15 

ii;g  of  fractions 

(x  +  V3-V5)*-(-r- V3  +  V5)^=  (xH8-  2Vl5)(x<-8  +  2V15)  ; 
or  Gx'-'cVa  -  V5)  +  2(V3  -  Vi>/  =  |^'  +  ( V3  -  V»)'}|-^'  -  (V3  -  V^)"!  i 
Let  ^3  -  V^  =  ">  ^^"^'i  ^*  ~  ^'''''^"  ~  "*  ~  -''^  whence 
*  =  i  V3.'  ±  a\Ja'-  +  3a  +  9  where  a  -  ^JZ  -  y'S 


EXEBOISK   LIV. 


1.  (X  +  J/)(x  -  y)  -  45,  but  X  -  1/  =  5   .-,   j(x  r  9)   -   46, 
or  X  +  1/  =  9  and  x  -  y  =  5  .-.  2x  =  14,  ic. 

2.  (x  +  J/)(x-j/)=  105,  but  X  +  y  =  21  ,',  2l(x  -  y)  =   l"5,j 
or  X  -  1/  ==  5  .-,  2x  =  26,  Ac. 


Ex.  LiV.]  ALGEBRA.    *  89 

3.  x2  4-  2x1/  +  y'^  =  81,  but  x^  +  y2  -  4^  ._  2xy  =  40,  and 
.-.  j^  -  2x!/  +  y'^  =  1,  whence  x  -  ?/  =  +  1,  .-.  2x  -  10  or  8,  &c. 

4.  X-  -  2xy  +  y^  =   225,   but  x^  +  1/  =113  .-.  2xi/  =  -  112 

.-.  x^+  2x1/  -(-  ?/2=  113  -  112  =  1,  whence  X  +  ]/  =  +  1  and  x  -  ?/  =  15 

.-.  2x  =  16  or  14,  &c. 

40       40" 

5.  x^4-2/2=89andx=  —  .-.  -^-  +  ^^-89;  or  y*-89!/^=  -  IGOO  ; 

t/*-  89(/2+  (8/)'''=  -  1600  +  i^^i-i-  =  ^Y-^  .-.  y2  -^i-  =  ±  ^-^t  whence 

40       40  40 

y  =  +  8  or  +  5.     And  x  -  —  -  — —  ;  or  -  — r  -  +  5  or  +  8 
-  J/+8'+5- 

72      24       24^ 

6.  x^-y^=  55,andx  =  --  =  —  .-.  — ^--i/Z- 55  .  ovy*+55y'  =  5l6 

whence  1/^-9  or  -  04,  and  .-.  y  =  +  3  or  i  8-^  -  1 

24       24  24 . 

And  X  =  —  =  -— T-:  or =^  -  f  8  or  +  3V  -  1 

y      ±  3  '        ±  8V  -  1 

7.  x2  -f  3y^  =  14?,  and  y  =  24  -  2x  .-.  x^  +  3(24  -  2x/  =  148  ; 
or  x'^-  1728  -  28Sx  +  12x2=  148.  ^r  Ux' -  288x  =  -  1580, 
whence  x  =  12i^j  or  10 

And  y  =  24  -  2x  :=  (24  -  24-,ij)  or  (24  -  20)  =  -  -^^  or  4 

2  +  3v  n  +  3])Y 

8.  3x^-2y2=  115,  and  x  =  — - —  .-.  31 — ^|   -  2y^ ^  115; 

or  19y2  +  36y  =  448,  whence  y  =  4  or  -  5f  J 

2  -  I7|i 
or ^  =  7  or  -  7} I 

2V  /         2y\^ 

9.  4x2  +  3y2  =  511,  and  x  =  9  - -r-  .-.  4    9  -  —      +  Sy^  :^  511, 

or  43y'''  -  432y  =  1C83  ;  whence  y  =  13-^2^-  or  -  3 

2y       /        26-AA 
And  X  =  9  -  -3'  =  f  9  -  -3- j  or  (9  +  «)  =  {}  or  11 

10.  x"  -  y'  =  26  ;  also  from  2nd  cquat.  x'-  Sx^y  +  3xy^-  y'  =  8 
.-.  by  subtraction  3x'y-  3xy''=  18  ;  or  xy(x  -  y)  =  6,  butx  -y  =  2 
.-.  2x1/  -  G  or  xy  =  3.  Then  xy  =  3  and  x  =  2  +  y  .-.  y(2  +  y)  =  3 
or  y'^  +  2y  -  3,  whence  y  =  1  or  -  3.     And  x=2  +  y  =  3or-l 


90  KEY   TO  [Ex-  Ltv. 

11.  X  +  2/  =  4  .•.  (x  +  yy  =  16  .-.  x'+  t,-'  =  16,  and  from  1st 
equat.  a'  +  Zxh/  +  3xy''+ 1/  =  64, .-.  by  subtraction  Zx'^y  +  Sxy'*  -  48  ; 
orxy(x  +  y)  =  16, but r  +y=4.-.x!/  =  4  .-.1/(4-1/)  =  4ory--4i/  =  -4, 
•whence  y  =  2,  and  x  =  4-i/  =  4-2  =  2 

12.  Squaring  the  1st  equat.  ^^x  +  2  "/xy  +  ;^y  -  9,  but  4  ^xy  =  8 
.-.  subtracting  we  have ^x- 2  ^xy+i^y  =  1 ;  whence  V^-Vl/  =  H:  1 
and  "/x  +  \'y  =  3,  .-.  by  addition  2  V^  =  4  or  2,  .-.  V-r  =  2  or  1, 
whence  x  =  25G  or  1,  &c. 

13.  2/H4x-2j/=  ll,andx=  14-4y  .-.  y2  +  4(14-4y)- 2;/=  11, 
or  y^  -  ISy  =  -  45  ;  whence  y  =  15  or  3,  and  x  =  14  -  4y  =  -  46  or  2 

3y  +  1 

14.  2r^  +  xy  -  Sy-  =  20,  and  x  =  — ^— 

/3y  +  ly        /3y  +  1\ 
•  •■  2    —z —      +  y    — ^ —      -   5y»  =   20 ;    or  2y'  +    7y  =    39, 


3y  +  1 
whence  y  =  3  or  -  6J  and  x  =  — r —  -  5  or  -  91 

15.  9x  +  5y-4xy  =  0,  andx  =  2  +  y  .-.  9(2  +  y)  +  5y-4y(2  4-y)  =  0, 
or  2y^  -  3y  =  9  ;  whence  y  =  3  or  -  ?,  and  x  =  2  +  y  =  5orJ 

16.  x^y2  +  4xy  +  4  =  100  .-.  xy  +  2  -  +  10  ;  whence  xy=  8  or-  12 
From  second  equation  x  =  G  -  y  .•.  y(6  -  y)  =  8  or  -  12 
That  is  y^  -  6y  =  -  8,  whence  y  =  4  or  2  ;  and  y' -  6y  =  12, 
whence  y  =  3  iV^l  .-.  x  =  6  -  y  =  2  or  4,  or  3  fv/21 

17.  9x^  +  36xy  -  85y^  =  0,  and  x  =  2  +  y 

.-.  9(2  +  y)'''  +  36y(2  +  y)  -  SSy^  =  0.     That  is  10y»  -  27y  =  9 ;  j 
whence  y  =  3  or  -  -|'„,  and  x  =  2  +  y  =  5  or  1-,'j  ! 

12  +y*  I 

18.  From  second  equation  x  =  and  substituting  thii' 

n2  +  t/2\a        /12  +  f\ 
for  X  in  the  Ist  equat.  we  get [  — T"^)    +  ( Z — iV  ~    '''^  ii 

144  +  24y'  +  y*  ^  ^  ' 

or r +   12  +  y»  =   77;    or  2y*  -  41y^  =  -    144 

.-.  y*  =  16  or  Y»  whence  y  =  ±  4  or  f  ^^2 

12 +y*      23        1  of  66  33  11 V^ 

Andx-.-^^  =  ---or^^^;   -  i  7  or  j^^;  =i7orl-2  - 


Ex.  i-iv.l  ALGEBRA.  91 

19.  Let  X  =  V  +  z  and  y  =  v  -  z 

Then  x^  +  xy  =  (v  +  zf  +  (y  +  z)(v  -  z)  =  2v~  -(-  2vz  =  6G  (i) 
Also  x^-  y^=  (y  +  z^  -  (v  -  zf  =  Avz  =  11  .-.  2vz  =  V  (") 
From  (i)  we  get  2v'^  =  66  -  ^  =  ^2^  •"•  '"^  =  -^t'-  or  t;  =  ±  i^^ 
From  (ii)  we  get  by  substitution  z  =  ^-r±\l  =  ±\.  Then 
a;  =  v4-s  =  ±Y±i=±6.     And  y  =  'i;-2  =  +il  +  i=±5 

20.  From  1st  equat.  by  clearing  of  fractions  x^  +  i/^  =  ISxy  (i) 
and  cubing  the  2nd  equat.  we  get  x*  +  3x^y  +  3xj/^  +  ^^  =  1728  (11) 
and  taking  (i)  from  (11)  we  have  2x^y  +  Sx?/^  =  1728  -  18xy; 
or  xy(x  +  y)  =  576  -  &xy\  or  since  x  +  y  =  12,  we  have  I2xy 
=  576  -  6xj/  .-.  ISxy  =  576,  and  hence  xy  =  32.  Then  x  =  12-7/ 
.-.  7/(12  -  y)  =  32,  or  7/^  -  12i/  =  -  32;  whence  7/  =  8  or  4. 
And  x=12-7/=4or8 

21.  Let  X  =  v  +  2  and  y  =  «  -  2 

Then  x«  +  y''  =  (7;  +  2)^  +  (y  -  2)"  =  27^^  4.  20r82^  +  lOyc*  =  3368  ; 
or  7'*^  +  10j)V+  57J2*  =  1684.  But  x  +  y  =  i;  +  2  +  v-2=27;  =  8 
.-.  V  =  4,  and  substituting  this  for  v,  1024  +  6402^+  2O2*  =  1684 ; 
whence  2*  +  322^  =  33,  .-.  2^  =  1  or  -  33  and  2  =  +  1  or  +  ^-33 
Then  x-f  +  2  =  4+1;  or  4  +  ^-33  =  5  or  3  or  4  +  V^33 
y  =  v_s=4:fl;  or  4  +  V  -  33  =  3  or  5  or  4  +  V  -33 

22.  From  1st  equat.  x'  + 3x^1/  + 3x!/^  + 7/^=  343,  and  x^+y^-  133 
.-.  3x^1/  +  3x1/^  =  210  ;  or  xy(x  +  7/)  =  70,  but  x  +  y  -  1  .-.  xy  -  10 
And  X  ^  7  -  7/  .-.  2/(7  -  7/)  -  10  ;  whence  7/  =  5  or  2,  and  x  =  2  or  5 

23.  Let  X  =  T  +  2  and  y  =  v  -  z 

Then  x*  +  7/*  =  (v  +  2)*  +  (y  -  =)*  =  27'*  +  127)V  +  2:*  r-  97 

But  x-7/  =  7;  +  r-r  +  2=2s=l.-.  2  =  i 

Hence  2v*  +  3t)*  =  97  -  i  -  9C|  ;    whence  t-^  -   «/   or  -  Y 

.-.  f  =  ±  '5  or  ±  iV-31 

Thenx  =  7;  +  2  =  i5  +  J  ;  or±  iV-31  +  1  =  3  or-  2  or  J(l  ±^-31) 

And7/-r-2  =  i5-i  ;  ori  J-/- 31  -  J  =  2  or-."?  or  }(- 1  +V-31) 


92  KEY   TO  [Ex-  i-i^- 

24.  Multiplying  the  Ini  equat.  by  3  and  adding  it  to  iho  1st 
equation  we  have  x^  +  3x^y  +  Sxy-  +  y^  -  343  ;  whence  x  +  y  ~  7 
.-.  X  =  1  -  y.  Also  x-y  +  xyi'  =  xy(x  +  y)  =  Ixy  =  84  .-.  xy  =  12 
•"•  VO  ~  V)  -  12,  OT  y'^  -  ly  =  —  \2\  whence  y  =  4  or  3, 
and  a;  =  7-w  =  3or4 

25.  a;^  +  y^  +  j;  +  y  =  26,  adding  2xy  to  each  side  of  tlie  eqimt. 
we  have  (x^  +  2xy  +  y"^)  +  (x  +  y)  =  26  +  2xy,  or  completing  the 
square  (x  +  y)^  +  (x  + 1/)  +  ^  --  26j  +  2x1/ .-.  x  +  y  +  ^  =  ±-Ji^i  +  2xy 
orx  +  y  =  +  V2Gi  +  2xi/-  ^,  but  4(x  +  y)  =  Zxy  .-.  Zxy  =  ±  4V^i  +  -J^y 

-  2  ;   transposing  and  squaring  these  we  have  Qx-y^  +  12xy  +  4 

-  420  +  32xy  .-.  9xV  -  20xj/  =  416;  whence  xy  =  8  or  -  V 
Then  4(x  +  y)  =  3xy  =  24  .-.  x  +  j/  =  6,  and  x  =  6  -  5 
•■•  !/(6  -  y)  =  8,  or  y''  -  Gy  =  -  8 ;  whence  y  =  4  or  2,  and 
.-.  X  =  2  or  4.  Also  y(6  -  y)  -  -  V  'i  whence  y  =  i(  -  13  4  v^377) 
and  .-.  X  -  ;(-  13  +  '^31l) 

20.  Clearing  the  first  equation  of  fractions  we  liave 

5{(x  +  y)-  +  (X  -  y)^}  =  26(x='  -  y")  :  i.  e.  lOx'^  +  lOy*  ^  2C.x-  -  20.,-' 

2x 
Hence  36y^  =  IGx'',  or  6y  =  +  4x,  or  y  =  ±  -- ;  substituting  thia 

/    2x\'  4x« 

in  the  2nd  equation,  we  have  x-  +  f  +  -  -  j    =  52  ;  or  x^  +  —  —  =  52 

1 3x^  X  *  X 

or  — ^  =   52   .-.  y  =  4  ;    or  Y  =  ±  2  i    or  x  =  f  6.     And 

2x 
y  =  i  -3-  =  ±  4 

2x  +  36 

27.  7y  =  2x  +  36  .".  y  =  — » — ;  substituting  this  in  the  first 

2x  +  36 
equation,  we  have  x  +  s —  =  x*,  or  7x''' -  9x  =  36  ;   whence 

2x  +  36 
X  =  3  or  -  1^.     And  y  =  ,j —  -  G  or  4^1 

28.  Let  X  =  r  +  s,  and  y  ~  v  -  z;  then  we  have  from  the  first 
equation  2r*+ 12p22»+  25*=  14i'*  -  28v^z^  +  14i*,  and  thia  bj 
transposition  gives  AOc'z'  =  12«'*  +  12c*  ;  but  x  +  y  =  v  +  z  +  v-i 


Ex.  Liv.]  ALGEBRA.  93 

-  2o  -   in,  .-.  V  -  s,.i:    siibsLitutiog  Ibis  for  i',  v/e  have  lOm'-z'^ 

3  in  V 

2V3"'-V3 


3;/i*  3m 

-  Im*  +  12=*,  or  6z*  -  5in^z"  =  -  —^  ;  whence  s  =  ±      ,-  or  +  — 


3  m  m       m  m 

Thenx-  t,+  2.  iOT  +  .^^-^  ;  or  im±-:^,,^  =  -~{\  +  V^)  i  or-(l  i  JV3) 

m  _  3m  m  _    m       ni     _  "^      _ 

Andi/..-x  =  -  +  -^^-^;  Ory+-^3  =  -(l-W3);  or-(l  +  W3) 

74 

29.  Letx=^vy;    then  t)  V  +  2(/- +  i-;/- ^  74,  or  y^  =;  t^^i-^T^ 

73  74  73 

•^      •'  ■'  '      •'      2y''+2D  +  l      V'+v  +  2     2i;''+2r  +  l 

74 
or  by  reduction  v^  +  v  =  l'^  ;  whence  v  =  j  or  -  f  ;  i/^  =  -2"T — T^ 

74  74 

=  A\.  +  ■■>  +  2  ^  25  .-.  y  =  i  5,  also  i/  =  ^^  _  g  ^  ^  =  25 ;   whence 

as  before  y  =  ±  5.    Therefore  J-'  =  J'y  =  ±  5  x  ^ ;  or  f  5  x  -  &  =  +  3 
or  +  8 

30.  Adding  the  two  equations  together,  we  have  a"*  +  2x^y^  +  y* 

=  169,  whence  x^  +  y'^  =  ±13;    but  x^+  2xV  +  2/^  =  85  .-.  by  sub- 

ti action  2xV  =    i^2  or  98  .-.  x'^if  -  36  or  49,' and  .ry  =  ±  6  or  +  7 

±6         4   7 
Then  x •  or ,  and    substituting  this  for   x,  we    have 

y         y 

QY  36 

±  -;-  )    +  2/^  =  i  13,  that  is  —  +  y^  =  i  13  ;    or  y*  ^r  13y2i  =  _  36, 

whence  y'^  ~  9  or  4,  and  ,-.  y  -  +  3  or  i  2.     (Impossible  values 

being  rejected.) 

,  ,  ,        ,  108 

31.  Letx^ry;  then  3i'-'jr+ 2i'y- 4y2- 108  ;  ory-  =  7>-T; — -„ . 

81 


Also   v'^y^  -  3ey^  -   7y^  =  -   81  ;    whence   y^  =   ^ 


3o   -  v^  +   1 
108  81 

•■•  3v^+~2^'^i  =  3^^v^Vl  °'  ^^  reduction,  13i;^-  6d  =  40 ; 

whence  t;  =  2  or  -  t§ 

81  81  81 


'^^^'^  '-''  -  3i;  -  v^  +  7  '   -  6  -  4  +  7  "'   _  ^^  ,  :j-^v  +  7  ' 
or  169  X  27  .-.  y  =  +  3  or  =  i  39V3  - 


94  KEY  TO  [Ex.  Liv. 

And  X  ^  vy  -  ±  3  X  2  ;  or±3x-3S=±6or  +  4-,Sj 

Also  X  ^vij  ^  ±  39V3  X  2  ;  or  ±  39V3  x  -  Ta  =  i  '^8V3  or  iF  60V3 

32.  Factoring  the  first  equation,  we  have 
(i/-x^)-(y  +  x)=12.-.(ij  +  xXy-x-l)=U;  ory  +  x  =  ^_^_^ 

48  12  48  1  4 


+  ","  +  2/  +  -^  +  Vi/  =  20  ;  that  is  (  ^  +  Vl/  ]    +  (  —  +  Vl/ 


but  y  +  X  =  .         >2  •  •  1  -  ,         N-.!  1  "»  ,  -  ^ 

•^  (y-x)"      y-x-1     (y-xy       y-x-l     (y-xy^ 

or  (y  -  x)2  =  4(i/  -  x)  -  4  .-.  (y  -  x)^  -  4(y  -  x)  +  4  =  0  ;  whence 

48 
j/-x^2,  and  y  +  x^  .a=^^-  12,  .-.y--^  14  or  2(/=  7,  and  x^  5 

33.  Transposing  the  first  equation,  we  have 
2x 

y'^^y  '  '  y  ■  ^^-""'  -—^y 

X  3 

.•.  —  +  Vi/  =  4  or  -  5  ;  that  is  x  +  y^  =  4y  or  -  by 

^  3 

Taking  x  +  y^  =  Ay  ~  x  +  8,  (by  2nd  given  equation)  we  have 

a. 
y'^  =  8  .-.  y  -  4,  and  x  =  4)/  -  8  -  8 

3 

Taking  x  +  j/^  =  -  5i/,  and  subtracting  this  from  the  2nd  given 
equal.,  we  have  8  -  y'^  =  9y  .-.S-Sy-y  +  y^  ;  or  8(l-j/)  =  y(l  +  i/^) 
and  dividing  each  side  by  1  +  y  ,  we  have  8(1  -  y^)  =  y  ;  that  i? 
y  +  8y-  8,  whence  y  -  -  4  +  2V6,  and  .-.  i/  =  40  ^f  ISVtJ.  and 
j; .;  4y  _  8  =  152  T  64V6  ;  also  l+y^  =  0.-.y=l 

34.  x^  +  J/*  =  35  (I),  x^  +  y^=  13  (ii). 

From    (i)  (X   +   y)    (x'^  -  xy   +   y^)   =    35 ;    but   x^  +   y^  =    13 

35 
.-.  (X  +  i/)(13  -  xy)  =  35  .-.  X  +  y  =       — — -,  squaring  we  have 

1225 
x^  +  2xy  +  y^=  ^^  _  26xy  +  x'-y- '    substituting  (ii)  in  this,  we 

1225  1225- 2197 +  338xy-l.TTV 

have  2xy  =  -^  _  oQxy  +  xh/  ~  ^^  ^  169  -  2Gxy  +  x^y^ 

338xy  —  \3xhj^  —  972 
.-.  2xy  =  -  j^gg  -'26x1/ TiV^  '    clearing  of  fractions,  we  have 

2xV-39x-j/*  +  972  =  0;  and  factoring (xy-6)(2xV- 2 7xy  -  162) 


I 


ti.  Liv.)  ALGEBRA.  95 

=  0  .-.  xy  -  6  =  0  (III)  ;  aad  also  2x^j/  -  27xj/  -  162  =  0  (iv) 
From(iii)i;i/  =  6,  andfrom  (iv)xj/-  18  or  4J  .-.  2.ry  =  12,  or  36,  or  9 
Hence  adding  these  to  (ii)  and  extracting  the  square  root,  we 
have  X  +  y  =  ±  5,  or  +  7,  or  +  1/2! ;  similarly  subtracting  these 
from  (11),  and  then  extracting  the  square  root  x  -  y  ~  i  1. 
or  i  V  -  ^3,  or  ±  2.  Hence  by  addition  and  subtraction  we  have 
a;  =4  3;  or  ±  i (1  +  ^~23}  ;  or±^(2+'/22) 
y  =  ±2;  or±i(7-V'^r23);  or+^(V^-2) 

Otherwise,  thus : 
Let  X  =  w  +  s,  and  y  =  v  -  z 

Then  from  (i)  2v^+  Qvz^  =  35  (iii),  and  from  (11)  2v^  +  2=^  ^  13  (iv) 
Multiplying  (iv)  by  3v,  and  subtracting  from  (in,)  we  have 
4v^  -  39i;  =  -  35.  Multiplying  by  0,  we  have  4y*  -  39u^  -  -  35y 
Dividing  by  4,  v*  -  ^^v'^  =  -  \-v  ;    add   v'-'  to  each  side,  and 

.-.  v'  -  V  -  0  (v),  and  v^  +  v  ~  ^^  (vi) 

From(v),  u  =  0orl ;  and  from  (vi),  t>- ^  or-^.     But  2«^+22^- 13 

.-.  z  =  iV26;  or  i  iV22;  or  +  J  ;  or  +  JV^23 

Then  x  =  v  +  z=\±  ^22;    or  g  i  i  ;    or  -   ^    ±   sT^Yi 

y  =  u  -  z  =  values  as  obtained  above. 

Note— The  values  w  =  0,  and  a  =  JV26  are  derived  from  the  v,  by 
Which  we  multiplied  equation  (iv). 

^/x  +  9 

35.  By  Algrebra  Article   106,  we  have  Vi/^  +  1  =  — 5 — ; 

whence  x  =  9y^;   sabstitutiag  this  in  the  2nd  equation,  we  get 
^'A'/  +  2y  +   I)   =   36!/'  +   64 ;    or   9y*  -    18y^  +  9y^  =    64 ; 
or  9yX>/  -  21/  +  1)  =  64  .-.  3y(ij  _  1)  =  ±  8 ;   or  i/^  -  y  =  +  f 
whence  y  =  ^(3  ±  VTo3),  or  ^(3  +  V"^^^) 
And  X  =  9i/2  =  1(19  ±  ./l05) ;  or  |(  -  13  +  V^^^^^) 

36.  Multiplying  the  2nd  equation  by  x,  we  have  x*  +  x^'*  -  x, 
butx*+ ?/*  =  x.-.x*+)/*  =  x*  +  x/.-.i/*-x^^  =  0  ,  that  is  2/'(i/ -  x)  =  0 
whence  y^  -  0  .-.  y  =  0,  and  hence  x  =  1.  Also  y-x  =  0.-.y-x; 
whence  2x'  =  1,  and  y  =  x  =  l^i 


93  KEY    TO  [Ex.  Liv. 


(^^+^y=f^ 


37.  Dividing  the  1st  equation  bjx*,  we  have 

,      1              1 
.-.  X*  +  --  =  7/  H (i).     Again  dividing  the  2nd  equation  bv  if, 

X  y 

we   have  h/  +  ~]x  =  9(x^  +    1)   .-.  t/^  +  —  =  ofx  +  — 


-3-(!/'v)=3(x  +  i)(n) 


Now  adding  equations  (i)  and  (u)  together,  we  have 

1  /  ,         1\  1  ,  3         1/1 

i/  +    —]  +   y   +   —   -x*+3x   +   —  +  -^=X4-  — 

1  3       /  1\'         /  l^ 


{y^\)  =3(. 


-■•  2/  +  ^  +  3i/  +  y  =  (  y  +  —  J     =  3  (  X  +  —  )  ,  or  extracting 

1         /  I'X  1 

the  cube  root  of  each,  y  +  —  =   I  x  +  —  1^3.     But  (i)  y  +  — 

1  1  /  1  \ 

«=  x'  +  -3-   .-.   x^  +   — a"   =    /     x   +   —-jv'^.    and    factoring 

^  +  4-V-^'-  1  +  il  -  (^^  +  tV^  ■•-  -^  +  T  =  0'  or  x*^  -  1, 


.  1  1 

or  X  =  4  V  -  1  •     Also  X-  -  1  +  -J  =  -^3  ;  or  j=  +  2  +  -j  =  3  +  ^'3  (in) 

1 


by  adding  3  to  each  side,   .-.  x  +  —  =  4V3  +  V^j  similarly  by 


taking  1  from  each  side  of  (in)  and  then  taking  V;  ^  - ~  =  ±  V^3  -  1 
Then  by   addition,  we   have  x  =   +   i{V3  +  V^    +    Vv^~^~^ 

And  2/  +  —  =    (-r  +  -- )\/3  -  +  V3  X  V\/3T~3    =    V^   +   3(/9 

1  ,  1 

.■.«■■*+  2  +  —r  =  3  +  3?/9,  or  taking  4  from  each  y-  -  2  +  — 

=   3^9  -  1    ■'•   y =   i  V3\/^   -    1-      Hence    by    additioa 

1 

y  =   ±   i{V3  +  3^9  +  V3V9  -   !}•     -^^so   since   x   +  —  =  0| 


{--) 


v'3  =  0  .-.  y  +  —  =0,  whence  y  -  ±*J  -I 


Ex.  Liv]  ALGERRA.  97 

X'       y'^        X        y 
38.  By  transposition  -r,  +  —  +  —  +  —  =  W  ov  adding  2  to 
•'  '  y^x-        y         X 


y 

+ 

—    ^  'a' 
XJ         ^ 

•  compi 

letu: 

'S' 

the 

S(| 

[uare 

+ 

i 

=    9    .-, 

X 

.    —  + 

y 

L 

x 

+ 

\ 

= 

-1- 

3 

-y 

2  _ 

5xy 
2     '''■ 

Ixy 
2 

each  side,  (  —  +  2  +  -7^  j  +  ( 

Again  by  squaring  the  2nd  equation,  we  get  x^  +  /  =  2x^  4-  4 

bxy  Ixy 

.'.  2xy  +  4  =  — .^  or  -  -y- ;  -whence  xy  =  8  or  -  -,  ,-.     Then  since 

x^  -  2xy  +  1/^  =  4,  and  Axy  -  32  or  -  f^  ;  we  hare  by  addition 

i''  +  2xy  +  2/^  =  3G  or  W  .-,  x  +  y  =  +  6  or  +  S^^,  and  x  -  y  =  2 

.-.  X  =  4  or  -  2;   or  1  +  t^-,-V33;   and  similarly  y  =  2,  or  -  4, 

or  -  1  +  -,'-,-V33 

39.  Prom  the  1st  equat..  we  get  {-Jx  +  Vy)  +  V^(V-^  +  Vy)    =  '  0, 

_  completing   the   square   (V-c  +  Vy)  +  V^C/^  +  Vy)     +  4  =  ^^ 

.-.  (Vx  +  Vy)*  +  W^  =  ±  2V5  ;  whence  (V-i"  +  Vy)'  =  V5  or  -  2V5 

.*.  V^  +  Vy  =  5  or  20  (11).     Taking  the  former  of  these  values, 

and  raising  to  the  5th  power,  we  have 

X'  +  .5x'''y'  +  10x*y  +  lOxy-  +  5x^y''  +  y-  =  3125 
But  x'-  H-  y^=    275 

.-.      5x^2/^  +  10x*y+  lOxy^  +  Sx^y^  =  2850 

x^y^(x^ +  2xy^  +  2x^y  +  y=)  =    570  (iii; 

But  cubing  equation  (1),  and  multiplying  it  by  x*y*,  we  have 

x*y^(x'^ +  3xy^  +3x^y+i/^)  ^  125x^y»'  (iv) 
Subtracting  (iii)  from  (iv),  we  have 
x^y^{xi}  +x^y)  ;  that  is  xy(x^  +  yO  =  125x^y^  -  570 
But  x^  +  y^  =  5  .-.  5xy  =  125x^y*  -  570 
Hence  xy  -  25x^y^  =  -  114  .-.  x^y^  =  19  or  G 
Then  x  +  2x^y^  +  y  =  25  ;    and  4x^y*  =  24  or  76 

a 


laxy  -  2  Jj/  _       /(g-  2jr//  _       /a-  2 
V  aa-i/  +  2xy  ~    V  (a  +  2)xi/  ~    V  a  +  2 


9§  KEY  *0  t£x.  LiV. 

.-.  X  -  Ix'y^  +  y  =  1  or  -  51  ;  or  x'  -  ^*  =  +  I  or  +  V  -  51  ;  and 
x'  +  2/*  =  5  .-.  x'  =  3,  or  2,  or  ^  (5  +  V  -  51)  j  whence  i  =  9,  or  4, 
or  ^  -  13  i  V -"sT)  ;  similarly  y  =  4  or  9,  or  ^  -  13  ?  V  -  51) 
By  using  tlirougbout  the  value  'Jx  +  ^y  =  20,  other  values  of 
X  and  )/  may  be  similarly  found. 

40.  From   the    Ist  equation   (x  +  y)(x^  -  xy  -i-  y^)  =  x  -  y 

X  —  y 
.•.  x^  -  xy  +  y"  =  —-—,  and  from  the  2nd  equat.  x^  -  axy  +  i/-  =  0 
X  T  y  ,_ 

X  -  y  I  ^^  ~  2-^y  +  J** 

.-.  by  subtraction,  we  have  (a-  l)xy  =  — -  ::  V  'xU^iii^y' 

{\     '^ 

b  62 

suppose.     Then  y  =  —,  and  ^^  =  — r;  substitcang  these  values 

in  the  2nd  equation,  we  have  x^  +  —.,  =  ab;  or  x*  -  abx^  =  -  I-  ; 
whence  x^=  -(a  ±  i/a^  -  4)  ;  and  therefore  x  =  ±  \/ -- (a  ±  ^a-* -  4) 

=  4  iV6{2(a  1  V'^^^)}^  =  ±  JV^CV^^^t^  i  V-^"^^)  by    Algebra 

Art.  189.  Aud  y  =  't  =  ±  '2.-^ b{^j'a+l.  T  V''  -  -)•  See  Algebra 
Art.  181 

41.  From  the  1st  equation  xy  -i-  ax  -  ay  -  or  ]  that  is  (x  -  a) 
()/  +  a)  =  0  .•.  X  -  a  =  0  ;  or  x  =  a.  Also  y  +  a=Q.-.  y  =  -a 
From  the  2nd  equat.,  substituting  x  =  a,  we  have  a  +  y'^  +  a^  =  t 
.-.y  =  ±  a^'J  -  (u'^+  I).  Again  substituting  -  a  for  y  in  the  2n(l 
equation,  we  have  x  +  a^  +  a^  =  0  ;  whence  x  -  -  d^{a  +  1) 

42.  Squaring  the  first  equation,  we  have 

X*  +  7/*  +  a*  +  2x2(y*  +  d^)  +  2a'^-  =  0  ;  and  subtracting  this  fron 
the  2nd  equal.,  we  getx2(y--a')  -  2aV  =  0  .-.  x\y^-a-)  -  2dY  (i 
From  the  1st  given  equat.  x'^  -  -  (y*  +  a')  .-.  i^(y^  -  a^)  =  -  (y*  -  tt*) 
.-.  -  (y*  -  a*)  '=  2(iV>  or  y*  +  2a''y''  =  a* ;  whence  y*  -  u='(  -  1  ±  V^) 
and  y  =  ±  flV  -  1  ±  V^-  -'^Iso  x-  =  -  (y*  +  a=)  =  f  "V^ 
.•.  X  =  y  f  u"Y- 


Ex.  LiV.,  ALGEBRA.  99 

43.  From  2ad  equat.  x^  -  3y^  +  a^  +  3x*t/  -  xV  -  a^J^  -  Sct'x^ 
=  0  (i),  and  1st  equation  x  (x^  -  y^)  gives  x^  -  Sf  +  3.c*y  -  x^y'^ 
+  a:^x*-a^y'^=0  (ii);  then  {i)-(ii)  gives  a^ -2a^x'^-2aPx^+a?y^=0 

or  dividing  by  -  a^,  we  tiave  2x*  +  2x^  -  a?-y'^=  0  (iii).    But  from 

f       {x^  +  0^)1  2        /x^  +  a^\ ' 
first  of  tlie  given  equations  ?/^  =  <    -  ^ f     ~    (  — j —  ) 

.-.  substituting  this  in  (in),  2x*+  2x^-  I  — - —  j    -«'  =  0 ;   tha/ 

is  when  clear  effractions  18x*  +  ISx^-x*-  2aV  -  (i^-  9a'=  0, 

that  is  I7r*  +  2(9  -  a^)x^  =  a»(9  +  a') 

a'  -  9  +  3V9  -  15a^  +  2a'* 
whence  x'^  = r- ,  and 

.-.  a-  =  i  -^f{  17("^  -  9  +  3V9  -  Voa?  +  2a^)\ 

And  y  =  -  —r^ —  =  -  -}-:(Q,a^  -  3  +  V9  -  15a^  +  2a6) 

44.  Raising  tlie    1st   equation   to   the  4th   power,   we  have 

a-*-  4x^i/  +  GxV-  4x!/^+  ?/*  =  x*  + 1/*-  2x'^f~  4xyCx^-  2xy  +  ?/^)  -  a* 

But  x''  -  2xy  +  1/^  =  a^,  and  x*  +  y*  =  6*  .-.  6*  -  2xh/  -  4d^xy  -  a*  ; 

.       ,                       -  2a2  i  V2a*  +  26* 
that  is  2x^y^  +  4f;^x^  -  6*  -  a*,  whence  xy  =    -^ 

Then  x-  -  2xi/  +  ?/  =  a",  and  multiplying  value  of  xy  by  4,  and 
adding,  we  have  x-  +  2xy  +  ;/-=:  -  'ia^  ±  2V2a*  +  26-*  =  ??i2^  suppose 

77t  +  a                7/1  -  a 
.-.  x-y  -  a,  and  x  +  y  =  m;   whence  x  =  — ;^ — ,  and  y  = 

where  m  =  4  V2 V2«*^26* -  Sa^ 

45.  From  1st  equation  x^  +  y^  =  a^  +  xy,  and  squaring  this,  we 

have  X*  +  2xV  +  y*  =  a*  +  2a-xy  +  x^y^  .-.  x*  -  x-y^  +  y*  -  2a-'xy  -  a* 

subtracting    the    2nd    given    equation    from    this,    wo    have 

d^  +  V3tt*-  26* 
2r^//^-  2(f-.ry  =  «*  -  6*  ;   wlience  .ry  = ;p =  r^,  8upj>o«e 

Then  xy  =  c^,  and  x^  -  xy  +  y^  =  </^  .-.  x^  -  2xy  +  y^  =  a^  -  c^, 

and  X  -  y  =  i  V<i^-^-     -^^^o  ^"  +  2.  y  +  y^  =  a^+  Sf^;  whence 

X   +   y   =   +  ^/oF+lic^  .-.   X   -   ±   iQ\/a^+  Sc-  ±   V^"  -  c-),    and 

a^  +  V3a*  -  2b* 


y  =  ±  2  (^1^^  +  3c*  +  V'*"  -  '-'0>  where  c^ 


100  KEY  TO  .'Ex.  Liv 

4G.  From  the  2nd  equation,  we  have 
X*  -  2xy  +  2x^(a  -  1)  +  3/*  -  2y\a  -  ])  +  (a  -  l)^  =  a''  -  47  ■(-  4 
•••  extracting  the  square  root  of  each,  we  have  x^  -  j/^  +  (a  -  1) 
=  +  (a  -  2).  To  find  the  values  of  x  and  y  which  are  independent 
of  a  use  x'^  -  y^  +  (a  -  1)  =  +  (a  -  2).  Thi-n  x^  -  /'  +  1  =  0,  o: 
x'^  =  y'^  -  I  ;  x^  -  3  =  ?/^  -  4  (hi).  Ag:iin  from  the  first  g' v  n 
equation  Sx^  -  18x*  +  27x«  =  3x«(x2-3)«  =  2j/6  -  lly*  + 52^«  +  27 
But  (III)  x^  -  3  =  y2  -  4,  and  x^  =  z/«  -  1  .-.  3(.y«  -  l)(y'  -  4>- 
=  2j/'5  -  111/*  +  52^2  +  27;  that  is  3^^  _  27^/*  -l-  727-  -  13 
^  2y«-  ni/+52y2+27  .-.  y"  -  16»/*+ 20y«  -  7,5  =  0  ;  multiplying 
by  -  4,  we  have  -  4y^  +  64y*  -  80y-  +  300  =  0,  and  adding  to 
each  side  y'  -  2Qy*  +  100  we  got  y^  -  4i/^  +  44^^'^  -  80y-  +  400 
=  J/'  -  20!/*  +  100.  Then  taking  the  square  root  j/*  -  2y*  +  20 
=  i  (y*  -  10);  tli^t  is  !/^  =  15,  or  y*  -  y'  -  -  5,  whence  y-  =  13, 
or  i(l  +  V^T9),  and  x"  =  y'  -  1  =  14,  or  i(  ±  V~  HJ  -  i;' 
whence  y  =  i  V^^i  or  i  V^l  4  V  -  i''^)  =  ^c. 

47.  From  the  1st  given  equat.  1/*  -  2x-y-  +  x*  +  4(y-  -  x-)  +  .'S 
=  2  V4(y*  -  x^y  +  5(?/«  -  x«)«  ;    that   is   (y*  -   2x«y'  +  x*) 

-  2(y^  -  x«)V4(y-~^~iO +^  +  {4(y'  -  X-)  +  5}  =  a  .-.  extracting 
the  square  root,  we  have  y'^  -  x*  -  V4(y'^  -  ^*)  +  5  =  0; 
or  y^  -  x2  =  V4(y*  -  x')  +  5  .-.  (y*  -  x^*  -  4(y*  -  x")  =  5 
whence  y*  -  x^  =  5  or  -  I,  taking  y*  -  x*  =  5,  we  have  from  the 
2nd  given  equation  y*  -  3y*  +  1  =  5x-  -  8x  +  8x^x*  -  2x  +  5  +  4 
=  5x'^  -  8x  +  4xV4x--8x-t-20  +  4  =  .Ix*  -  8x  +  Axy/iy-  +  x«  -  8xT5"+  4 
since  15  =  3y'  -  3x-.  Hence  by  transposition,  we  have 
1/*  -  3y*  +  x«  -  8x  +  5  +  Ax-^2y^Tx^  -  8x  +  5  +  4x*,  and  taking 
the  square  root  y*  =  i  (V^y"  +  x"  -  8x  +  5  +  2x);  usinj 
the  positive  sign,  y*  -  2x  =  -^Zy-  +  x*  -  8x  +  5 ;  squaring 
■>/  -  4xy2  +  4x-'  =  3y-  +  x*  -  8x  +  5  .-.  y*  -  4xy«  +  4x«  -  4y«  +  8x  +  4 

-  a-i  _  y2  +  9 ;  but  y«  -  x«  =  5  .-.  y*-  4x-y-  +  4x*  _  4y*  +  3x  +  4 
=i  9  -  5  =  4  .-.  y*  -  2x  -  2  =  i  2  .-.  y*  -  2x  =  4  or  0  ;  but  y«  =  x*  +  5 

•.  x«  -  2x  +  5  =  4,  or  x«  -  2x  +  5  =  0;  whence  x  =  1,  or  1  ±  2V  -^ 
and  y  ^  i  V-^*  -  5  =  ±  V«3,  or  ±  V2  i  4V^ 


Zx.  Liv,  LV  ]  ALGEBRA. 


101 


48.  From  the  2nd  given  equation,  we  have 
t'^ij-  -  dxy^y^  -  x^  +  9(;/^  -  x")  =  16(y«  -  x^)  .-,  extracting  the 
tquare  root,  -ne  have  xy  -  3^/y'^  -  x^  =  ±  4'Jy^  -  x* 
■••^i/=  VF^^^or-v^^-a;-,and.-.x'y*-49(!/*-x^)or(?/--xO 
From  the  1st  given  equation  x*  -  y'^  -  4x'  +  4i/^  =  4x^  -  12 
.-.  X*  -  8x2  +  16  =  ^*  -  4!/*  +  4 ;  whence  x^  -  4  =  +  (y^  -  2) 
Ihat  is  !/2  =  x'^  -  2,  or  6  -  x2  .-.  1/2  -  x^  =  -  2  ;  x^y*  =  49(y2  -  x^) 
or  {y-  -  x");  that  is  x-(x^  -  2)  =  49(  -  2)  or  -  2  ;  that  is 
X*  -  2x2  =  _  98  or  -  2  ;  whence  x^  =  i  4  V^97  or  1  +  V^2 
and  !/2  =  a:*  -  2  -  -  1  i  V  -  •''^i  or  -  1  +  V^-~2.  Also  since 
j/2  -  6  -  x2,  we  have  by  substitution  x-(ti  -  x^)  =:  49(6  -  2x2) 
or  6  -  2x2  ;  that  is  x*  -  6x'^  =  QSx^  -  294,  or  =2x^-6 
that  is  X*  -  104x2  =  -  294,  whence  x'  =  52  ±  V2410  ;  or  x*  -  8x2- 
-•  -  0,  whence  x^  =  4  4  VlO.  And  i/*  =  6  -  x'^  =  -  46  +  V^IO 
or  2  f  a/To  ;  whence  x  =  &c. 


Exercise  LV. 

1.  Let  X  -  one  part,  then  19  -  x  =  other,  and  x(19  -  x)  =  84 

361-336 
x2  -  19x  =  -  84,  x2  -  19x  +  (Y)^ 1 =  ¥  •••  ^  --"^^  ±  i 

X  =  12  or  7;  whence  the  numbers  are  12  and  7 

2.  Let  X  =  greater,  then  17  -x  =  less;  x  -  (17  -  x)  =  2x  -  17 
=  difference;  then  r(2x  -  17)  =  30,  2x2  -  I7x  =  30^  whence 
X  =  10,  and  the  numbers  are  10  and  7 

3.  Let  X  -  length,  then  x  -  12  =  breadth,  and  x(x  -  12)  -  2080 
that  is  x2  -  12x  -  2080,  whence  x  =  52  and  sides  an;  52  and  40 
rods. 

4.  Let  X  =  greater,  then  x  -  9  =  the  less,  and  x*  +  (x  -  9)2  =  353 
that  is  2x"  -  18x  +  81  =  353;  x'  -  9x  =  136,  whence  x  =  17  or- 8 
and  the  numbers  are  17  and  8,  or  -  8  and  -  17 


102  KEY  TO  [Ex.  Lv. 

5.  Let  X  -  one  part,  then  16  -  x  =  other,  then  x(16  -  x)  +  x^ 
+  (\Q  -  X)-  =  208 ;  that  is  x^  -  l€x  -^  -  48,  whence  x  =  12  or  4, 
and  the  numbers  are  12  and  4 

X 

6.  Let  X  -  gain  per  cent.  =  buying  price  of  wheat ;  then  — TJ 

X  X* 

=  gain  per  dollar  on  buying  price,  and  x  x  -—-  -  -—  =  gain  on  x 

dollars,  i.e.  gain  on  whole  transaction  ;  but  171  -x  =  whole  gain, 

x^ 
whence  t^  =  1 71  -  x  ;  or  x'  +  lOOx  =  17100,  whence  x  =  $90, 

buying  jirice  of  wheat. 

Proop.  $81  +  $90  =  SlTl ;  also  if  he  gain  S81  on  S90,  he  gaius  at  the 
rate  of  $9  on  $10,  or  $90  ou  $100. 

80 

7.  Let  X  =  number  of  sheep  bought,  then  —  =  price  in  dollars ; 

80         80 
then  -     -  -  —  -  1,;  that  is  x^  +  4x  =  320,  whence  x  =  16. 
X  +  4        X         "  ' 

8.  Let  X,  y  and  z  be  the  digits,  z  being  the  right  hand  one; 

then  x«  +  ?/«  +  s«  =  104  (i) ;   ?/«  =  2xr  +  4  (ii) ;    lOOx  +  lOy 

■V  z-  594  =  100c  +  lOi/  +  X  (III).     Substituting  (ii)  in  (i),  we  have 

X-  +  2x2  +  c*  =  100  ;  whence  x  +  s  =  10  (it).     Reducing  (iii),  we 

have  99x  -  99s  =  594  ,  or  x  -  2  =  6  (v).     Adding  (iv)  and  (v) 

together,  we  have  2x  =  16;   whence  x  =  8.     Also  x  +  2  =  10 

.-.  z  ■■-  2,  and  y*  =  Ixy  +  4  =  32  +  4  =  36  ;   whence  y  =  6.     Hence 

the  required  number  is  862 

240 

9.  Let  X  =  number  of  sheep  bought,  then =  price  per  shei 

240 
X  -   15  =  number  sold,   and  ■ —  +  f  =  selling  price;    thea 

/240        2  \  /600  +  x\ 

(X  -  15)  f    .J.-  +  -A  =216,  that  is  (  — g;^](-f-15)  =  108; 

or  X*  +  45x  =  9000  ;  whence  x  =  75,  and  '-^^  =  $:,-20.     Hence 
number  bought  was  75,  and  price  per  sheep  $3-20 

10.  Let  X  =  one  number,  then  10  -i  =  other,  and  x'+  (10  -x)', 
that  is  x^  +  1000  -  300x  +  30x-  -  x'  =  280,  that  is  30x*  -  300a 


Ex.  LV]  ALGEBRA.  103 

=  -  120  ;  or  x'  -  lOx  =  -  24,  whence  j  =  6  or  4,  and  tbe  required 
numbers  are  6  and  4 

11.  Let  X  =  less,  then  21  -  x  =  greater,  and  x(24  -  x) 
=  35(24  -  X  -  x),  that  is  24x  -  x^  -  35(24  -  2x)  -  840  -  70x  ; 
or  x^  -  94x  =  -  840,  whence  x  =  10  or  84,  and  24  -  x  =  14  or 
-  60  .-.  the  required  parts  of  24  are  10  and  14,  or  84  and  -  60 

12.  Let  X  and  y  be  the  numbers,  then  x  +  y  =  xy  =  x-  -  y- ; 
X  +  1/  =  x^  -  2/^,  whence  dividing  by  x  +  y,  we  have  x  -  y  =  1  ; 
or  r  =  1  4-  y.  Also  x  +  y  =  xy,  that  is  I  +  y  +  y  =  y(l  +  y);  or 
\  +  2y  =  y  +  y^  .-.  y-  -  y  =  I ;  whence  y  =  i(l  ±  V^)?  ^u<l  x  t  1  +  ?/ 

13.  Let  x  =  circumference  of  hind  wheel,  and  y  -  that  of  fore 

120          ]20 
wheel  in  yards;    then  —^  and =  revolutions  made  by  each 

120  120 

in  coiner  120  yards.     Also  by  second  condition r  and 

b      b  J  J  x+ly+1 

=  revolutions  made  in  120  yards. 

120       120  20         20 

-:-  = Q'\  —    =    —    -1  ~\  (i)  by  dividing  by  6 

Then     ^  ^  (-;  or     "^  ^  (- 

120       120  i  30         30  I 

x  +  1     J/+1         -^  x  +  1     y+1         ''  ^   '' 

.-.  20x  -  201/  =  xy  (III)  and  29x  -  3\y  =  xy  +  1  (iv).     Subtracting 

117/+1 

(m)from  (iv),  we  have  9x-  lly  =  1 ;  or  9x  =  11?/+  1  .-.  .r  =  — -- — 

20(11?/+ 1)  /iiy+i 

Substituting  this  in  (in),  we  have ^ -  20y=y  (  — - — 

that  is  220]/+  20  -  180?/  =  \ly^  +y  .-.  \\y^  -  39y  =  20;  whence 

11?/+  1 
y  ~  4,  and  x  =  — r —  =  ^^  =  5.   Hence  circumferences  of  wheels 

arc  4  and  5  yards  respectively. 

14.  Let  X  =  one  fraction,  then  It  -  x  -  other,  and  —  +  i.c, — 

^      ii  ~^ 
1  15 

=   sum    of    their   reciprocals    .•.    —  +  rr rz~  -  i?,  ;    that  is 

'  X        29  -  lox        '  ^  ' 

12(29   -  15x)  +  ISOx  =  29x(29  -  15x),  whence  by  reduction 


101  KEY    TO  (Ex.  LV 

435r'-841.r  =  -343;  orx«-|H  =-  Uz  i    ^*  -  UU  +  (5?J)* 

~    746900  736»ll(>   -  TsbSuOl   ^"^'*'''=  •'•         »70   i   870  870     "^   j  J-J 

Hence  fractions  are  i  and  § 

•*  46800 

15.  Let  X  =  number  of  children,  then =  share  of  each  ; 

4'5300         46800  24  24 

then —z-  =  +  1950,  that  is  =  —  +  1 ;  whence 

X  -  2  X  '  X  -  2        X  ' 

r-  -  2x  =  48  .•.  X  =  8  =  number  of  children. 

16.  Let  X  =  number  of  hours  the  clock  is  too  fast,  then,  since 
the  shadow  on  the  dial  moves  from  1  to  5,  the  clock  will  strike 
the  hours  from  2  +  x  to  5  +  x  inclusive  ;  i.  e.  will  strike 
2  +  x  +  3  +  x  +  4  +  x  +  5-i-x  =  14+  4x  strokes,  and  last  stroke 
will  be  5  +  X.  Then  (5  +  x)'^  -  41  =  number  of  minutes  the  clock 
is  too  fast  above  the  x  hours;  i.e.  25  +  lOx  +  x*  -  41; 
i.  e.  X*  +  lOx  -  16.  But  hours  too  fast  +  minutes  too  fiist 
=  whole  number  of  strokes ;  that  is  x  +  x*  +  lOx  -  16  =  14  +  4x, 
whence  x'^  +  7x  =  30  .-.  x  -  3  or  10,  and  x'  +  lOx  -  16  =  9  +  30 
-  1 6  =  23  .-.  the  clock  is  3  h.  23  m.  too  fast.  The  second  answer 
10  is  excluded  by  the  limitation  that  the  clock  does  not  strike 
12  during  the  time. 

17.  Let  X  =  hours  travelled  bj  each  -  miles  per  hour  travelled 
by  slower,  then  x  +  3  -  miles  per  hour  travelled  by  faster; 
x'  +  x(x  +  3)  =  2x*  +  3x  =  324,  whence  x  =  12.  Hence  slower 
travelled  12  x  12  =  144  miles,  and  the  faster  12x15  =  ISO  miles. 

144  144  144 

IS.  Let  X  -  number,  then =  share  of  each  .-.  -  , ^,  +  I  =  — - 

'  X  X  +  J  X 

whence  x-  +  2t  =  288  .-.  x  =  16  =  number  at  first. 

lOx  +  y 
IM.   Let  X  =  left  hand,  and  y  =  right  band  digit,  then  — — — 

•=  2  (0,  and  10x  +  y+  27=  10i/  +  x(ii).  From  (i)  10x+ y  =  2xi/ (in) 
From  (11)  9x  -  9^  =  -  27,  whence  x-y  =  -3,  orx  =  y-3; 
substituting  this  in  (m),  we  hav«  10(y  -  3)  +  y  =  2y(y  -  3) 
.-.  2y'  -  Uy  =  -  30,  whence  y  =  6,  and  x  =  y  -  3  =  3.    Hence  the 

required  number  is  30 


Ex.  Lv.]  ALGEBRA.  105 

20.  Let  X  =  price  of  cf>ffeo,  and  y  •  prico  of  s  igar  per  lb.  in 

800 
cts. ;  then  60x  + 80]/ =  2500  (i).    Also =  lbs.  of  sugar  for  $8, 

1000                                                            800       1000 
and =  lbs.  of  coffoe  for  $10:    then    —  = +  24   (ii) 

125-4y 
From  (ii)  by  reduction  100.c-12.'')i/  =  3x7/ (iii).  From(i)x  =  — r — , 

100(125-  4y) 
substituting  this  for  x  in  (in),  we  have  ^ 125y 

(125 -4y) 
--  3)j  X — -,  whence  by  reduction  Gy^  -  575)/  =  -  6250 

125  -  iy 
.-.  y  =  12i  cents,  and  x =  25  cents. 

21.  Lot  X  and  y  =  number  of  days  required  by  B  and    C 
respectively  to  finish  the  work ;    then  in  1  day  .4  does  -j^sth; 

1  1  11  36     x  +  18 

JB,   -  th  ;  and  C,  — th  of  the  field  ;  -,' •  +  —  :  —  : :  $36  :  -  -^  -  ^  — 
'  X       '  '  y  '   ^^      X       X  X        18x 

o"4?i 

=       ,    ,o  =  what  B  would  have  received,  had  C  not  been  called 
X  +  18  ' 

10                 360 
in;    but  B  worked  10  days    .•.   he  did  receive  —  x  36  =  

/    648  360\ 

Then   j  -  • —  I  dollars  -  $1-50  =  $3 ;    whence  by  reduc- 


X  +  18 
tion  x«  -  l74x  =  -4320  .-.x^  -l74x  +  (87)^  =  7569-4320  =  3249 

4 
.-.  X  =  30  =  days  B  would  require.     And    ^8+50+  —  =1 

•••  7^1-  Cf'  +  3")  =  1  -  iH  h  ■■■  2i/  -  T2,  and  y  --  36  =  days 
''.'would  require  tocradle  the  field. 

Troof.— If  C  had  not  been  called  in,  they  would  have  taken  11 1  days  to 
finish  the  work,  and  ^'s  share  would  have  boon  S2  x  Hi  =  S22-50. 
Uenco  ZJ's  share  would  liave  been  $13'50,  but  since,  when  C  ie  called  in, 
Ii  only  works  10  days,  he  receives  only  i^  =  i  of  $33—  fl2  =  SISO  less 
than  he  would  have  otherwise  received. 

22.  Let  X  and  y  =  the  number  of  feet  in  the  side  of  the  base  ; 
then  5x1/  -  4ry  -  xi/  =  80  +  x  +  1/  (i)  ;  also  V25  +  x*  +  j/* 
=  V'oC^'  +  y*)(n).     From  (n),  we  get  3(x''  +  y^)  =  40V25  +  i"«Ty* 


I 


106  KEY   TO  [Ei-  Lv. 

That  is  (x^  +  y'  +  25)  -  ^^V^^  +  y^  +  25  =  25 

(x*  +  I/*  +  25)  -  *32(x»  +  y2  +  25)^  +  *5°  =  ^5"  +  »5^  =  *|^ 

.-.  'Jx'  +1/^+25  -  "3<^  =  1  V-  •••  V^*^  +  y*  +  25  =  15  or  -  ^; 

squaring,  we  have  x^  +  y"  +  25  =  225  or  ^^  .-.  taking  the  former 

value,  x2  +  1/2  =  200  (iii),  but  by  (i)  2xj/  =  160  +  2(x  +  y)  (iv). 

Adding  (iv)  and  (iii),  we  have  i*  +  2xy  +  y*  =  360  +  2(x  +  y). 

Hence  (x  +  y)"^  -  2(x  +  y)  =  360  .-.  (x  +  y)*  -  2(i  +  y)  +  1  =  361 ; 

(x  +  y)  -  1  =  +  19,  X  +  y  =  20  or  -  18  .-.  again  taking  the 

former  value,   x  +  y  =   20,   and   hence  xy  =  80  +  20  =  100 

x^  +  2xy  +  y"  =  400  n 

J    .-.  X  =  y  .-.  2x  =  20,  and  x  =  10  =  y 
4xy  =  400    '  ' 


x^  -  2.ry +  y''  =       0 

1    .-.  the  base  is  a  square  whose  side  is  10  ft. 

.•.  X  -  y    =      0 

23.  Let  X  =  distance  B  has  travelled  when  he  meets  »4,  then 

X  +  15  =  distance  j1  has  travelled  ;    Also  since  .f  has  yet  to 

travel  x  miles,  and  accomplishes  it  in  2   hours,  his   rate  of 

X  X  +  1 5     2x  +  30 

travelling  is  —  miles  per  hour  ;  also  B's  rate  is  — —. —  =  — ~ — 

X  +    15       2x   +   30 

Then   time  ^  travels  before  they  meet  =  — , =  

•^  ix  X 

X  9x 

time    B   travels   before    they   meet  =   n^"^??;^   =   .; r^ ; 

^  2x  +  30  2x    +   30' 

9 

2x  +  30  9x 

.-.  — - —  =  2j.^3o»  <hat  is  X*  -  24x  =  180  ;   whence  x  =  30  =  rate 

X,  ,.  K        .  ^  2X  +  30       90 

Hence  distance  =  x  +  x+15  =  75  miles.     B's  rate  = =  — 

9  9 

X 

=  10  miles  per  hour.     J's  rate  =  —  =  15  miles  per  hour. 

24.  Let  X   and   xy  be   the  two   numbers,    the   latter   being 

the  greater;    then  x-y  =  x'y*  -  x*,  whence  y*  -  y  =  1,  and 

y  =  J(l  i  V5).     Also  x*y*  +  x«  =  x'y^  -  x»  .-.  y^  +  1  =  xy^  -  x  ; 

y-  +  i      i(3±V5)+l      K5±V5)      W5(V5+1)        ,  , 
whence  x  =  ^j-_^  =  ^^-^  -^^j—  =  -— ^-  =  "T±Y5-  =±  ^^^ 


Ex.  Lv.]  ALGEBRA.  107 

25.  Let  X  and  y  ~  hours  required  bj  Bacchus  and  Silenus 

•       1 

respectively  ;   then  Bacchus  would  drink  —  of  it  in   1  hour, 

y 
hence  in  y  hours  he  would  drink  --ths  of  it,  and  in  ly  hours 

2y  2y 

Bacchus  would  driuk  ^ths  of  the  cask  full  .•.  1  —  —  =  part 

1 

drunk  by  Silenus,  and  since  he  drinks  —  of  the  cask  in  1  hour, 

/       2y\      1 
the  time  he  required  to  driuk  part  remaining,  was  j  1  -  —  1-r  — 

2v* 
-  y  -  .1  %     Had  both  drunk  together,  Bacchus  would  only  have 

/         2  (A  V 

consumed  i  (  I  -  ^-;  )  =  i  -  r-,  and  Silenus  would  have  taken 

3x 


y 

i  +  r-  ;   hence  when  drinking  together,  time  taken  by  Bacchus 


was  [\  -q~)-^  —  ~~ q">  ^°*^  ^^®  ^™^  taken  by  Silenus 

/       y  \      1       y      3/*  ^      y       y      v* 

2y       /  2.y2\ 

Also  "^  +  ( 2/  ~  ~o  ~  1  =  tioie  taken  when  drinking  separately 

=    ^^  -  "3"  +  2    (n).     From   (i)   Sx'  _  Zyx  =   2y^.     From   (11) 

57/        257/2       25y«       2y2 
12j:;/  -   \y'^  =   3x'^  +    12x   (in)   x'  -   ^^  +  '  36^  =      9      "^  "F 

40(/'''  5?/         7?/  12y 

=     o  .'  ;   •■'^  — TT  =  ±-T.     Hence  x  =  -—-  =  2y ;  substituting  this 

00  DO  O  "^ 

value  of  X  in  (iii),  we  have  24y"  -  4y^  =  3(2y)'*  +  12 (2y),  that 
h24y' -  4y^  =  I2y^+  24:y  .:  By-  -.24y^0;  y^=3y;  y  =  3, 
ivhcnce  x  =  U 


108  KEY   TO  [Ex.  Lvi. 


Exercise  LVI. 

a         c       ab        I 

1.    ,  •  X  --  X  —  =  —  =  1  :  (f 
0        a^        cd        d 

a2_63     (a-bf    a=«-£f6  +  6*_  (a-6)(a+4)(o-ft)2(a2-a6  +  6») 
"•  a^~+b*^       a       ^    (a -by    ^  ""a(a+l()(a«  raAT6')(o-6)» 

I 

—  ==   1  :  a 


4.     ,  ,  ■■■  ■-  5-  — riTi  according  as  a*  +  ^''ft  +  ab'  +  i*  -c   -j* 

+  2(i^b''  +  li* ;    or  as  a^t»  +  a6*  ^  2o'6* ;    or  as  a*  +  6*  ^  2a6 ; 
but  a«  +  6*  is  greater  than  2ab  (Algebra  Art.  134,  Note  2) 

a'  +  b*        a'  +  b^ 
■  ■   a>=  +  6^  '^     0  +  6 

X*  -^  y'  >  (j;  +  .v)* 

x^  -  x'y  +  2j*^'  -  2x^1/'  +  2x"y*  -  xy"^  +  y^  ^  x"  +  4x'y  +  5x*// 
-  5.c»t/* -  4jy '  - y** ;  or  as  Tx^  +  3xy "  +  2y^  >  Sx'y  +  2x V  +  3x V ; 
or  as  Ixh/  +  3xy*  +  2y'  ^  Sx"*  +  2x»y-  +  Sx^  ;  or  as 
y3(7x'  +  3xy  +  2y')  ^  x'(5x«  +  3x.v  +  2y-) 
Now  since  x^5  >  y^7,  cubing  we  have  5x^  >  7y^  .-.  x*  >  y^ 
.-.  ?/S(3Ty+  2y*)  <  x«(3x,y  +  2y')  ;  also  7xV  <  5x»  •.•  7y»  <  5x^ 
.-.  i/\Tx''  +  3xy  +  2y»)  <  x*(5x«  +  3xy  +  2.v-) 

x°  +  »/«    ^ (xj-  y)* 

'■x--y*  '^  X*  -  x'y  +  ^"y*  -  3ry*  +  y* 

6.  Let  X  =  the  quantity  to  be  subtracted  from  each  term; 

o-x       c  be  -  <uC 

then  r =  —r  .' .  ad  -  dx  =  be  -  ex  ;  ex  -dx  =  bc-ad  .•.  x  = -j- 

0 -X      a  '  c -  ft 


Ex.  Lvi.  Lvn.]  ALGEBRA.  109 

7.  Let  X   =   the   qaantity  to  be  added  to  each  term  ;    then 

'"^^^l,whence7n  +  x  =  n  +  a:;  x-x^m-n;  i(l-l)  =  m-n; 

71  -T  X  ' 

IP    —  n       m   —  n 
T =  =  oc 

(«^6)       o^^       _J1_  _*  t  :  a  4-  6 

-  («.-b)  i»^  (a +  6)'       a +6 

_  (a  +  c)«  _  oM^^  +  c* 

y.  Since  a  :  c  : :  c  :  6,  c  =  V"'' ;   then  T^TjT^  ~  6M-  2bc  +  c^ 

.7*  +  lo'Jab  +  o&  _  a((H-  Vajjf  M  ^  ^  ^  ^  .  j 
&■'  +  26Va6  +  ff^       ^(^  +  2V"''  +  0)        '' 

a'  +  fr*  ^  a  +6'  °  ^ 

+  a&«  -  6^    or  as  a^b  -  ab^  >  a6^  -  a«6  ;    or  as  2an  J.  2a-6«  ; 

or  a3  a  ^  6 


Exercise  LYII. 

1.  Let  x:^  the  quantity  to  be  added;  thena  +  T:?>  +  x::  c  +  a'irf +  x 
.-.  (1,1  +  flx  +  rfx  +  x«  =  60  +  6x  +  ex  +  x«  ;  ax  +  (/x  -  6x  -  fx  =  tc  -  fl<2 

6r  -  cul 
■'■  ^^  aZb~-Wd 

2.  Ua:  b  ::  c:  d ;   then  y  =  "J*     ^^  '*"  ^^  P'^ssil'le,  let  x  be  a 
quantity  added  to  each,  so  that  a  +  x  :  b  +  x  ::  c  +  x  :  d  +  x ; 

a-'r  X       c  -^x  be  —  ad  . 

^'"="  i  +  x  "  </  +  x'  "'^^"^  ^^  ''^°''''  ^  "  o~6^^7^~J'   ^"*  "°" 

g  C  0 

-r-  =  -7,  "we  ^ave  be  -  ad  .-.  bc-ad  =  0  .-.  x=  ^j  _  ^  _  ^  +  ^  =  0 


110  KEY  TO  [Kx   LVit. 

a       c        a"^      c^  m       p        m        p 

3.  T  =  T  •'•  r7  =  F  !    ^'so  —  =  ^  •■■  r-  =  „-.     Multiplying 

ma^  pc^ 

equals    by    equals    „  vi    =    ,  T2  >     ^^^^     Algebra    Art.     10 : 

ma*   -   2ni«       pc^  -  2qd'^ 

7/ia'*  +   2n6'       pc*  +  2^^'  '  ^ 

+  271^2  :;;c*  +  2(/(/'2 

24 

4.  Let  X  =  one  number,  then  —  =  other 

And    x'-  (  —  )    :   (  X  -  —  )    ::  19  :  1 


.       /24\'             r                     1728       /24\'l 
Hence  x'*  -  (  —  j     =  19-^  x'  -  72x  + (  ~~  )   f 


24\'  32332  /24 

—  j     =19x'-1368x  +  — - —  -:9|  — 


,__.                         32832 
18    —       -   18x3  _  _   i3gg_j. 


24 


Dividing  each  side  by  -7  -  r,  we  have  (  -    )     +  24  +  x*  =  76, 

576 
thatis-y +  x«=  52;  x*  -  52x'' =  -  576  ;  x*-52r  +(26)*  =  100; 

x'2  =  26  +  10  =  36  or  16  .-.  x  =  ±  6  or  ±  4,  and  the  n  Jinbers  are 
4  6  and  ±  4 

5.  Let  X  =  one  part,  then  20  -  x  -  other  part ;  then 
a-  :  20  -  X  ::  9  :  1  .-.  X  =  180  -  9x;    or  X  =  18,  and  20  -  x  =■  2 
Let  y  be  the  mean  proportional  between  these ; 
then  18  ;  1/  ::  1/  :  2  ;  or  ?/*  =  36  .-.  y  -  6 

X        «^  a        ^r  +  X         a^       c  +  x  x        c  +  x 

'    1/    ~  l>'  '      "      b    ~   »/,y  j^.y  '  '  b^  ~  d  + ;/   '  '    y        ''+.</ 
.•.  dx  +  xi/  ~  rt/  +  XV,  or  dx  =  ry 

7.  Dividinyf  the  equation  by  (u  + 1  -  r  -  r/)(<i  -  b  -  c  +  d), 

a  +  b  -i-  c  +  d       a  -b  +  c  —  d 

we  have      ,  ,  ~'"    /  =   — 1. 7~,-     Then  Art.   IOC,  wo  have 

a  +  0  -  c  -  d       a-  b  -  c  -t  d  ' 


Ji-v.  i.vtj.i  aLgsbrA.  Ill 

2a+2b       2a -2b  a  +  b       c  +  d  2a      2c 

2^Tld  "  2rru  ■■■  Art.  106,   — ^  =  ^^^  .-.  Art.  106,  ^-^  =  ^^  ; 

(I  c 

oc  ',-  =  —  .•.  a  :  b  ::  c  :  d 
b        d 

8.  Let  or  and  y  =  the  numbers  ;  then  x  +  y:s::x  -  y  :  d::xy  :  p 

s      x  +  1/  d      x—y 

■■  P{^  +  y)  =  sxy,  and  p{x  -  y)  =  dxy  .-.  —  =  -^y,  and  —  =  -^ 

s  1  1    -x  s  +  d        2  2v 


+  —  M  By  addition  =  — ,    wbenco    y  = 


p         y         X    F     •'  p  y'  -^        s  +  d 

d  I  I    C  2        s-d  2p 

—  -  — 1  By  subtraction  —  =  ,  whence  x  -   -; 

p        y         X    J     ■'  X  p    '  s  —  d 

9.  Let  X  =  speed  in  yards  of  faster  train  per  second,  and 
y  =  speed  of  slower;  then  in  2"  the  former  passes  over  2x,  and 
the  latter  2y  yards,  consequently  2a;  +  2y  =  length  of  the  faster 
train;  also  36a; -30^  =  length  of  faster  train,  .-.  30x-30^- 2z-f-2y, 
or  28t  =  32»/,  or  1x  =  8y,  .-.  x  :  y  ::  8  :  7 

10.  Let  X  =  ^'s  money,  and  y  =  5'3  ;  then  a;  4- 150  :y- 50  ::  3:2, 

whence  2x  +  300  =  3y  -  150  ;    or  2x  -  3y  =  -  450  (i).     Also 

X  -   50  :  y  +    100  ::.5  :  9,  whence  9x  -  450  =  5y  +  500;   or 

9x  -  5y  =  950  (n).     Multiplying  (i)  by  9,  and  (n)  by  2,  we  Lave 

18x-2  7y  =  -4050  (w)) 

^.Subtracting  (ui)  from  (iv)  17y  =  5950, 
18x  -  lOy  =      1900  (iv)J  ^   ^  v     /       :/  > 

whence  y  «  $350  =  JS's  stock;    2x  -  3y  =  2r  -  1050  =  -  450, 

whence  2x  =  COO,  aud  x  =  $300  =  ^'s  stock. 

11.  b  =  V«c"  .-.  b'^  =  ac;    2b^  =  2ac;    6'^=  2ac  ~  b^  .-.  adding 

a*  +  r*  to  each,  a*  +  b'^  +  c^  =  a'  +  2ac  +  c^  -  ¥ ;    or  a^  +  t^  +  c' 

(a  +  c  -  b)(a  +  c  +  6) 
^  {a  +  cY  -b-'=(a  +  c-  bXa  +  c  +  6)  .-.  1  =  ^ ^rf^l^^ 

1  a+c-b  a+c+b  a-b+c 


a  +  c  +  b      a*  +  6^  +  c^'   "'   (a  +  c  +  b)'^      a^  +  b^  +  & 
a  +  b  +  c  :  (^a  +  b  +  cy  : :  a-b  +  c  :  a^  +  6^  +  c* 


Il2  KEY   TO  [Ex.  Lvti. 

12.  a  :  c  ::  a  :  c,  .-.  multipljing  each  term  of  the  latter  ratio 
b}'  fl  -  c,  we  have  a  :  c  ::  a(^a  -  c)  :  c(a  -  c) 

.-.  a  :  c  ::  a{^a  -  'Jc){\la  +  VO  :  f(V''  -  V<^)(V<'  +  V<^) 
.-.a  :  c  ::  V«(V"  -  V'^)(V'^  +  VOV<^  •  V-CV*^  -  V'^)(V''  +  V'')V'^ 
.-.  «  :  c  : :  (a  -  '^'uc)(a  +  -jHc)  :  (i/ac-  c)  ('Jac  +  c)  :   but  V<^  =  * 
.-.a  :  c::  (a-  6)(a  +  6):(6  -  c)(6  +  c)  siacc  by  hypothesis  V<'^=  ^ 

13.  Let  X  =  the  number  ;  then  x  +  3;x  +  8::x  +  8:x+l7 
.-.  (x  +  3)(x  +  17)  =  (x  +  8)^  that  is  x«  +  20x  +  51  =x^  +  16x  +  64; 
or  4x  =  13  .-.  X  =  3i 

14.  Let  D  and  d  =  diameters  of  a  sovereign  and  shilling  res- 
pectively, and  t  and  T  -  thickness  of  a  sovereign  and  shilling 
respectively  ;  then  mii  -  iiD,  and  pt  =  qT,  and  since  circles  are 
to  one  another  as  the  squares  of  their  diameters,  we  have 
quantity  of  metal  in  sovereign  :  quantity  of  metal  in  shilling 

D«         t 
::  D^T :  d*t,  or  ::  -^—  =  -y" 

Dm  t  q 

But  md  ^  nD  .-.  D  :  d  ::  m  .  n  .-.  —r~  =  - — ;  similarly  ~rfr  =  — 

a  n  •  ''I  p 

:n*         q 
.'.  quantity  of  gold  in  sov.  :  quaa.  silver  in  shil.  ::  — y   :  — - 

But  a  sovereign  =  20  s.,  .-.  quantity  of  gold  in  bulk  equal  to  a 
shil.  :  quantity  of  silver  in  a  shil.  ::  ."  —5-,  or  ::  20f»'^  :  m'p 

a        c  -42  -420      -Ale 

^^-  y  "^  T  •■•  "^"'''PlyinS  both  by  —  ,  we  liave  -y-^  -  ,  j,^ 

•42a+n;6       •42c+llirf 
.-.  Art.   100,   — j-^7j^ —  ^^  — llld"^'   °^  multiplying   by    llf 

•42a+lU6       •42r+lUd  a         c        4a       4c       4^-5b 
b           =  "~~d <'^i  '^''^  y  "  T  •■•  56  "  W   ■•  ~56 

4c  -  f>d  4(1  -5b     4c  -  5,/ 

=  —5 J—  ;  or  — ^  -  =  —J, —  (11).     Dividing  (1)  by  (11),  we  b:iTe 

•42a+lU6  b  •42c+nlrf  d  .    •42a -HI 7/ 

b  ^  4</  -  56  "  d  '^  4c  ^TW  '  ^^'^^  ^  '4a  -  bb 

•42a+U\d 
~      4c-5d 


Es.  LVII,  LVIII.] 


ALGEIiKA.  113 


16.  If  a:6::fc:c::c:</,  «c  =  6^  and  bd  =  c-  .-.  d  =  -^  =  ——  =  -- 

/      V^\  <^ 

(a  +  /))  (c  -  r/)  =  (rz  +  V«c)  ['^--j-j  =  *J(^W'^  +  VO  (V'^  -  V'-)-T^ 

«       ft        c        a^       a       6        c        a         a        ^/a 

=  c'ia  -  c) ;  y  -  y  =  -5-  •••  ^3  =  y  ^  ~  ^  -^  -  "J  •■■  y  =  ^J 

.-.  a  :  ya  ::  b  :  yd 


EXBBCISE   LVIII. 

1 .  v.x^  ■{■  vy  cc  cx^  -  dy  .• .  mx*  +  ny  =  p(cx^  -  dy)  =  pcx^  -  pdy 

.'.  pcx'  -  mx^  =  ny  +  pdy;    that  is  x^(pc  -  m)  =  (re  +  pd)y 

Oi  +  />':^\  /     In  +  pd\ 

^;r:^,Jy  ■■■  ^  =  (^V^^ri^^JVy;    But  since  n,p,  m,  d 


and  c  lire  all  constant,  Y  I  ^  — ^»'  )  ^^  constant  .-.  x  az  \Jy 

2.  X  -  my;  1  -  3/u  .-.  m  =  I  .-.  x  -  ly 

m  m 

3.  X  =  p  +  ;    then   1  =  /)  +  — -  (i),  and  2  =  p  +  m   (11) 

y  ^ 

Subtracting  (1)  from  (11),  1  -  liti  .-.  m  =  I,  and  p  +  m  =  p  +  ?.  =2 

7/1  I  3 

.-.  ^  =  i  .-.  X  =  p  +  --  =  i  +  y  =  J  +  2-  =  i  +  ,33^  =  i  +  Vo  =  t 

4.  X*  =  mr/* ;  4  =  64m  .'.  m  =  -,V  ;  x«  =  iigj/*  .-.  x  =  ly^/y 

5.  X  =  ??i  +  nxy  .•.  2  =  m  +  6?i  (i),  and  3  =  in  -  9n  (11) 
Subtracting  (i)  from  (u),  we  have  1  =  -  15/i  .•.  7i  =  -  -,V 
771  -  /j  =  2  .-.  OT  =  2  +  -j^s"  =  V'     Then  x  -  7ixy  =  m ;  x{l  -  ny)  =  zn 

-       ^^  V  _  ','         _       36 

■  •  "^  ~  1  -  7iy  ~   i-(--,J3  2/)  ~      \2  iJ'     ~    ^"  +  y 

15 
G.  .v  =  7n  +  72X  +  px^ ;  then  0  =  m  +  3n  +  9^9  (i) 
-  12  =  »i  +  5/1  +  25/>  (11),  and  -  32  =  m+  1n  +  49p  (ill) 
Subtracting  (1)  from  (11),  we  get-  12  =  2/i  +  I6p  (it) 
"  (I)  from  (III),     "        -  32  =  4n  +  40/) 

11 


iv)-l 
fv)J 


114  KEY  TO  [Ex.  Lviii 

Dividing  (iv)  by  2,  and   (r)  bj  4,  we  have  -  6  =  n  +    8/j    (vi) 

-  8  =  ?*  +  ICtp  (vii) 
Subtracting  (vi)  from  (vii),  we  have  -  2  =  2p  .-.  p  =  -  1  , 
_  6  =  ra  -  8  .-.  n  =  2  ;  0  =  m  +  6  -  9  .-.  7H  =  3  .-.  y  =  3  +  2x  -  X* 

n  n  n 

1.  y  =  mx"  +  — ;    then  1  =  25m.  +  —  (i) ;    5  =  Sim  +  —  (n) 

25a        n 

Dividing  (ii)  by  5,  and  (i;  by  9,  we  have  I  =  — ■  +  —  (in), 

81»i        n  2        604/11 

and  1  -  -   —  +  —  (iv).     Subtracting  (iii)  from  (iv)  —  =  '—.r,' 

n      5x*     9945 
•■•  "i  =  ii;-i  •••  n  =  35-125/«  =  3235i;  theny  =  ;«x2 +  _  =  _  +  ^^^^^ 

a'^  1 

^  b  b 

b^     x^  x" 

y  =  y  +  T  =  ^  +  T 

9.  I  -  X  -  1/  =  HI,  and  (x  +  y  +  :)(x  -  i/  —  :)  =  nyz ;    thai  is 

{x  +  (y  +  s)}{x  -  (y  +  =)}  or  x*  -  (y  +  :)'  =  nyr.     Adding  4y=  to 

each  side,  x*  -  (y  -  2)*  =  (n  +  4)y=  ;  that  is  (x  -  y  +  r)(x  +  y  -  :) 

=  (/I  +  4)i/z  ;  rmt  :  -  X  -  y  =  7n  .•.  X  +  y  -  2  =  -  7n  .•.  -  ;/i(x  -y  +  z) 

n  4-  4 
=  (/t  +  4)y2  .■.x-y  +  z  =  -  — rr-y=  .-.  x  -  y  +  c  oc  ys 


10.  Let  X*  =  number  of  cars  attached,  then  decrease  of  speed 
oc  x'^,  and  is  .*.  =  ttix*  ;  then  24  -  jnx  =  speed  of  train, 
.-.  20  =  24  -  2m  .-.  2in  =  4,  or  7/1  =  2  ;  then  24  -  2x-  =  speed  when 
x^  waggons  are  attached.  Now  if  speed  is  reduced  to  0,  we 
have  24  -  2x*  =  0  .•.  x  =  12,  and  .-.  x-  =  144  =  number  of  cars 
required  to  completely  stop  the  train,  .-.  greatest  namber  it  can 
move  =  143 


Ex.jLix.1  ALGEi3RA.  115 


Exercise  LIX. 

1.  53,  ={126 +  (31  -  1)2]^J  =  ^^ — -  X  31  -  2883 

„      ,  n  ra      124  +  2n 

5.=  {126  +  («-l)2}  -  =  (126  +  2«-2)— = x«  =  7i(G2  +  n) 

2.  S.,2  =  {-400  +  (22  -  1)  X  12}"/=:  (  -400  +  252)11  ^~  1G28 

„      ,  n      (-40O  +  12n-12) 

^S',  ^  {  -  400  +  (11  -  I)  X  12}  -  =  ^- — xn=  Ji(6n  -  20G) 

3.  S,,  -{4 +  (17-1)^]'^  =  (4 +  24)7  =  238 

2ot  +  p  2in  +  p 

S2:*,  =  {4  +  (2/;i  +  p  -  i)-.}  — ^—  =  {•  +  (2/ft  +  ;0l'}— y— 

=  5(2//i+p)  +  £(2?/i  +  ;0- 

4.  ^.,-(^  +  (n-i)x-s}^  =  (3-^')¥=-¥x-:/=-^F=-29J 

5.  17">=  2  +  (l7-  1)3  =  2  +  16  X  3  =  2  +  48  =  50 
28"'  =  2  +  (28  -  1)3  =  2  +  27  X  3  =  2  +  81  =  83 

n">  =  2  +  («  -  1  )3  =  2  +  3/i  -  3  =  3?i  -  1 

6.  17"'  =  3  +  (17-l)x-5  =  3+16x-5  =  3-80  =  -77 
28'"  =  3  +  (28  -  1)  X  -  5  =  3  +  27  X  -  5  =  3  -  135  =  -  132 
n"-  =  3  +  (n  -  1)  X  -  5  =  3  -  5/1  +  5  =  8  -  5/1 

7.  17"'  i=  2J  +  (17  -  1)?  =  §  +  16  X  ^  =  ^  +  80.  =  Y^  =  13,3^- 
28">=:2J  +(28-  1)5  =  5  +27x?  =  s  +iJ5-=  21^4 

utu  =  2J  +  (/I  -  1)H  ?  +  {;;i  -  ^  =  ^  -  ^  +  ^«  =  s^  +471  =  ^^(5  +  27/) 

33  —  3 

8.  d  =  ~fzr\  -  ^"  ~  '^^  ;  hence  series  =  3  +  10^  +  18  +  25^  +  33 

-66-9 

9.  r/=  ~^g  _Y~  =  ~  V  -  ~  ^"^  i    bonce  series  =  9-0-21-36 

-51-66 

100  -  (  -  1)        100  +  I 
10.  d  =  ^~Z~[ ~  ' — 8 —  ~  "^^ '  =  ^-5  )  hence  scries  =  -  1 

+  1  l|  +  24i  +  36|  +  49J  +  62i  +  74^  +  87^  +  100 

11.  5,3  =  |2  +  (73-l)l}'/  =  (2  +  72)^/  =  37x  73-2701 

12.  ?»">  term  =  1  +  {n  -  1)2  =  1  +  2n  -  2  ^  2a  -  1 


116  KEY  TO  lEx.  .uv 

]  3.   .S,  =  {2  +  (n  -  1)2] .  ■  =  (2  +  2u  -  2)  -  =  2«  x  -  ■  =  Ji« 

<  /  / 

If.   S,^  {la  +  (/  -  I;2,/j  -  =  (2a  +  2a/  -  2a)  -  =  2at  x  -  -  ^  iiV- 

1  .">.  20"'  term  =  a  +  (20  -  ) )2(i  =  rt+19x2a  =  a  +  38a  =  30a 
t'''  term  =  a  +  (/  -  l)2rt  =  a  +  2af  -  2a  =  2af  -  a  =  fl(2/  -  1) 
1  (3.  Let  .r  -  3^/,  x  -  y,  x  +  >/,  x  +  3y  represent  the  numbers 
Then  (x  -  3'/)^  +  (x  +  3y)2  =  2x-  +  \9y^  =  200 
(X-    2/)'  +  ('^+    y)-=2x'«+    2y'  =  136 
r^Tey*  =    64 
Ilcncc  ^-  =  4  or  2/  =  +  2   .-.   2x''  =   136  -  2y«  =  136  -  8  =   128; 
or  X-  =  64;  or  x  =  +  8  .-.  tlie  series  is  ±14 +10  +  6  +  2 

1 7.  Let  X  -  3//,  X  -  y,  x  +  i/,  x  +  3//  represent  the  numbers ; 
tlieu  (X  -  3y)(x  -  7/)(x  +  y)(x  +  3^)  =  (x*  -  9y«)(x*  -  y^) 
^(x"-36)(x2-4)*  ;  x*-40x2+  144=  1680,  orx*-40x'=  153C; 
x»  -  40x''*  +  400  =  1936  .-.  x'^  -  20  =  +  44  .-.  x*  =  +  64,  or  -  24 
Ri'jectiug  the  latter  value,  we  have  x-  =  64,  or  x  =  +  8;  hence 
till.'  series  is+14+10±6i2 

18.  Let  X  -  2v,  X  -  y,  x,  x  +  y,  x  +  2y  rejiresent  the  numbers, 
tlu'u  x-2y  +  x-y  +  x  +  x  +  y  +  x  +  2y=5x=25.*.  x=6 
(X  -  2i/)(x  -  »/)(x  +  yXx  +  2y)x  =  (x«  -  4j-)(-r*  -  2/')-c 
=  5(25  -  47*)(25  -  y')  =  o(iy*  -  125y*  +  625)  =  945,  or 
47*  -  123y'-  =  -  436  ;  y*  -  ^  a  V+  Ci^)'  =  ^el^  -••  v'  "  ^j'  =  i  V 
\vlience,  y-  =  4  and  y  =  +  2.  lieuce  the  series  is  1,  3,  5,  7,  9  or 
9,  7,  5,  3,  1 

10.   .S  =  ("  +  0t  =  (60  +  Oy  =  61  X  30  =  1830,  i.  e.  since  tlie 

fi.  incipal  on  interest  is  $60  the  first  day,  and  only  $1  the  60th  day, 
the  whole  interest  is  equivalent  to  that  of  $1830  for  1  day.  Interest 
of  $60  for  360  days  =  33-60,  or  of  $1  for  360  days  =  $0-00,  or  of 
Si  fur  1  day  =  jg^  =  ^';  of  a  cent ;  hence  the  Inle""»^st  of  ^1  for 

•  Tlie  couimoii  difltTcncc  is  given  =  4  .•.  y  —  2. 


Ex.  LIS.]  ALGEBRA.  117 

1830  dMvy;,  i.  e.  of  $1830  for  1  day  =  -^-j^  cents,  and  siuce  this  is 

1830 

to  be  divii'.ed  into  CO  pavmcuts,  eucb  iivill  be  — r; — -—   =   ,-/„-  of 

a  cetif. 

n  n  n      -nin  A-  ]) 

20.  r,  =  {la H-  (n  - 1)./]  2  =  {^  +  («  "  D} y  =  («  +  i)y  -  -  -r- 

21.  Let  .9=  1-  +2^  4-3'2  + n^ 

Now  7!^  -  (n-  iy  =  3n^  -  3n  +  I 

(n-  l)«-('i-2)8=3(«-  1)«  -3(/i-  1)  +  1 

(n  -  2y  -  (n  -  3)3  =  3(?J  -  2Y  -  3(?t  -  2)  +  1     Hence  by  nd.Jiticn 

u*  =  *13(»  -  2)'  +  3(ra -l0*^^«^'}  -  {3(«  •^)  +  3(«  -  1)  +  3/)}  +  H 

»;'  =  3(1°  +  2^  + n")  -  3(1  +  2  + n)  -^-n 

F.ut  by  supposilloii,    l-   +  2-  +   ....   7i^  =  &',  and  it  hag   been 

n{n  +  1) 
shown    in  que.stion    20,    that    1    +2   +   3    ....    n ; 

2n(n  + 1) 
Therefore  n'  =  3  S  -  —x +  n 

.3n(n+l)            2n'  + 3?i(«  +  1)  -  2ra 
5,9  =  «3  + _._    _  „  .  ^ 

w(2n'+3'?  +  3-2)     «(2n2+3n+l)     7»(7i+ 1)(?7?  + 1) 


3S- 


2  2  2 

»z(n+ 1)(2«  +  1)       7i(77  +  l)(2n+ 1) 


2-3  6 

22.  S  =  [2a  +  (n-  l)''}^;    517  ^  [4  +  {n  -  l)^}"^;    10^4 

=  4?i  +  9n(n  -  1)  ;   1034  =  9/1^- 5?^ ;  324«^- 180n  +  25  =37224+ 25 
=  37249;  18n-  5  =  ±  193;  18/i  =  198  ;  n  =  11 

Note. — The  negative  value  is  inadmissible. 

23.  M-  Z  -  rf  +  /  -  2'/  =  3/  -  3/  =  9G,  or  /  -  rf  ^  32; 
i  _  3,/  +  /  _  4'Z  +  Z  -  5rf  +  i  -  erf  =  4/  -  18f/  =  86,  or  2/  -  9f/  =  43  ; 
2/-9i  =  43,  and  11-2(1=  G4  .-.  7rf  =  21,  andrf  =3,  whence/ =35; 


*JsOTE.— The  student  must  hore  read  n  as  number,  n  -  1,  one  lei's  than 
number,  &c.  Thus  taking  n  =  3,  then  »t-l  =  2,  n-2=:l;  if  n  be  taken 
as  4,  we  sliould  Lave  to  take  four  addends  as  above. 


118  KEY   TO  TEr.  lix. 

and  inverting  the  series  35,  32,  29,  26,  Ac  ,  wb  have  2,  5,  8,  11, 
14,  17,  20,  23,  26,  29,  32,  35 

24.  1  =  5,  and  I  -  2d  =  7  .-.  the  sixth  term  or  last  term  but 

7+5 
one  =  I  -  d  =  —- —  -  6,  and  the  series  is  found  by  reversing  the 

series  5,  G,  7,  8,  9,  10,  11 

n  n 

25.  S  =  bn  +  cn^  =  (6  +  ai)n  =  (2b  +  2cn)—  =  (26  +  2c  +  2cn -  2c) y 

=  {2(6  +  c)  +  (71  -  l)2r}—.    But  by  formula  5  =  {2a  +  (n  -  1)'/}^, 

whence  it  is  evident  that  a,  the  first  term  of  the  series  =  6  +  c ; 
d,  the  common  difference  =  2c.  Then  the  Z"'  term  =  a  +  (t  -  l)d 
=  b  +  c  +  (t  -  l)2c  =  b-c  +  2ct 

26.  The  (m  -  n)^^  term  =  a  +  (m  -  7i  -  l)d; 

the  (7/1  +  m)"i  term  =  n  +  ()n  +  n  -  l)d  .-.  the  sum  of  the  two 
terms  =  2a  +  (27/i  -  2)d  =  2{a  +  (//i  -  l)J\.  Also  the  m">  term 
=  a  +  (771  -  l)'i.     Therefore,  &c. 

27.  (p  +  7)"'term=a  +  (p  +  5-l)(/-77n  m~n 

>.•.  2qd  =  in-n  .-.d  -  —r — 
(p-qy^term  =  a  +  (p-q-\)d=nj  ^1 

m  —  n 
But  a  +  (q  -  l)d  +  pd  =  7n  .-.  a  +  (q  -l)d  =  in  -  pd  =  m  - p  X  —,- — ■ 

P 
=  111  —  (in  -  11)--- ;    but  a  +  (^  -   !)</  =  9""  term   .•.   9"*  term 
^? 
P 

28.  p^^  term   =    7  -  y   =    V   "  y  +   1    =   ¥  -  (P  -    1)1 

=  V  +  (P  ~  1)  ^  ~  i  i   ^"*  P"*  ^^rm  =  «  +  (p  -  l)i/,  wbcn-e  a  =  \3, 

n 
and  (/  =  -  J.     Then  sum  of  n  terras  =  5  =  {2a  +  (n  -  l)(f}— 

n  n  n         /27        n\  n 

=  {13  +  (a  -  1)  X  -  J}-   =  (13  -  --  +  i)y  =  K-  -  yJY 

n 
=  -(27-n) 


Ex.  Lix,  Lx.]  ALGSBRA.  119 

29.  Letx-y,  X  and  x  +  j/  =  the  numbers;  then  (x-i/)2  +  x(xJ-i/) 

=  x^  -  2xy  +  y^  +  x^  +  xy=  2x*  -  xy  +  2/^=10,  and  x^  +  (x  -  y)(x  +  y) 

=  x^  +  x^  -  y-  =  2x2  _  y-.'  _  j4      Subtracting  the  second  from  the 

2/_  2 
first,  2i/2  -  xy  =  2,  or  X  = .     Substituting  this  for  x  iu  the 

2(2y2  _   2)2 
equation   2x2  _  ^^2  _   ^4^    ^g   l,^yg   _ ^    =    14, 

or  7]/*  -  30y2  =  _  8,  196y*  -  840i/2  +  900  =  -  224  +  yoO  =  676  » 

14)/2-  30  =  ±  26  ;   14^^  =  53  or  4,  t/^  =  4  or  f  ;  rejecting  this  hitter 

8-2        6 
value  we  have  3/  =  ±  2.     Hence  x  =  — —  =  — —  -  ±  3,  and  the  three 

numbers  are  1,  3  and  5,  or  -5,-3  and  -  1 

30.  Let  X  -  3y,  X  -  y,  X  +  y  and  x  +  Sy  repn  sent  the  numbers  ; 
then  X  -  3y  +  X  -  y  +  X  +  ?/  +  X  +  3i/  =  4x  =  20  .•.  X  =  5  ; 

1111  4x»- 20x1/2 

X  -  3y  "*"  X  -  y  ^  x  +  y  ^  x  +  3y  ~  x*  -  lOx^  ^  gy*  =  li 
.-.  25(625  -  250y2  +  Sj/*)  =  24(500  -  100y2)  ;  or  9i/*  -  1 54i/2  =  -  145  ; 
324y*-5544y2+ 23716  =-  5220  +  23716  =  18496  .-.  18!/2-154 
=  +  136  ;  18i/2  =  290  or  18  .-.  y2  =  1  or  ^^^,  and  y  =  +  1  or  ±  JV145 
Rejecting  the  latter  value,  we  have  5  +  3,  5  +  1,  5  +  1  and  5  i  3  ; 
that  is  2,  4,  6  and  8  or  8,  6,  4  and  2  for  the  series. 


EXKRCISB  LX. 

\.  6"*  term  ^=  3  x  3^  =  3  x  243  =  729 ; 
i>(36  -  1)       3  x(729-  1) 
-'a  =  -T=m   =—^2 =1°»2 

2.  9"'  term  =  1  x  2"  =  1  x  256  =  256  ; 
1(28  .  1)     512  _  1 

S9--2-rT— — [-  =  511 

3.  7">  term  =  ?  X  26  =^x  64  =  -i?8  =  I8f; 
?(2'  -  1) 


120  l^EY  TO  [tx   r.i. 

4.  12"»   tonn   =    3x(-2)'i    =   3x-   2048   =   -  6141; 

3[(- 2/^-1}       3(4096-1) 
A  ,  J,  =  -     _^^1 = —^ =  -  409o 

5.  6">  term  =  4  x  (  -  ;)5  =  4  x  -  ^^^  =  -  V.V  =  "  12, /e  ; 

4  ^  4  *  4 

6.  8"'  term  =  30  x  (  -  i)'  ^  30  x  -  yi-g-  =  -  y^-  =  -  ii  ; 

3o[i  -ny\ 

^8   -  1   +  J  -      S4      -    '^64 

-   1  J  -  1  -    J 

«=     I  -(-?)         1       t         * 

2  -2 

7  7  7 

^-     'oc  l-(-J)  l  +  J  3  3 

64  64         64 

^    •      «         1  -(-i)        l  +  J         2  3  *^* 

11       o       _        m  0 "        _        Tun  IT        _  1  iTo  ?  _    g  g  j 

'  10         To 

76  7  6 

io       c        _     Q      .     Jllll-  -     "     X   T"*"^  -     1      .     _;r      _    or.7 

i.-J.     Oj^    ~    I  u   ■""    1  _     l_  ~   1>p    ■*■      9  •      ~  T5  ■*■    ysTo'  ~   35i) 
*       1 1.  u  loo 

Id        <?       -     9!i,2     ■    TTT?.n7rff  _      ,,.„  T.Tp.f.jVjJ  _    ,^^_^       3S_-91VT 

•^    ToJj  loir 

li'S"-  1)        3"  -   1 

15.  s,  =  '\zr  =  —Y"  -■  ^<'^  -  ^> 

iG.  s.  =  -'-— pn—  ■-  - — I = ':'{!  -  ( -  n"! 

SlCV-)^"  -  1}       2(32   -   1)  P2  62v'2  +  62 


G'2(l  +  V-) 

1Q  C        _     ^'J'       S !     _     i .     -  I 


Ex.  tx.j  ALOEBRA.  121 


19.  T  =  (j-j  =(','■)«=  5  .-.  series  -~  1  +  J  4- ^  +  ./^  +  Tf 

20.  r  =  (i2J-!)'~''  =  (65G1)*  =  3  .-.  series  =  2  +  6+18 
+  54  +  102  +  4fcG  +  1458  +  4374  +  13122 

21.  r  =   f  -     j  ^  (A  )•»  =  1  .-.  series  ^  P  +  3  +  1  +  J  +  !, 

22.  Let  X,  xy,  nj^  and  xi/  represent  the  four  numbers  ;    tlieu 

148  8S8        148       838 

.-.  1  +  ?/2r.  and  1  +  ^^=  .-.  =  .-.   148y  ^^  888 

a;  '         ay  x  xv 

148 
.-.  y  ^  6;    then  1  +  ?/' =  1  ■»-  36  ^  37  =  .-.  37j-  =  148  .-.  x  =  4, 

and  the  series  is  4,  24,  144  and  864 

23.  Let  r,  xy,  xy^  and  xi/  represent  the  n umbers  ;  then  x  +  xy 
~  15  (i),  and  xy' +  xy^  =  i/''(x  +  xy)  =  GO  (ii).  DividinGT  (ii)  by  (r) 
we  have  2^- =  4  .-.  y  =  +  2,  and  since  x(l  +  ?/)  -  15,  we  have 
X  =  K^,  or  -J-'\  =  5  or  -  15  ;  hence  (he  numljers  are  5,  10,  20  and 
40,  or-  15,  30,  -  60  and  120 

24.  Leta-y^,  ay  and  x  represent  the  nnmbor  of  dollars  they 
severally  had  ;  then  xy'^  =  x  +  135  (i),  and  xy^  +  xy  +  x  =  315  (ti) 

135 

.-.  xy  +  2x  =  180   (ill).     From  (i)  x  =     .^  _     ,   and  from  (in) 

180  135  180  3  4 

X  = .-.  -, r  =  — -  .-.  -5 ~  =  .-.  4'/^  -  3'/  -  10, 

2/  +  2       y,*  -  1       y  +  2       y*  -  1       y  +  2 

180  180 

whence  y  =  2  or  -  -^  ;  hence  x  = =  -ilc  =  45,  or  x  =  ^     , 

180 
=  ---  =  240  ;  hence  the  shares  were  $180,  $90  and  $45. 

Talcing  the  negative  value  as  above,  gives  us  x  —  ?240,  and  the  sha<e.s 
would  bo  S375,  -  $300  and  S240,  which  implies  that  tlie  Fecond  icceives 
S300  less  than  nothing  for  liis  share,  or  in  other  words,  instead  of  receiv- 
ing anything  lie  gives  S300  to  bo  divided  in  addition  to  the  S315  among 
the  other  two. 


122  KEY  TO  [Ki.  LS. 

25.  Let  X  =  the  first  number,  and  y  =  the  common  ratio  of  the 
1st  three  numbers  ;  then  the  numbers  are  x,  xy,  xy^,  xy'^  +  xy,  and 
xy^  +  2xy 

And  xy  +  ary^  +  (xy^  +  xy)  +  (xy^  +  2xy)  =  3xy'  +  Axy  =  40  (i) ; 

also  xy(xy^  +  2xy)  =  xh/^  +  2xhj^  =  64  (ii).     Multiplying  (i)  b^ 

xy  and  (ii)  by  3,  and  subtracting,  we  have  2xV  =  192  -  -lOjv 

.-.  x'l/  +  20xy  =  96  ;  whence  xy  =  4  or  -  24.     From  (i)  xy(3y  +  4) 

40 
=  40  .-.  Sy  +  4  =  —  =  V  =  10  .-.  3y  =  G,  and  y  =  2  .-.  x  -  2  : 

hence  the  numbers  are  2,  4,  8,  12  and  IG 

2G.  S  ^  a  +  (a  +  b)r  +  (a  +  2&)r*  + {a  ■{■  {n-  l)t('"-  ' 

Sr     =ar +  («  +  &)?-=+ {a  +  (n-2)ft}r"-i+[«  +  (n-l)M'"- 

5  -  Sr  =       a  +  6r  +  6r''  +  6r5  + fcr""-^  -  [a  +  (n  -  Oftlr""" 

ir(l  -7'*-») 
5(  1  -  r)  =  a  +         ^_^         -  {a  +  (7.  -  1  )t}r'' 

irCl-r^-i)       {a  +  (n-l)6jr" 
1  -  r  "^        (1  -  r)»  1  -r 

a-{a  +  (n- l)6jr»      6r(l -»"-») 

27.  (I)  «2  +  6^  +  c'  -  (a  -  ft  +  f)^  =  a^+  h'  +  r»  -  (a=  +  b'  4-  r-) 
+  2a6  +  26c  -  2ar  =  2ab  +  2bc  -  2ac  =  2ab  +  2hc  -  2'j-,  (siuce 
ac  =  h^)  =  2b(a  +  c  -  b).  Now  (a  +  c)^  -  i'  =  a*  +  2af  +  c-  -  6* 
=  a^  +  26^  +  c^-  b-  =  a^+  b'^+  c^  a,  positive  quantity  .•.  (a  +  e)-  >  6*, 
and  .-.  «  +  c  >  6,  and  .-.  a  +  c  -  6  is  a  positive  quantity,  p.n.l 
.-.  26(a  +  c  -  6)  is  positive,  .-.  u*  +  6^  +  c^-  (a  -  6  +  c)^  is  positive, 
.-.  a^  +  b^  +  c^>  (a-b  +  c)» 

(n)  (a  +  6  +  c  +  d)^  =  (a  +  /))»  +  (c  +  </)'  +  2(a  +  6)(c  +  <Jj  ;  but 

a       6        r        a  +b     b  +c     c  +d       (a  +  b)(c  +  d)       (A  +  cY 

b       c       d  '  '     b     ~     c     ~     d     '  '  bd  ~        <r 

bd 
.-.  (a  +  b)(c  +  </)  =  -.,(t  +  cy  =  (6  +  f )»  •.•  bd  =  c'.:(a  +  b  +  c  +  dy 

=  (a  +  by-  +  (c  +  J)^  +  2(6  +  0» 


Ex.  Lx,  Lii.]  ALGEBRA.  123 

28,  (i)  (p  +  gy^  term  =  ar^*^-'^  =  in 
(P  ~  y)""  term  =  ar^''^'^  =  rt 

.•.  a'r'^P-'  =  mn,  .-.  ar^'^  =  ^7/i»  =  p"' term 

(n)  Also  -TT^i  =  r2«  =  —  .-.  r  =  ( )    ,  and  —  =  { )  ^ 


the  (?"'  term  =  o;'"^  =  — — s —  =  -;r  =  m  x  — r  =  ?« 

-1  'j'f  yi*  ^i* 


2« 


29.  Let  X,  xy  and  x?/^  represent  the  numbers  ;  then  x  +  xi/  +  xy^ 
=  35,  and  xy  :  xy'^  -  x  ::  2  :  3  .-.  y  :  if  -  \  ::  2:3, 
Dr  3y  =21/^-2  .-.  2?/*  -  3y  =  2,  whence  y  =  2  or  -  J 
.-.  X  +  XT/  +  XJ/'''  =  X  +  2x  +  4x  =  7x  =  35  .-.  X  -  5  ;  or  x  +  xy  +  xy'' 
=  X  -  Jx  +  ix  =  |x  =  35  .•.  X  =  ^^'^  =  46j,  hence  the  numbers  are 
5,  10  and  20  ;  or  46?,  -  23J  and  1 1§ 

30.  Let x,x?/ and  xi/^ represent  the  digits;  then  100x+  10x7/  + xy^ 

:  the  number,   and  x  +  xy  +  xy^  =  sura  of  its  digits ;    then 

lOOx   +    lOxy   +   xy^    :    x   +   xy    +  xy^    ::     124    :    7,    that    is 

100  +  10y  +  y2 
100  +  lOy  +  y^  :  1  +  y  +  y^  ::  124  :  7,  whence  — r— — ^■/~- 

124  99  +  9y  117  11  +  y  13 

=  -„-    .-.    Art.    106,    7—- — ■ — -—2  =  -„  ,  or  7 — , — 5  =  -=r 

7  'l+y  +  y-*        7'         l+y  +  y*        7 

.-.  77  +  7y  =  13  +  13y  +  ISy^,  or  ISy^  +  Gy  ^  64  ;    whence  y  =  2. 

Also  lOOx  +  lOxy  +  xy^+  594  =  100xy2+-  lOxy  +  x  ;  or  99x  -  99xy' 

=  -  594  ;  or  X  -  xif  =  -  6  j  or  x  -  4x  =  -  G ;  or  -  3x  =  -  6  .-.  x  =  2, 

hence  lOOx  +  lOxy  +  xy^  =  248,  the  number  required. 


Exercise  LXI. 

1.  (i)  ^.S.  =  7,  5,  3  ;  hence  d  =  -  2  ,-.  13,  11,  9,  7,  5,  3,  1, 
-1,-3  inverted,  give  H.S.  -,V,  Vf,  i,  h  h  J,  1,  -  1,  -  i 

(II)  J.S.  =  18,  14,  10 ;  hence  d  =  -  4  ,-.  30,  26,  22,  18,  14,  10, 
6,  2,  -  2  inverted,  give  H.S.  ^^^,  -/„.,  ^V,  r,;  ii,  fo",  i,  i,  -  i 


124  KEY  TO  [Ei.  LXi. 

(in)  .';.."?.  =  2,  4,  G;  lience  d  =  2  and  -4,-2,  0,  2,  4,  6,  P,  n, 
12  luverte'l,  give  H.S.  -  J,  -  J,  oc,  i,  i,  ^^  i,  Vui  A 

(IV)  A.S.  =  i\-,  ^2f,  |i  ;  heace  cZ  :=  A  and  -  3i,  _  if^  _  ./,,  y,, 
1^..-,  li,  ll,  ?i  and  ti  inverted,  give  i/.S.  -  a,  -  ij,  -  2,  14, 
n,  If,  i:^,  if  and  it 

(V)  J.S.  =  -^S  *,  -  ?  ;  hence  d  =  -laud\',  V,  V,  V,  S  "  •. 
-  V  and  -  y  inverted,  give  H.S.  =  /;,  j,  J-,  /„  li,  -  Ij,  -  i, 


(vi)  ^.5?.  ^  -  2,  0,  +  2  ;  hence  '/  =  2  ;  then  -8,-0,-4,-2,  0, 
2,  4,  C,  8  inverted,  give  U.S.  ^  -  J,  _  |,  _  i,  -  j,  cc,  J,  i,  I,  i 

2.  (i)  Insert  3  J),  means,  T)etween  i   and   i-     Ikre  d  -  \r^ . 
=     4"  =  -  iV ;  hPnc*  -^^  series  ^  i,  Jl,  {1,  v,-,  -/^  =  i,  ii,  yV,  i,  i, 

(II)  Insert  3  J.  means,  between  J  and  \.     Here  d  -  ^  _ 

=  ._^n_  =  _  J. ;  hence  .?.S.  =  4 J,  f;l,  Ig,  }i  and  ig,  and   invert- 
ting  these  we  have  /T.S.  =  5,  5^^^,  ^%,  C/,-  and  7 

(III)  Insert  3  ..i.  means,  between  -i^,-  and  \.     Here  d  -  -  • 


5-  1 

=    4-  =  A;  lient^e  Jl.S.  =  ./^t  fi,  "la'i  -/it  ii.  ""^^  inverting  tbes 
we  have  7/.5.  =  11,  6?,  4?,33,  3  , 

"if  ~  I 
(iv)  Insert  3  ^.  means,  between  ^  and  J^.     Uerc  d  =  —    _   • 

_  .ae. 

inverting  these,  we  have  H.S.  =  2^+2  -i,«\,  +  2  iVt  +  2  |?f  +  3| 
(v)  Insert  3  .d.  means,  beiween  J  and  -  ?•     Here  rf  =     ,  _  .■ 


_  d 


--  ^  -  3  ;  hence  J.S.  =  J,  -  g,  -  J,  -  V,  -  V,  ar.d  inverting 
these,  we  have  H.S.  =  6,-  2,-  ^,  -  ,<>,,  -  I 

3.  Corresponding  ^I.S.  =  I,  1,  f.  Hcuco  </ =  2;  S""  terra  ol 
^.S.  =  J  +  (5  -  1)J  =  2  +  '4*  =  V ;    hence  5"*  term  of  H.S.  =  1*^- 


£x.  Lxi  )  ALGEBRA.  125 

11'    u-nu  ui'^i.S.  -  i  +  (^li-  l)j=  I  +  ^'^i  =  ■>/;  hunce  ll^^  lei 

s 

5       *  ~      5 


m 


3«  3n-l 

i.f  //.S'.  =  /,- ;  7t'i'  term  of  ^.S.  -  I  +  («  -  1)^  =  ^  +  y  -  ^  = 


5 

hence  «"■  term  of  U.S.  =  „ 7 

3/t  —  1 

4.  Of  corresponding  J. S.  -^^^-,  f^ ,  ^ij  the  6«>  term  ^  f  j  +  (G  -  1)  ,i,f 
=  Vj  +  ^  ^  ^  5    10'Merm  =  ^^  +  (10-l),V=:YV  +  -.^=l!.  ^^d 

71  2   +   7?. 

7J"'  term  =  ,'3  +  (n  -  1)  ,'3  =  i^^  +  13  ~  "i^s  =  "73-   ■••  required  G'", 

13 

lO'b  and  ;(">  terms  of  H.S.  =  1| ;   IV2-  and  ^^-^^ 

5.  Of    the    corresponding    ^l.S.     10,    12,    14,    the    4'"    term 

=■   10  +  (4  -  1)2  =  16,  and  the  8"'  term  =  10  +  (8  -  1)2  =  24 

.-.  tlie  4"'  and  8">  term  of  the  H.S.  =  -i^  and  ./^ 

1  —  }       ^ 
o.  Insert  2  •■?.  means,  between  i  and  1.     Here  d  -        .  =  -^-  =  i 

hence  .'?.5.   =   J   +   i   +   i  4-  ^,  and   inverting  these,  we  have 
7^  <S'.  =  4  +  2  +  I3  +  1  ;  hence  unknown  terms  are  2  and  Ij 

1         1 

"1.  Of     the     corresponding     ^.S.     — ,     -r-,     the     S""     term 

1  /  1        1  \        1  7         7         7        6        7a  -  6^* 

--+(8-l)-T--—      =—  +  - =---—=    —    —  ■ 

u       ^  ^  \o        a  J        a         b         a         b        a  ab      ' 

ab  1  /  1        1 

hence  8''»  term  of  H.S.  =  iz -,7  ;  /jthterm  -  —  +  (n-l)l  -7 

"a  -bo'  a      ^  \o       a 

1  /I         1  \        1         1        2        1        n        n       2-n      n-  I 

~    a         [b       a  J       b        a  ~  a       b        b        a  a  b 

h{2  -  n)  ■¥a{n-  1)  ab 

=  , .-.  /t'li  term  of  H.S.  =  -r-/x r— — 7 — -. . 

ab  6(2  -  7t)  +  o(/i  -  1) 


2ab  m^ -  n 

8.   H.M.  =  — 7-T  = 


a+  6  ~       1  1 

771  +  71  771  —  71  m^  -  7t^ 

9.  jJ.3I.  =  i(a  +  6)  =  ^(4  +  9)  =   ^^  =  61  ;    G.3I.  ^  ^jTb --  V4  x  9 

—  2ab         2x4x9 

V3G  .  6  i  H.M.  =  ^y^  =  -  -^—  =U^  5A 


126  KEY   TO  [Ex.  LXJ. 

10.  j^.M.  =  i(<i  +  i\)  =  i  of  lOj  =  5,ij;    G.M.  =  -^6x4^  ^  ^JH 

2  X  6  X  4i        50 

=  5  •  H  M  - =  3ca  =  444 

'  ""■     •  6-1-41  lOi        «>'        *15T 


11.  a:c..u-b:b  —  c.-.ab-ac  =  ac-bc  .-.  2ac  =  ab  +  bc-  b(a  +  c) 


2ac  /  2ac  , 

a  -f  c  V "-t-  c 


/  2ac  \  ^  r  4</c    ") 

\a+  cj        ^         ■'  ^  (a  +  c)M 

=  (a  -  f)^  +  2ac^  —— — r_,L  :^  a  positive  quantity  if  a  and  c  have 
like  signs  .-.  a^  +  c^>  26* 

12.  b  =  l(a  +  c),  and  7»6  =  ^ac;    substituting  the  value  of  b, 

VI  —        m^ 

we  liavc  ~(a  +  c)  -  ^jac  .-.  "X~("  "^  ^^^  ~  "^  ■'•  "'■^i^  +  0"  -  ^''^j 

4ac 
and  dividing  each  by  a  4  c  we  get  vi'(a  +  c)  -  — — ,  but  a  +  c  -  2l> 

4ac  2ac 

.-.  2bin^~  -— — ,  or6m^=  — ; —  ;  hence  Art.  261,  bin''  is  the  H.M. 

between  a  and  c  .-.  a,  bni^  and  c  arc  in  H.  Prog. 

13.  Let  rt,  b  and  c  be  any  three  quantities  in  H.  Prog.,  and 

let  X  be  the  quantity  which,  when  subtracted  from  each,  leaves 

remainders  in   G.P. ;    then  (a  -  x)(c  -  x)  =  (6  -  i)*,  that  is 

uc  -  ex  —  ax  +  x'^  =  b^  -  2bx  +  x-  .-.  26x  -  ex  -  ax  =  b-  -  uc 

b'^  -  ac 
.-.   X   =    2b  -  c  -  a  '     ^^^  since   a,   b   and   c   are    in    H.P., 

a  :  c  ::  a  —  b  :  b  -  c  .-.  ab  -  ac  =  ac  -  be;   ab  =   2ac  —  be 
Substitute  this  for  c  in  the  above  value  of  x. 


•  •  "-  ~  2a  -  6- 

a-b  2tib^  -  6»  -  a-'j 

^'~  2a  -  b  2a-  6 

and  we  have  x 


ab            4ub  -  2u-  -  26^  H-  ah  -  <ib 
-^  -  '^  -  27^b       2736 


2n6»_    b^-a^b       b(2ab  -  b'*  -  a')        b 

4ub  -  26'  -  2a*  =  2(1^163  i^a^  -  y  =  1  of  middle  term. 


nx.  Lxi,  Lxii.]  ALGEBRA.  127 

14.  J.M.  =■  i(a  +  5),  and  G.M.  =  V^  •■•  K"  +  ^)  +  V«^  =  !<'  (0 
and  i(ff  +  6")  -  ^/ab  =  4  (ii).  From  (i)  a  +  2Vi^  f  6  =  32 
.-.  \'a  +  ^/b  =  +  4V2.  From  (ii)  a- 2^06 +  6  =  8  .-.^/a-^Jb  =  ±2-^2 
.-.  2Va  =  i  6^/2,  whence  a  =  18 ;    and  2^b  =  ±  2V2,  whence  6  =  2 

2nh  46 

15.  .i.M.  =  i(a+  6),  and  H.M.  =  ^-^  .-.  J (2  +  6)  =  '/  x  ^-^ 

646 
^  r.--;-^  ■••  28+  146  +  146  +  762=  j286  .-.  76^-  IOO6  =  -  28,   or 
14  +  76  ' 

2500-  196 
t2_i^06  =  _4j  J2_xa(i5  +  t£§c^ -_ =  zjg^   •••    6  -  ^a 

=  +  V  •■•  6  =  14  or  f 

2((6  2a6 

16.  a  +  b  =  30,  and  -—-^  =  13J   .-.  -5^-  =  13J   .-.  2«6  =  400 

d^+  2ab  +  62=  900,  and  4a6  =  800  .-.  a^-  2«6  +  6-  =  100,  or  a  -  6  =  +  10 
a  +  6  =  30,  and  a  -  6  =  +  10  .-.  2«  =  40,  or  20  ;  a  =  20  or  10 
26  =  20  or  40  .-.  6  =  10  or  20  .-.  the  numbers  are  20  and  10 

17.  a  -  b  =  16J,  and  'Jab  =  9,  since  the  G.M.  between  the  J. 
and  H.3I.  of  a  and  6  =  G.M.  between  a  and  6,  (see  Art.  261) 
Then  a^- 2ab +b^=  m^,  and  ab  -  81  .-.  4«6  =  324;  0^+206 +  6=2 
•=  <i^^  +  324=  ^Iga  ,..  a  +  b  =  Y-  Hence  a  -  6  =  \^,  and  a  +  6 
=  2^^  .-.  2a  =  V  •••  a  =  V  =  20i ;  26  =  \2  =  8  .-.  6  =  4 


EXBRCISB  LXII. 

1.  Fg  =  l-2-3-4-5-6=  720 

2.  (i)    V^  =  8-7-6-5  =   1680;    (11)    Fg  =  8-7-6-5-4-3  =   20160; 
Fg  =  l-2-3-4-5-6-7«8  =  40320 

3.  We  are  to  find  the  permutations  of  13  letters  of  which  5  are 
a's,  4  are  6's,  and  3  are  c'a 

\^  1-2-3-4-5-6-7-8-0-lOn  1213 

Then  N=  7 — r-r-  =  -     .  „  , ,  .,  ,^  . ,  .,  ,^      =  360360 

[2*  |9  k         1-2-3-4-5  X  l-2-3'4  x  1  2-3 


128  KEY   TO  [Ex.  Lxii. 

4.  F,  2  -  1-2-3-4-5O-7-8-9101M2  -  whole  number  of  changes 

1-2-3-4-5-6-7-8-9-10-1112 

,  rTTTrA =  49896   =  number    of  days   required 

Id'dU'  10 

-  136  years  222  days. 

5.  F,  =  n(n  -  l)(n  -  2)(n  -  3)(n  -  4),  and  V^  =  n(n  -  l)(/i  -  2) 
Thea  n(n  -  \)(n  -  2)(n  -  3)(»  -  4)  =  6  x  n(ra  -  l)(n  -  2) 
.-.  (,i  -  3){n  -  4)  =  6  ;  that  is  «-  -  7/j  -  -  6,  whence  n  =  6 

6.  F, f,    =    l-2-3-4-5-6'7-8-9-10     =    whole    number    of    days 

.-.  1-2-3-4-5-6-8-910  -    518400   =    number  of  weeks  be  had  to 

board  them,  and  since  board  is  worth  $5  per  week  for  one  pcr- 

Bon,   it  is  worth  $50  per  week  for  10.     Hence  total  value  uf 

board  =  $50  x  518400  =  $25920000;   and  $25920000  -  $5000 

=  $25915000  =  loss  when  the  $5000  is  not  paid  till  the  e.xpira- 

tion  of  the  term  of  the  board.    Andamountof  $5000at6  per  cent. 

3G28800 
for  — 3,.y, —   years,    i.e.    for   9935-112    years   =    5000(1    +   rl) 

=     5000(1     +     596-112)    =    5000    x    5:)7-112    =    $2985533-60. 

Hence  his  loss  when  the  $5000  is  paid  at  once,  and  put  out 

at   interest   until    the   expiratiou    of    the    term    =    $25920000 

$2985533-60  =  $22934466-40 

7.  F,i=  15- 14-13 (15  -n  +  2)(15  -/i+  1) 

and  F„.i  =  1514  13 {15-(«-  1)  +  1};  then 

15- 14- 13 (15-n+2)(15-;H-l)=151413    ••-(15-nr2)>:  li> 

.-.  cancelling  same  factors  of  both  sides,  we  have  1 J  —  »*  4-  I  =  10 
.-.  rt  =  6 

8.  (i)  Pernmtiitions  of  14  letters  wheri-of  2  arc  o's,  3  are  u's. 

If  1-2-3-4-5-6-7-8-9  10-n-l2-].^-14 

and  two  are  Vs   =    , — j — |—  = t-z — nr^'T'.-r 

[P    \^    \£  1"2  X  1-2-3  >  1-2 

=  3632428800 

(ii)  Permutations  of  12  letters  whereof  5  are  i's 
If         1-2-3-4-5-6-7-8-9-10-1M2 


[p    ~  1-23-4-5 


=  3991680 


£x.  Lxn,  Lxiii.]  ALGEBRA.  129 

(in)  Permutations  of  8  letters  whereof  4  are  o's 

\n         1-2-3-4-5-6  T8 
\p  1"2'3'4 

(iv)  Permutations  of  13  letters  ivhereof  3  are  o's  and  3  are  Jrt's 
\n            l-2-3-4-r)-6-'7-8-9-10-lM2-13 
=  17T7  -  123  X  1-23 =  ^^2^*^2800 

9.  (i)  Permutations  of  7  letters  of  which  2  are  a's 
l«         1-2-3-4-5-6-7 
=  -^  =  1^2 =  2520 

(n)  Permutations  of  13  letters  whereof  2  are  o's,  2  arc  n's, 

\!}          1-2-3  4-5-6-7-8-9-10-1M2-13 
:and  2  are  /'s  -  ^^= IT^<T2Vr2 =  ^l^^^^^m 

(ni)  Permutations  of  7  letters  whereof  2  are  fs  and  3  are  o's 

I?  l-2-3-4-r)-6-7 

jiL-  = =  420 

\p    \q  1-2  X  1-2-3 

/5m  -  2\  /5n  -  4\         290/1  /2n  -  3\  /2n  -  6 

w5(5ra-  2)(5n-4)  =  ^J^(2n  -  3)(2n  -  6)  ;  or  135(2571^- 30«  +  8) 
=  2320(2n*  -  9n  +  9)  ;  or  25371^  -  33667t  =  -  3960,  whence  n  =  U 


Exercise  LXIII. 

-   10-9-8  10-9-8-7-6 

1.  0)    C,   =    -—    =    120  ;     (II)    C,   =   -^^:2¥?5   =   ^53  ; 

10-9 
(in)  C,  =  C2  =  —^  =  45 

15-14-13-12-11 

2.  (0  ^5= -^7273:^:5—  =  3003; 

15-14-13-12-11-10-9 

151413 
(MI)  C,,^  C,=     j.2.3   -  =  455 


loO  KEY  TO  [Ex.  Liiii. 

12-1M0-9-8 
^-    '^'  "      1-2-3-4-5      "   '^^^ 

4.  Whole  number  of  combinatio-ns  of  2n  things,  1,  2,  3,  4, 
kc,  ...  2n  together  =  2^"  -  1  ;  similarly  the  whole  number  of 
combinations  of  n  things,  1,  2,  3,  4,  kc,  . .  n  together  =  2"  -  1. 

2^-1  2^-1 

Then  2^"  -  1  =  (2"  -  1)  x  513  .-.   gn^TT  =  513 ;    or  since  -gn^ry- 

=  2"  +  I,  we  have  2"  +  1  =  513  .-.  2"  =  512,  and  .-.  by  inspection 

n  =  8 

36-35-34-33-32 

5.  (i)  C5  =  — 1  o  o~^  e —  =  439824  =  No.  of  different  selections 

(11)  Taking  away  one  man  from  the  36  there  remain  35,  and 

35-34-32-31 
these  combined  together,  4  and  4  give  — T.h.-i-i —  ~  52360  com- 
binations to  each  of  which  the  reserved  man  must  be  attached. 

6.  Number   of  combinations   of  21    consonants,    4  together 

21-20-19-18 
=   — 1  o  o  < —  =5985;  also  number  of  combinations  of  5  vowels, 

5-4-3 
3  together  =  "^T^  -  ^O-     Hence  there  can  be  formed  5985  x  10 

=  59850  different  sets  of  seven  letters,  each  set  containing  four 
consonants  and  three  vowels.  But  each  of  these  59850  set5 
can  be  permutated,  l-2-34-5"6'7  =  5040  ways,  each  forming  a 
different  word  .•.  the  required  number  of  words  =  59850  x  5040 
=  301644000. 

7.  The  different  arrangements  of  9  of  the  persons  while  the 
tenth  remains  fixed  =  9-8-7-6-5-4-3-21  =  362880  =  whole  number 
of  different  arrangements  of  the  ten  persons,  so  that  no  one  has 
the  same  neighbours  in  any  two  cases.  But  one  half  of  these 
arrangements  will  be  similar  to  the  other  half  if  the  position  of 
neighbours  on  the  right  and  left  hand  sides  be  not  regarded 
as  making  a  difference.  So  that  if  .4  is  said  to  have  the  same 
neighbours  in  the  arrangement  B.tC  that  he  has  in  the  arrange- 
ment CJB^  then  the  correct  answer  will  be  \  of  362880  =  181440 


Ex.  i.xni.]  ALGEBRA.  131 

n(n  -  l)(rt  -  2)(7J.  -  3) 
8.  n(rt  -  1)(«-  2)  :  -^^ vVfl '■'■  ^  '■   ^  •■•  caucel- 

n-  3 

ling,  — -.—  =  1,  whence  7i  =  7 

y.  «(«-!)(«- 2)  ....  (7i-7>+l)=  10«(«-l)(/i-2)  ....  (/i-7)  +  2); 

or  dividing  each  by  «(ra  -   [)(«  -  2) {a  -  p  +  2),  we  get 

n-p+l  =  \0.-.n-p-9  (i).     Again 

«(/',-l)(ra-2)....  (n-p+l)     n(n-lXn-2) . .  .  (n-  p  +  2)  ^  ^ 

\P  \P-  1 

or  multiplying  each  side  by  |;>  -  1  we  have 

-^ '-^ '— ^ '- =J«(/i-l)(/i-2)  ....  (n-p  +  2), 

and  dividing  each  side  by  n(n  -  l)(n  -  2) («  -  p  +  2),  we 

n-p+l 
liave =  5  ;  or  3n  -  3p  +  3  ^  5p  .-.  3n  -  Sp  =  -  3   (ii). 

Now  multiplying  (i)  by  3,  and  subtracting  from  (lO,  we  havf. 
5p  =  30  .-.  /)  =  6,  and  similarly  n  =  15 

10.  1-2-3 (n  -  I)  =  \n  -  1,   or  ^  | ?i  -  1   according  as  BJC 

and  CyiB  are  regarded  as  different  or  the  same  arrangement. 

10-9-8-7  6 

11.  Number  of  5  flag  signals  with  10  flags  =    1.0.0.4.5  ~  ^^"^ ' 

10-9-8-7 
number  of  signals  with  four  flags  out  of  10  =         ,^  „     =  210; 

10-9-8  10-9 

number  with  3  flags  =  -t7^^=120;  number  with  2  flags  =  —2  =  '*^ 

and  number  with  one  flag  =  10.     Therefore  whole  number  of 
signals  =  10  f  45  +  120  +  210  +  252  =  637 

12.  There  arc  in  all  nine  coins  and  they  may  be  combined, 
any  number  together,  to  make  a  sum ;  then  the  combinations  of 
9  things  1,  2,  3, 9  together  =  2"-  1  =  2''-l  =  512-1  =  511 


132  KEY    TO  [Ex.  Lx.v. 

ExEiiciSE  LXIV. 


1.    (1    +  x)-3  =    1  -  — X   +   -J— X*  -    T-^X*   +    \   ..  ^   .X*  -    &C. 


3-4  3-45  3-4-5-6 

— I*  —    x*  + 

•2  1-2  3  1-2-3  4 

=  1  -  3x  +  6x2  _  ioj;3  +  15J.4  _  &c. 

2  2-3  2-3-4  2-3-4-5 

2.  (1  +  x)-^  =   1  -  -x  +    ~2^-^  -  -p^-gx'  +  p^T^x*  -   Ac. 

-  1  -  2x  +  3x^  -  4x^  +  5x*  -  &c. 

1  1-2  1-2-3  1-2-3-4 

3.  (1  -2x)-i.  1  +-(2x)+  -  (2x)=+  1.2-3<^-''>'"^  l-2-3-4<^-''^*'^  *^- 
=>  1  +  2r  +  4x2  +  gj.3  +  jgj.4  +  ^c. 

5  5-6  5-6-7  5-6-7-8 

4.  (1  -  Jx)-  «  =  1  +  y  (Jx)  4-  -(ix)2  +  jTr3(Jx)='+  ^..^T^^CIJ^)* 

+  &C.   =    1   +  |X  +  Y^^  +  ^8^^^  +  ^t^*  +  &C. 

2  23  2-3-4  2-3-4-5 

5.  (l  +  3x)-^=l-y(3x)  +  — (3x)2-^2V3x)'+  1.234(3-^)* 

-  &c.  =  1  -  6x  +  27x2  _  108x3  +  405x*  -  &c. 

5  5-6  5C-7  5-6-7-8 

6.  (1  -  2x)-«  =  1  +-^(2x)+  — (2x)2+  i.-2T-3(2^)'+  1:^374(2^)* 

+  &c.  =  1  +  10x  +  60x2+  280x3+  ii20x*  +  <!tc. 

4  4-5  4-5-6  4-5-6-7 

7.  (1  -  X)-  =  1  +  yx  +  -x2  +  ^..^.3x3  +  ^,2,3-x*  +    &c. 

=  1  +  4x  +  10x2  4.  20.c3  +  35x*  +  &c. 

8.  (1  -  4x)i  =  1  -  --(4x)  +  -~^W^  -  -^.3.8—  W 

+  '^^-^-2¥4^^^(^-^>*  -  &C.  .  1  -  2x  -  2x2-  4x»  -  lOx-'  -  ,^,. 

1  2  2-5  2-5-8  2-5-8-11 

9.  (l+x)   '-l--3-^+i:T-9^'-r2T^:^^+r2.3-T4T8i^''-«c. 

=  I  -  3x  +  *x»  -  f  Qx3  +  \\%x*  -  &c. 

1  4  4(-l)         ,      4(-l)(-6) 

10.  (1  -  Jx)  •  =  1  -  y (Jx)  -  yy^  (Jx)2  -       1.2.3.125      (*'^ 

4(-l)(-6)(-ll) 
+  1-2^3^4^5— (i^>-**^- 


Ex.  Lxiv]  ALGERKA.  133 

I  1  U-2)  ,      U-2)(-5) 

11.  (i  +  ?x)*  =  ^  +  '^(U)  +  -jT^G^)'  +  —l:^:^^-'(i'^y 

1(  -  2)(  -  5;(  -8) 

■*"  1^2^"^! ^"^""^  "^  *^- 

=  1  +  -.(.•  -  -i^x-  +  -.J^-x^  -  rlth^^*  +  &c. 

i  4  4-9  4-914  4-9-14-19 

1  2.   (1  -  r)     -  --  1  +  -X  +  -^-as^'  +  r2~3-125^'  +  r2^Tt325-^ 

-1-  &c.  =  1  +  ix  +  i"x2  4-  -f^x»  +  3^fx4  +  &c. 

13.  («-x2)-3  =  |,;(i  _a-ix2)}-3  =  a-*(l  -  a'^x^)-'^ 

3-4  3-4-5  3-4-5-6 

=  a-^{l  +  3-»-ix2  +  Ga-^x*  +  lOfj-^^x"^  +  15a-*x«  +  &c.} 
=  «■  *+  3a -■'x^  +  6ff-*x*+  10a- 6x6  +  15a- '^x*  +  &c. 

14.  (a^  +  x^)-!  =  {a==(l  +a-2x3)}-i  =  «-2(l  -a-^x^)-! 

I  1-2  1-2-3  1-2-3-4 

=  a  2{i -  y(o  V)  +  —  (a-^xO^-  i:y.3(«'-^'/+  f:2r3::i(«''-''/  +  -S'c.l 

=  a-2(l  -a-^x^  +  a-4x«  -a-6x''  +a-''xi^-  &c.) 
=  a-=*-a-4x''  +  a-''x'^  -a-'x?  +a-i"xi^-  &c. 

15.  (a^   -  x^)'^  =:   {a^(l  -n'^x*)  } '^  =  a -i(l  -  a"  ^x' )"' 
,(  2/    .1    IX  2-3/    .1    i\2  2-3-4/    .J    INS 

2-3-4-5/     .1    i\*  ) 


1  +  2a"^x'  +3(Z-ix'  +  4a"^x+  Sa'^x^  + 


_  3 


-3     1.  ■!  &  A 

«-i  +  2a    -x'^  +  3a--x'  +  4a    ^x  +  5a-^x'  +  &c. 

IC.   («*-x=')"*  =    {«^(l -a-*x3)}^  =  J{l-a   *x«)^ 
a  2  2(  -  1)  2(  -  1)(  -  4) 

2(-l)(-4)(-7)  3,^      ,     , 

+  — T2^-4-^l ("    ''^•y-&c.} 

=  a'{l- 3a-<x^  +  J,a-8x6  --^i,-a-i2x«  -  ^'^^a-iSxi^  -  &c.} 
=  a^  -  §a    ^x'  -  ia     3  ^^.e  _  A.a     3  x"'  -  jj^a     •»  x^^  _  &c. 


134  KEY  TO  [Ex.  Lxrc 

17.  (a*  +  x-^)-*  =  {a^(l  +  a-^x-^)\-*  =  a'^Xl  +  a'^x-^)"* 

4  4-5  ,  4-5-6         ,       ,  3 

=  a-i2{l    -  y(a-3x-0    +    — (a-'x-O'    -    iVa-sf"  * '^  ") 

4-5-6-7  . 

r:  a -12(1  _4a-3x-2+  10a- fix-*- 20a- s"!- 6  +  35a-i2x«  -  &c.>  | 
=  «-i^-  4a-i'^x-^+  IOa-i»x-*-  20a-2ix-6  +  35<t-2*x-«  -  &c  ' 

18.(a'^-xO'*={a^(l-a'*x-^)}'*=a"''«{l-a'*x  *}  ^ 
.J.,  1  -   I  1-4  .  a  1-4-7  .  3 

1-4-7  10  .t 

+  r2-y4¥i(«^>  '  +  *^{ 

=r  a ■  "'^'^{1  +  i((zx) '^+1  (ax) ■  ^  +  i4(ax) ' ^  +  ^VsC^-^) ' *  +  &c.| 

=  a  1*^+  Ja  ^«x  «  +  5tt  ^»r  *  +  J|a  'x  *  +  ^35^   i»x  »  +  &c. 

19.  (a^/n   -  x^)"        =      {ahn(l   -   a  "  ^/n  "  ^x' )  }  '* 


a    '/ft 


a~%ir^  { 1  +  -l  -.^— j    + 
'  3  \u'in/ 

2-5-811   fx^ 

+ 1-2^3^81  Vii^y  +*^-} 

=  n   'm  '{■-  +3 


2-5  /x^  Y  2-5-8    /x^  Y 

r2-9Va^/      "*"    l-2-3-27Va^/ 


^x*\     5/    X    \     40/  X*  \     110/  x'  \ 
\^^J ■*■  9  (^^*^  J  ■*■  81  [pHiy  ■*"  243  (^i«^  j  ■*■  **^-f 

=  a    *m    '   +    5    a"  ^  m    ^x'    +   §a     ^  m    ^x  +  I'Ja     •*  m     ^  x* 

m  "  •*  x^  +  &c. 


V    --^^^ 


20.  (a  +  x-')»    =   {(1(1    +   a-^x-')}^    =    a^(l    +  a-^x-*)* 

?f,        VM         2(   -   3)  /  1  y       2(  -  3)(  -  8)/  1  Y 
~  ^^     '*'  5\axy   '^      1-2-25     [ax*)    "^        1-2-3125       [ax-> J 
3)(-8)(-13)/  1  V      ..    . 


.^-^ 


fix.  LXiv,  Lxv.]  AtGEBRA.  13^ 


2  2/   1  \ 

•        5  \ax^J 


■^  25(^^7  ^  U5\a^x^j  ~  625[a*x^'J  ^"^'^  ^ 


2  .3  8  .11  _1«,., 

21.  (a  -   6x)  ■  ^    =   {a(l   -  a-^ix)}  '  ^    =  a  "  ^(1  -  a'^x)  '  ^ 

.1         l/bx\      1-3    /bx\'     1-3-5    /6x\»      1-3-5-7    /bx\*    ,     , 

=  ^  '{^  +  2(^>)+r¥iU)  +1-2-3-81t;  +r^3^4T6U;  +*^! 

I  1  /bx\        3  /ft^x^N         5  /6«x3\         35  /b*x*\        ,     ^ 

=  ^  '^^  +  2U;  +  sl^O  "*"  ^^ra^-j  ^  r28(^;  +  ^^-j 

=  a'  ^  +  ia'^bx  +  ia'h'^''  +  T^^a'Uv  +  -■'.l^-a'^b*x*  &c. 


Exercise  LXV". 

n(n  +  l)(n  +  2) (n  +  r-1) 

1.  Gen.  term  of  (1  -  x)-»  = r: x' 

3.4.5 (2  +r) 

=  j x*" 

(11)  Since  general  term  =  the  (r  +  l)tli  term  =  Gth  term  .-.  ?•  =  5 

3-4-5-6-7 

Hence  Gth  term  =  -nrTT^K^'  =  21x5 
l-2"3'4'5 

n(n+lXn  +  2y--(n  +  r-l) 

2.  (1)  Gen.termof  (1+x)  *=(-l)''x-^^ ^ r^ x'' 

4-5-6 (3  +  r) 

=  (-irx 1^-^^ X' 

(11)  Since  general  term  =  6*1^  term  =  (1  +  ry^  term  .•.  r  =  5 

4-5-6-7-8 
Hence  6'^  term  =  (-  1)"  x  ,  „  „  .  .x^  =  -  56x^ 

^  1'4"D*4'0 

3.  (i)   General  term  of  (1  -x)"' 

p(p   +    9)(p    +    2q)    [p    +    (r    -    1)?} 

^         ■'  \rxq' 

2-5-8 (3r-  1) 

=  (-l)^'<     ^3> ^' 

(11)  Since  general  term  =  {r  +  l)tli  term  =  6tl>  term  .-.  r  =  5 
2-5-811-14  308 

Hence  6th  term  =  (  -  1)=  x  r:2-:^.5V2r3^'  =  "  7^^' 


136  KEY  TO  [Ex.  txv 

4.  (i)  General  terra  of  (1  -x)^ 

,       ,,,       ;>(/'- 7)0' -2v)---}/>-(r- 1)7}  ^ 

-  (  -  17  X j^^-, X- 

.      ,  ^r      41(-2).---(7-3r) 

-(-1/x   -r:^-^, X^ 

4-l(-2)(-5)(-S) 
(II)  As  before  r  =  5  .-.  Gtl»  term  -  (  -  1  )*  x ,  ,-,,,„., x' 

5.  (0  General  term  of  (1  +x)'^ 

=  (  _  l)r  X  ^'^^P  +  9)(P  +  29)-  •  •  -{P  +  (>•  -  l)?}^. 
|r  X  g*" 

7-912 (5  +  2r) 

-  (  -  l)""  X  — x' 

7-91M315 
(ii)  As  before  r  ^  5  .-.  6tt  term  =  ('  -  1)'  x    ,  .,  „  .  ..  ^^x" 


Vip  +  ilXp^2q) (/'+(r-l)y}  , 

^        '  |r  X  5*^ 


C.  (i)  General  terra  of  (1  +  x)  "  ^ 

p(p +  (/)(/>+ 2^) 

[r  X  g*" 

81M4 (5  +  3r) 

^        '^  r  X  3*^ 

8-1114-17-20 
(ii)  As  before  r-b  .-.   6tb  term  -  (  -  1)'  x  y;^  j^ 

7.  («  -a)    1  =  {./(I  -  a-ia)|-^  =  a-\l  -a'^x)-^ 

n(n  +  \) (n  +  r-  1) 

.-.  (i)  Gen.  term  of  (i/  -  x)"^  =  a'^  x  j— ^ (a'^x)' 

1-2-3 r 

(ii)   .-.  6tlj  term  =  a*  ^x' 

6  1         '1  /         X  \  9 

8.  ((/+  Jx)'  =  {a(l  +  Ja-'x)p  =«'h  +  2;i) 

(I)  .-.  Geu.  term  of  ((i+  Jx)i 

= "  ^ ifiT^r l.2i) 


fcs.  Lxv  1                               ALGEBRA.  137 

6-l(  -4)(-9) (11  -57-)/  I*" 


=  a'   X 


\r  X  S-" 


ij        61-4----(5r- 11) 

^        '^  \r  X  10"^ 

s  61-4  9  14 

(ii)     .-.    Gth    term    -    (  -  1)'  x  a'  x  p^^sTiys^'  "^^ 

—  /J  6    V 6.^ /T  -  S  .J.  '<     —    1»  T /»        -^    7--'» 

u       -^         iSolPOO"         -"^  2.  110  Oil"  •*■ 

9.  (i)Gen.termof(l-2.rr^=  "'^ ^ r^ -(2a/ 

2-3-4 (r+l)  2-3-4 r(r  +  1) 

\r  •- -^         1-2-3--    -r  "^  "^       (.rri;.  i 

(ii)  Since  general  term  =  (r  +  l)'h  term  =  5tli  term  .-.  r  -  4. 
Hence  5^^  term  =  (4  +  l)2*x*  =  5  x  IGx*  =  80x* 

10.  General  term  of  (1  +  sx=)  '>= 

,      ,,,     K/^ +  «?)(/> +  2g)- •••{/» +  (>•-!)</{ 

5-7-9 (3  +  2r)     2V-''-  5-7-9 (3  +  2r)  , 

=  (  -  1  )'■  X   ; .z X  -„  -  =  (  -  l)'  X    ; -r 2  ■'" 

^        '  |r  X  2*^  3'        ^  \rx  3' 

5-7-911 
(ii)  As  before  r   =   4    .-.    5"i  terra    =   (  -  1)*  x    ,  .,  „  ,  p.x^ 

-4.1    v385T8_.3a/iT.8 


11. 


(a-2+x^)    '=  (rt-^(H-a^x"*))     ''  =  a^(l+a-'x"*) 


.-.  (i)  General  term  of  (a'^  +  x"') 

=  a'^  X  (  -  1/x   ■ 1^— , \^a'x    '  ) 


=  a'  X  (  -  P'  .< 
-(-l)'-x 


5 

2-712----(5r-3) 

a* 

jr  x  5*" 

2-712---(5r-3)  .,_  ,    . 


2-7-12-I7  ^    .a       119  a    .3 

CO  •••5^''^^'-""  =  (-^)*^12-3-4-625'^      "^    '  =  625'^''' 

.1  .1-2,-1  1      .  1       -.i  1  1     -a 

12.  (a    '-X    ^)     =  {a    ^(1-a'x    ^)}     =  a(l  -  a'x    *) 
.-.  (i)  General  term  of  (t^    '  -  x    *) 


KEY  to  l£x.  Lxv,  Lxri. 

n(n+l)(rt+2) (n  +  r-l)     i    .i   r 

\r  ^  ' 

2.3.4.. ..(.^l)     1                     l-2-34---Tx(r+l)^   J   .1   r 
=  ax ^ (a=x  5)  =ax p-^-^,^_---^:^_(«5^  5) 

ir+l     .i 

=  (r+  l)a'       X    » 

13.  r^"  =  1024        14.  r^  =  128        15.  0"  =  0        16.  r^^  =  40'J0 

17.  r     is    the   least  integer  equal   to  or  next  greater   than 

X  2 

(ra  +  1)   -,— ,  or  (4  +  1),—-;:,  or  5  x  J  or  ^r\  but  the  first  integer 

(X  "T  X  1.  "T  it 

>  '3'^  is  4  .-.  the  greatest  term  of  the  expansion  is  the  4lh  term 

=  32 

X  \ 

18.  r  is  the  least  integer  =  or  next  >(/i- 1).-^  ;  or(5-l)         ; 

or  4  X  1 ;  or  4  .•.  r  =  4'''  term  =  5tli  term  =  4j 

X 

19.  r   is   the   least  integer  =   or   next  >  (n  +■  1)^7—  ;    or 

3 
(20  +  I),  rsi  ^"^  21x3;  or  "ijS  which  is  13  .-.  the  greatest  tern. 

is  the  13'"  term  =  125970  X  2"  X  31^ 

X  I 

20.  r  is  the  least  integer  =  or  next  >  (n-  1),~;7",  or  (7-  1) — -\ 

or  6  X  5  ;  or  9  .-.  the  9tli  term  =  -^^^81$?'  =  the  10th  term. 


EXBRCISB   LXVI. 

1.  7x<35.-.  x<5     2.  16x-84>108,  or  16x>  192  .•.x>  la 
3.  4x<  12  .■.x<3     4.  4x+10>x-20;  3x>-30  .•.x>-  10 

5.  ax  +  56x  -  hah  >  a^ ;  ox  -  a'-  +  hbx  -  Bab  >  0  ;  a(x  -  a) 
+  5b(x  -  a)  >  0 ;  (x  -  a)((i  +  56)  >  0  .-.  x  -  a  >  0  .-.  x  >  a 
Also  fix  -  7'Jx  +  lab  <  b'^,  bx  -  b''  -  7ax  +  lab  <  0,  b(x  -  b)  - 
ta(x  -  6)  <  0,  (6  -  7n)(x  -  6)  <  0  .-.  x  -  6  <;  0  .-.  i  <  6 

6.  a*+  1  > a^+ a,  according  a*+  l>a(a+  1);  orasa*-a+  1  "a; 
oras  a*+  1  >  2a.  Now  if  a  =  1,  a^  +  I  »  2  =  2  x  1 ;  but  if  a  >  1 
then  Art.  134,  a^  +  1  >  2a  .-.  c:"  +  1  >  d^  +  a.  accord'"5f  as  o  >  I 


Ex.  Lxv»l  ALGEBRA.  139 

7.  As  above  a^  +  I  >  a^  +  a,  if  a^  +  I  >  2a;  but  Art.  134  for 
all  values  of  a,  except  a  =  1,  a^  +  1  >  2(i  .-.  a'  +  1  >  a^  +  a,  when 
a  is  a  neg-ative  improper  fraction. 

8.  -T-  +  —  >  2,  if  a^  +  i2  >  2ab  :  but  a^  +  b' >  2ab  by  Art,  134 

9.  Multiplying  each  by  12,  and  reducing,  we  have  7x  +  G  <  Cx 
+  12,  and  7x  +  6  >  6x  +  10  .-.  x  <  6,  and  x  >  4  .-.  x  =  5 

10.  a2  +  62>2a6.  Art.  134;  also  a2  +  c^>  2a^,  and  6^+ c2>  26c. 
Then  by  addition  a^  +  6^  +  a^  +  c^  +  b^  +  c^  >  2ab  +  2ac  +  26c; 
that  is  2a^+  2b^+  2c^>2ab  +  2ac  +  26c  .-.  a2  +  6^+c2>a6 +ac  +  6c 

11.  a^>  a^-  (6  -  c)^,  since  (6  -  c)^  is  necessarily  positive 
.-.  a^>  (a  -  6  +  c){a+  b  -c),  these  being  the  factors  of  a^-  (6-c)^ 
similarly  6^  >  (a  +  6  -  c)(6  +  c  -  a),  and  c^y  (^a  +  c-  b)(b  +  c-d). 
Multiplying  unequals  by  unequals,  a^6^c^ >  (a  -  6  +  c)^(a  +  b  -cy 
(b  -^c-aY;  extracting  sq.  root  abc >  {a-b  +  c){a-i-b-c)(b +  c-a} 

13.  Let  b  =  a  +  m,  and  c  =  (i  +  n^  a  being  the  least  of  the  tbree 
quantities ;  then  ab(a  +  6)  =  a(a  +  m){2a  +  m)  =  2a?  +  3u^/«  +  am^ 
ac(a  +  c)  =  a(a  +  n)(2a  +  7i)  =  2u^  +  ^a^n  +  a/i^ 
6cC6  +  c)   =   (a  -^^  m){a  +  7i)(2a  +   ?n   +  n) 
=  2a*  +  3a^(;n  +  n)  +  a(7n,  +  7j)^+  Hm(;?i  +  n) 

.'.  by  addition 

(I)  ab(a  +  6)  +  ac(a  +  c)  +  6c(6  +  c)  =  6a'  +  6a^(;/i  +  n) 
+  2a(;n^  +  v})  +  2amra  +  7?in(7/i  +  n) 

(ii)  Also  6aAc  =  6a(a  +  m)(a  +  n)  =  Qa?  +  6a^(;/i  +  n)  +  6a;/i7i ; 
subtracting  (ii)  from  (i)  we  have  (i)  -  (u)  =  2a(m^  -  2mn  +  n^) 
+  mn{m.  +  n)  =  2a(m  -  n)^  +  mn{in  +  n)  ;  but  since  by  supposition 
a  <  b  and  <  c,  it  follows  that  m  and  n  are  positive  quantities 
.-.  2rt(?/i  -  n)*  +  mn{in  +  n)  is  positive  .-.  ab{a  +  6)  +  ac(a  +  c) 
+  6c(6  +  c)  -  6a6c  is  a  positive  quantity  .-.  ab(^a  +  6)  +  ac(^a  +  c) 
+  6c(6  +  c)  >  6a6c 

(III)  Also  2 (a'  +  6»  +  c»)  =  2a'  +  2(a  +  mY  +  2(a  +  n)' 
=  6u'  +  6a^(7/i  +  7i)  +  6fl(wi''*+  rt^')  +  2(jft'+n') ;  subtracting  (i)  from 


140  KEY  TO  [Ex.  Lxvi. 

(m)  we  have  (iii)-(i)  =  4a(m'+n^)-2am»+2(m'+n')-.'nn(n)  +n) 
=  4a(in'  -2inn +11^)  +  8anin-2amn  +  2(tn  ■i-n)(m^  -tnn  ■^■})^  -  (m  --1^)11)11 
=  4a(OT  -  7i)^  +  6amn  +  (m  +  n)[2(m2  -  mn  +  n*)  -  mn] 
=  4t/(w  -  «)'  +  6anm  +  (m  +  n){2(m*  -  2mn  +  n^  +  mn] 
-  4(/(»i  -  n)'^  +  6am/J  +  (7ft  +  n){2(7ft  -  n)'  +  mn]  which  as 
before  is  a  positive  quantity  •.•  a,  m,  and  n  are  all  positive 
.-.  2(a'  +  6'  +  (T*)  -  ab(a  +  b)  +  ac(a  +  c)  +  bc(h  +  c)  =  a  po.^tive 
quantity  .-.  ah{a  +  6)  +  ac(M  +  c)  +  6c(6  +  c)  <  2(a'  +  i'  +  c*) 

12.  3(1  +  «^  +  «■»)  -  (1  +  a  +  a}f  =  2  -  2a  -  2a»  +  2a* 
=  2(1  -  a)  -  2a''(l  -  a)  =  2(1  -  a)(l  -  a').  Now  1  -  o  and  1  -  a» 
have  the  same  sign  whether  a  >  or  <  1  .-.  their  product  is 
positive.  Hence  3(1  +  a^  +  o*)  -  (I  +  a  +  a''')  =  a  positive 
quantity  .-.  (1  +  a  +  a^)    <  3(1  +  a^  +  a*)  unless  a  =  \ 

14.  aV  -    {0.C   +   W)»  =    (a^  +  6'0(c^  +   (i^')    -    (ac   +   bd)* 

=  u'^cF  -  2(ibcd  +  b^c'^  =  (^11:1  -  if)-' which  is  necessarily  positive; 

unless  ad  =  be;  but  x'^y^ -  (ac  +  bdy -  \xy  +  (ac  +  bd)]{xy  -(ac  +  bd), 

.-.  {xy  +  (ac  +  bd)\\xy  -  (ac  +  bd)]  =  (a I  -  be)'  .-.  xy  -  (ac  +  bd) 

(ad  -  bc)'^ 

= —r-,  ^  a  positive  quantity  .•.  xu  >  a<r  +  bd 

xy  +  ac  +  bd         ^  '  •'  ^ 

15.  V""-  b'^  +  'J'iab  -  b-  >  a,  if  'Jlab  -  i^  >  a  -  V^^T*"; 
or  if  2</6-62>a^-2aVtt^"62'+a*-62;  or  if  2.76  >2a*- 2aVa='-fr'; 
orif  6>a-'\/a^-6^;  orif^/ifi-b^ya-b  ;  or  if  a--i'>fl*-2a&+6'''; 
or  if  2a6>  26^;  or  if  a>  6  , 

IG.  Making  the  sam'e  supposition  as  in  Ex.  13 
(1)  (a  +  fr  +  c)»  =   (3a   +    m   +   ri)'  =    37a'  +    27<i*(«t    +    n) 
+  9a(m  +  7J)*  +  (/n  +  7j)* 

(11)  27aic  =  27a(o  +  in)(a  +  n)  -  27a'  +  2Ta'''(m  +  ti)  +  27«»n 
(ni)  9(a*+6»  +  r')  =  9{a'4-  (a  +  /«)»+  (a  +  ri)»{  :r  27a»+  27a«(:n  +^) 
+  27<»(7n'2  +  n')  +  9(m»+7J») 
.-.  (1)  -  (u)  =  9a(/rt  +  71)*  -  27amn  +  (m  +  n)* 

=  9a(/n  +  7t)^  -  3Ga/u7i  +  9a7«n  +  (m  +  ;i)' 

=  9a(m  -  /i)^  +  9amn  +  (m  +  n)'  =  a  positive  quantiiv 


£x   Lxiv.]  ALGEBRA.  141 

That  is  (a  +  6  +  c)*  -  27a6i:  =  a  positive  quantity 
.  .  (a+  b  +  cy>  21abc 
Again  (in)  -  (i)  =  27a(in^+n^) -  9a(m  +  ny  +  9(m^  +  n^)  -  (m  +  ny 
=  9a{3(rti2  +  n^)  -  (/n.  +  n)^}  +  (m  +  n){9(in'  -  mn  +  n^)  -  (ot  +  n)'^\ 
=  9a(2/?i^  +  2m^  -  2mn)  +  (/«  +  n){^m'^  -  Imn  +  8n^) 
=  18a(m^-  ;/!«  +  n^)  -  18a//m+  18a7ft7i  + (w  +  ?i){(8m^-  167ft?t+  8«^) 

+  9inn] 
=  18a(m  -  7t)^  +  ISamii  +  (w  +  H)(8(;n  -  ii)^  +  Oinn] 
=  a  positive  quantity 

That  is  9(0*  +  6'  +  c*)  -  (a  +  6  +  c)*  =  a  positive  quantity 
.-.  (a+6  +  c)»<9(a3+63  +  c3) 

17.  ((t  +  6)(6  +  c)(c  +  a)  ^  8a6r,  according  as     • 
a'^b  +  ab^  +  a'c  +  ac'-  +  6^c  +  ic^  ^  6a6c 

or  as  (a62  -  lobe  +  ac^)  +  6(0^*  -  2ac  +  rt^)  +  c{(i?  -  2a6  +  &^)  ^  0 

or  as  fl(6  -  c)^  +  6(c  -  fl)^  +  c(a  -  6)^  ^  0 

But  «(6  -  c^  +  6(c  -  a)-  +  c{a,  -  6)^  >  0  unless  a  =  b  =  c 

.-.  (a  +  6)(6  +  c)(c  +  a)  >  8«6c 

x2+34x-7l 

18.  Let     .^       — IT?""^'  then  a'''  +  34x- 71  =  7na;^+ 2rtia;  -  7;/i ; 

that    is     (/n    -    1)j:-  +    2(in    -    l7).r    =     7//i    -    71,    whence 


X  ~  :{17  -  m  +  VSO"'  -  5)(//i  -  9){,  where  if  x  is  to  be  real, 

m  —  5  and  /«  —  9  must  both  have  the  same  sign  :  i.  e.  wi  must 
be  >  9  or  <  5  .-.  the  given  expression  can  have  no  value  between 
9  and  5 

19.  First    -^— — -    >  — ,  if  3u2  -  3n  +  3  >  n^  +  71  +   1; 

or  if  2/1*  -  4^  +  2  >  0  ;    or  if  71^  -  2«  +  1  >  0  ;    or  if  7)^  +  1  >  2/i  : 

but  71^  +  1  is  >  2/1  .•.  &c. 

71^  -  7i  +    1 
Secondly  ^^z   .    ,t   .    i    <  3>   if  7i^  -  7i  +    1  <  3n^  +  3n  +  3 ; 

or  if  0  <  2/1''  +  -In  +  2  ;  or  if  0  <  n^  +  2/1  +  1  ;    or  if  0  <  (n  +  l)^ ; 


142  KEY   TO  [Ex.  Lxvi.  lxvii. 

but  {n  +  1)-  is  necessarily  positive  .-.  0  is  <  (n  +  1)^  .-.  &c. 

n^  -  n  +  1 

Hence  -s r ,  lies  between  3  and  i 

n^  +  71  +  1 

Note.— If  »  —  1,  the  expression  =  J. 


EXEBCISE   LXVII. 

1  —  x" 
1 . =  1  +  X  +  X*  4-  x^  +  &c.  •  •  •  •  to  /I  terms  =1  +  14-1  +  1 

1  -X 

+  &c.  •  •  •  •  to  71  terms  =  n 

(x  -  a) (x^  +  ax  +  a^)      x'^  +  ax  +  a^     a'^  +  «^  +  a^      3a'      3./ 
(x  -  a)(x  +  a)        ~       X  +  a       ~       a  +  o       ~2a~2 

x^(x^  -  o*)  X*  a^         1 


(X*  +  a^)(x^  -  a^)       x*+a^        2a^ 

(X  +  7)(x  -5)       ^+7_12_ 
(X  +  3)(x  -  5)  "  x+  3  "  T  ~      * 

(x-0(a;-3)       x+  3  3i      _        ^ 


(X-  i)(-»^-2)       X-  2         -  IJ 

x(x''  +  6)-a(x2  4-//)       (x-a)(i2  +  ft)       x^  +  6       a^  +  6 
®'  x(x-a)  +  6^(x-a)  ^  (x  -  a)(x  +  6*)  "  x  +  6»  ~  a  +  6» 

a(x  -  c)(x  -  c)        rt 
6(x~-'c)(x  -  c)  ~  X 

x(a  -  x)  a  a 


(«  -  x)(o»-  a*x  -  uxH  x»)  ~  a*  -  <r»  -  a»  +  a*       0    ~ 

x(x2-a»)  +  2a(x»-rt^  (x  +  2a)(x-a)(x  +  fl)  (x  +  2aK^  +  a) 
(x-a)(x'^+ax-  12a^)  ~  (x-a)(x''+ax-12a^)  ~  x'  +  ax  -  1 2a» 
3o  X  2a  6a'  3 


a»+  a*-  12a^  "  -  10a«  ~        5 


Ex.  Lxviii.]  ALGEBRA.  143 

EXBRCISK  LXVIII. 

X  X  -2 

1.  Dividing  by  3,  we  have  x  +  i/  +  — -  =  3+i.-.  —  —  is  integral 

:  /  say.     Then  x  =  3t  +  2  ;  substituting  this  for  x  in  the  given 

jquation  we  have  By  =  II  -  4(3f  +  2)  .-.  y  -  I  -  4t;  letting  t  =  0 

we  have  x  =  2  and  y  =  1. 

3y  1  +  By 

2.  Divide  by  5,  and  we  have  x-  -  2y  -  —  =  2  4-  ^  .-. —  is 

2  +  6y  2  +  y 

integral  .".  so  also  — r —  integral  .•.   — —  =  t,  whence  y  =  5t  -2. 

Substituting  this  for  y  in  the  given  equation,  we  have  5x  =  11 
+  I3(5t  —  2)  .*.  X  =  ISi  -  3.  Hence  taking  in  succession  ^=1,2, 
3,  &c.,  we  have  x  =  10,  23,  36,  49,  &c.,   and  y  =  3,  8,  13,  IS,  &c. 

y  y- 1 

3.  Divide  by  2,  and  we  have  x  +  3y  +  -  -  =  29  +  J  .-.    -  -  -  t ; 

whence  y  =  2t  +  1.  Substituting  this  for  y  in  the  given  equation, 
we  have  2x  =  59  -  7(2t  +  I)  .-.  x  =  26  -  Tt,  and  taking  in  snc- 
cession  t  =  0,  I,  2,  &c.,  we  have  x  ~  26,  19,  12  or  5,  and  y  -  1, 

3,  5,  or  7 

y  y  -  1 

4.  Dividing  by  5,  and  we  have  r  +  2y  +  —  =  5  +  ^  .-.  — ^—  =  t ; 

whence  y  =  5t  +  1.  Substituting  this  in  the  given  equation 
for  y,  we  have  5x  =  26  -  11(5^  +  1)  ;  whence  x  =  3  -  11^,  and 
hence  when  ^  =  0,  we  have  x  =  3  and  y  =  1 

8y      2        2  +  By 

5.  Divide  by  9,  and  we  get  x  -  y  -    -  =  — , .-.  — r —  is  integral, 


1  +  4y  7   +  28y 

—T —   integral,    .-.  so  als'     '  ' 


Bo   also   is   — - —  integral,    .-.  so  also  is   z integral. 


g  . — g,.„..  „..  9  -  ^  *^^°  y  =  9^  -  7; 
substituting  this  for  y  in  the  given  equation,  we  have  9x 
=  2  +  I7(9i  -  1)  .-.  x=  nt  -  13.  Now  writing  in  successiou 
^  =  1,  2,  3,  &c.,  we  have  x  =  4,  21,  38,  55,  &c.,  and  y  -  2,  11,  20, 
29,  &c. 


14-4  KEY    TO  [Ei    LXviH. 

8y  11       8y-ll 

6.  Divide  by  13,  and  we  get  x  +  y+  —  =  6  +  jz.-.  -is 

40y  -  55  y  -  3 

integral,  .-.  ~ is  integral,  .-.  3y  -  4.  +     .„  -  is  integral, 

y  -3 
.'.     ~,^  =  'i  whence  y  =  13t  +  3  ;  substituting  this  for  j/  in  the 

given    equation,    we    have    13j   ^   89   -  21(13f  +  3),    wbi-nce 
r  =  2  -  2lt.     Now  writing  /  =  0,  we  Lave  x  =  2  and  y  =  3 

51/  5 

7.  Divide    bj    12,    and    we    get  x-3y—    r^   =   —    1-t— 

5r/-5  y  -  1  y  -  1 

.•.    is  integral,  .-.  so  also  is  —_ —  integral.     Let  =  t, 

then  y  =  I2t  +  I  ;  substituting  this  in  the  given  equation  for  y, 

we  have  x  =  41(12i  +  1)  -  17,  whence  x  =  ilt  +  2.     Now  writing 

in  succession,  0,  1,  2,  &c.,  for  /,  we  have  x  =  2,  43,  34,  125,  Ac, 

and  y  =  1,  13,  25,  37,  Ac. 

'      '  61/  24       6y-24 

8.  Divide  by  37,  and  we  get  x  -i- 1/  +  —  ~  9  +  z-  .'.  — x^ —  is 

y-  4 
integral,  .-.  so  also  is    — —  w'jich  say  -  t ;    then  y  -  311   +    4 

Then  37x  =  357  -  43(37/  +  4),  whence  x  =  5  -  43/ ;   wherefort) 

taking  /  =  0,  we  have  x  =  5  and  w  =  4 

^  '  21y        6        21y  +  6 

9.  Divide  by  22,  and  we  get  x  -  y  -  -jjr-  =  —  .-.  — — —   is 

21I/+6                                 22v-21y-6 
integral,  .-.  so  also  is  y  -  — 22 —  integral ;  that  is ^ , 

V  —  6 
or  " is  integral  =  /,   say  then  y  =  22/  +  6.     Hence   22x 

=  6  +  43(22/  +  6)  .-.  X  =  43/  +  12.     Now  writing  in  succession, 
0,  1,  2,  &c.,  for  /  we  get  x  =  12,65,  98,  Ac,  andy  =  6,  2P,  50,  Ac. 

Ay  2        4v  -  2 

10.  Divide  by  7,  and  we  have  x  +  3y  +  y  =  25  +  -r-  .-.  — ^ — 

8y  -  4  y  -  4 

is  integral,  .-.  so  also  is  — » —  integral,  .-.  — - —  =  /,  whence 

y  =  7/  +  4.     Then  7x  =  177  -  25(7/  +  4),  whence  x  =  11  -  25/. 
Hence  taking  /  =  0,  we  have  x  =  11  and  y  =  4 


tx.  Lxriii.i  ALG2BRA.  i-±5 

61j/  38       61y+38 

11.  Divide  by  99,  and  we  get  ^  -  y  -  "^  =  ^  ■*"  99  •'•  — 99 — 

305'/  +  190 
13   integral,   .-.   so  also  is    '-^ integral,    .-.    so  also  13 

8y  +  91                                              104y  +  1183 
3y  +  1  +  — — —   integral,  .-.  so  also  is  r^ integral, 

5y  +  94                                     .    IQOy  +  1830 
.-.  so  also  is  ?/  +  11  +  — ^ —  integral,  .-.  so  also  is ^^ 

y  +  98 
integral,  .-.  so  also  is  y  +  18  +  — g^ —  integral,  .-.  ]/  =  99f  -  98  ; 

substituting  this   in  the   given  equation  for  y,  vre   have  99x 

=  335  +  160(99i  -  98),  whence  x  =  160^  -  155.     Now  substituting 

in  succession  1,  2,  3,  &c.,  for  t,  we  have  r  =  5,   165,  325,  485, 

&c.,  and  y  =  1,  100,  199,  298,  &c. 

X  2         X-  2 

12.  Divide  by  4,  and  we  have  4x-y  +  — -  =  5  +  -t-.".  — r—  =  ^ 

•'     '  ^4  4  4' 

whence  x  =  4f  +  2 ;  then  4y  =  I7(4f  +  2)  -  22  .-.y  =\1t  +  3. 
Taking  /  =  0,  1,  2,  3,  &c.,  we  have  x  =  2,  6,  10,  14,  &c.,  and 
J/  =  3,  20,  37,  54,  &c. 

13.  Multiplying  the  first  equation  by  3,  and  the  lower  by  4, 

and  adding  the  results,  we  have   13x  +  29y  =  123.     Divide  by 

lly              15       ll.v-  15 
18,  and  we  have  x  +  y  +  -^  =  6  +  —  •"• T^ —  is  integral, 

55y-75  y-3 

.-.  so  also  is  — T^ —  integral,  .*.  so  also  is  3]/  -  4  +  -yr-  integ., 

.-.  J/  =  18f  +  3.  Hence  18x  =  123  -  29(18/  +  3)  .-.  x  =  2  -  29^ 
Now  taking  f  =  0,  we  have  x  =  2,  and  3/  =  3,  and  consequently 
«-  4 

14.  Multiplying  the  upper  equation  by  11,  the  lower  by  6, 

and  adding  the  results,  we  get  56x  -  49i/  =  469,  or  8x  -  7y  =  67. 

X  4        X  —  4 

Dividing  this  by  7,  we  have  x  -  y  +  -^-  =  9  4-  -z-  ,-.  —  ^—  =  <. 

Whence  x  =  7f  +  4 ;  then  1y  =  8(7/  +  4)  -  67,  whence  y  =  8<  -  5. 
Now  taking  /  =  1,  2,  3,  &c.,  we  have  x  =  11,  18,  25,  &c.,  and 
y  =  3,  11,  19,  Ac;  but  since  s  must  also  be  positive  and  integral, 

K 


146  KEY  TO  [Ex.  Lxrui. 

we  find  upon  trial  that  the  only  admissible  values  are  z  =  11, 
and  y  =  3,  and  consequently  2=2 

15.  Lelx  =  the  number  of  $3  notes,  and  y  =  the  number  of 

2y  1        2y-l 

$5  notes  ;   then  3x  +  5y  =  697,  or  x  +  y  +  —  =  232  +  —  .-.  — o~ 

4y  -  2  y  -  2 

is  integral,  .-.  — - —   is  integral,  and  .-.  also      ,-  =  ^  that  is 

y  =  3i  +  2  ;  then  3x  =  697  -  5(3/  +  2),  whence  x  =  229  -  5/. 
Hence  5/  <  229,  or  f  <  ^^;  i.  e.  <  45^  .-.  the  given  sum  can 
be  made  up  of  $3  and  $5  notes  only  in  45  different  ways. 

16.  Let  X  =  the  number  of  25  cent  pieces,  and  y  =  the  numbor 
of  10  cent  pieces;   then  25x  +  lOy  =  2730,  or  5x  +  2y  =  54(;, 

X 

.-.  2x  +  y  +  —  =  273  .-.  x  =  2t.  Also  2y  =  546  -  10/  .-.  y  =  273  -  £/. 
Hence  5/  C  273,  or  /  <  54^  .•.  the  given  sum  may  be  made  ujt 
as  directed  in  54  different  ways. 

17.  Let  X  =  the  number  of  guineas  paid,  and  y  =  the  numbiT 

5y 
of  half-crowus  received  in  change;   then  21x  -  —  =  150^,  or 

2x  1         2x  —  1 

42x  -  5y  =  301,  .-.  8x  -  y  +  -r-  =  GO  +  —  .-.  — r —  is  integral, 

16x  -  8         X  -  3 

.•.   so   also   is    — z .'.  — - —   =   /,    or  X   =   5/  +   3.     Also 

5  5  ' 

5y  =  42(5/+ 3) -301  =  210/- 175  .-.  y  =  42/-35;  and  taking 
/  =  1,  we  have  x  =  8,  and  y  =  7 

18.  Let  X*  and  y'  be  the  two  square  numbers  required,  and 

assume  x^+  y''=  Qix  -yy  =  n-x'-  2}ixy  +y^;  then  x'=  n^x*-  2nxy  ; 

2ity 
or  X  =  Ti^x  -  2/iy  .-.  {n^  -  l)x  =  2Hy,  or  x  =  ~ — -,  where  n  and  y 

may  be  assumed  at  pleasure,  and  it  will  be  found  that  x-  +  y^  is 

a  complete  square. 

But  if  only  integral  values  are  required  assume  in  the  exprcs- 

2ny 
sion  x=  ~r, ,  that  y  -  li^-  1,  then  x  =  2n,  where  n  m.iv  be 


Ex.  txviii.]  ALGEBRA.  147 

taken  --  any  integral  number,  and  it  will  be  found  that  x^  +  y* 
is  a  complete  square. 

19.  Let  a:^  and  y^  be  the  two  squares  required,  and  assume 
x^  -  2/2  =  (I  _  7iy)2  =  x^  -  2nxy  +  nV-     Then  y^  =  2nxy  -  r^y'^  ; 

«^  +  1 

or  7/  =  Inx  -  n^y  ;   or  2«x  =  (n^  +  \)y  .-.  x  =  — x  y,  where 

n  and  y  may  be  assumed  at  pleasure,  and  it  will  be  found  that 
x'^  _  y2  ig  a  complete  square. 

But  if  only  integral  values  are  required,  assume  in  the  above 
expression  y  =  2n]  then  x  =  7i^+  1,  where  it  will  be  found  that 
when  n  is  taken  =  any  integral  number,  x^  —  y'  will  be  a  com- 
plete square. 

20.  Assume  that  the  basket  contains  x  parcels  of  4  with  2 
over,  or  y  parcels  of  6  with  2  over.     Then  4x  +  2  =  6y  +  2  ;  or 

y  y 

4x  -6y  =  0  ;  or  2x  -  3]/  =  0  ;  ov  x  -  y  -  —  =  0  .-.  —  =  t ;  or  y  =  2t. 

Also  2x  =  3y  .-.  x  -  Zt.  Hence  taking,  f  =  1,  2,  3,  &c.,  we  have 
X  =  3,  6,  9,  12,  &c.,  and  y  =  2,  4,  6,  8,  &c. 

But  X  and  y  must  be  taken  such  that  both  6y  +  2  and  4x  +  2 
are  >  90  and  <  100  .-.  y  =  16,  and  x  =  24,  and  the  number  of 
apples  =  16!/  +  2  =  98 

21.  Let  the  number  =  6x  +  l  =  8!/  +  5  =  10c  +  9 

y         2 
Then  6x  -  8y  =  4  ;    or  3x  -  4?/  =  2  ;    or  x  -  j/  -  — -  =  — 

.-.  y  =  3<-  2,  and  x  =  4^  -  2 

Also  6x  +  1  =  102  +  9 ;   or  6x  -  lOs  =  8 ;    or  3x  -  5=  =  4,  but 

2t 
a;  =  4f  -  2  .-.  Z(\t  -  2)  -  5=  =  4 ;  or  12/  -  5=  =  10  .-.  2/  -  =  4-  -  -  2, 

whence  t  =  U'  and  z  =  12^  -  2.  Then  x  =  4f  -  2  =  20/'  -  2  ; 
y  =  3/  -  2  =  15/'  -  2,  and  z  =  12/'  -  2,  whence  taking  /'  =  1,  we 
have  X  =  18,  y  =  13,  and  z  =  10,  and  .-.  the  least  number  divisible 
as  required  =  6x  +  1  =  (18  x  6)  +  1  =  108  +  1  =  109 


148  KEY  TO  t^x.  txviit. 

X  y  X         1/         33 

22.  Let  —  and  rr  be  the  two  fractions,  then  rr  +  tt  =  :rr  ; 

10  15  10         15        60 

X  1 

or  cleariug  effractions  3x  +  2i/  =  19  .-.  x  +  y  +  —  =  9  +  — ,  and 

consequently  i  =  2/  +  1,  whence  y  =  8  -  3^     Now  taking  ^  =  0, 

1  and  2,  we  have  x  =  1,  3  or  5,  and  y  =  8,  5  and  2 

.-.  the  required  fractions  are  -jV  *°<i  'h  >  "I'o"  *iid  -j^^  ;  and  ^^^and  Vj- 

KoTE.— We  cauuot  take  t  =  3,  since  then  j/=r8-3f  =  8-9=-l  — a 
negative  quantity 

23.  Let  X,  y  and  z  =  barrels  respectively  ;  then  x  +  y  +  z 
=  50  (i),  and  2x  +  5y  +  42  =  250  (ii).  Multiplying  (i)  by  2,  and 
subtracting  the  result  from  (ii),  we  have  3y  +  2s  =  150,  whence 
y=2t  and  z  =  15  -  3t 

Also  X  =  50  -  y  -  2  =  50  -  2/  -  (75  -  30  =  i  -  25 
Then  in  order  that  z  may  be  positive  75  -  3/  must  be  positive, 
and  .•.  Zt  <  75,  or  f  <  25,  and  in  order  that  x  may  be  positive, 
t  -  25  must  be  positive,  that  is  f  >  25  ;  therefore  t  is  both  less 
than  and  greater  than  25,  which  is  impossible. 

24.  Let  X,  y  and  s  ^  the  number  of  pieces  respectively ; 
then  X  +  y  +  s  =  100  (i),  and  lOOx  +  20y  +  5z  =  2000  (ii)- 
Dividing  (ii)  by  5,  and  from  the  result  subtracting  (i),  wc 
have  19x  +  3y  =  300,  whence  x  =  3/  and  y  =  100  -  ]9t 
.-.  2  =  100  -  X  -  y  =  100  -3t  -  (100  -  19/)  =  16/.  Now  taking 
/  =  1,  2,  3,  Ac,  we  have  x  =  1,  6,  9,  12  or  15 ;  y  =  81,  62,  43,  24 
or  5 ;  and  s  =  16,  32,  48,  64  or  80 

25.  2x  +  3y  =  25,  whence  x  =  11-3/,  and  y  =  2/  +  1.     Now 
taking  /  =  0,  1,  2  or  3,  we  have  x  =  11,  8,  5  or  2,  and  y  =  1,  3,  5 
or  7,  and  hence  the  parts  are  2x  and  3y  =  22  and  3 ;    16  and   9  a 
10  and  15,  or  4  and  21. 

2G.  Let  X,  y  and  z  be  the  three  parts  ;  then  x  +  y  +  2  =  24  (i), 
and  3C.X  +  24y  +  Si  =  516  (ii).  Dividing  (ti)  by  4,  and 
multiplying  (i)  by  2,  and  taking  the  difference  of  the  results, 


Ex.  Lxvin.]  ALGEBRA.  149 

wc   have  1x  +  4y  =  81,  whence  x  =  it  -  \,  and  y  =  22  -  It 

.-.  s  =  24  -  (4<  -  1)  -  (22  -  70  =  3  +  3t.     Now  taking  t  =  I,  2 

or  3,  we  have  x  =  3,  T  or  11  ;  y  -  15,  18  or  1 ;  and  z  -  6,  9  or  12 

2  7.  Assume  y"x  to  be  a  perfect  number ;  then  its  divisors  are 

1,  y,  y-, 3/",  X,  xy,  xy\   ....   x!/"-i  .-.  j/^x  =   1  +  y  +  y^ 

+  . . . .  J/"  +  X  +  xy  +  xi/2  +  . . . .  x!/™  -  ^.     Now  1  +  2/  +  2/^  +  •  •  • .  ^'^ 

yn  +  l    _    1  2/"    -    1 

=  — r— j— ,   and  X  +  xy  +  x(/2  +    xy"-i  =        ^  ^-     x  x 

^1  +  1  _    1    +   (,/n  _    i)x 
.-.   i/"x  -   ; ;    or   clearing   of  fractions 

y'*  +  ix- j/"x  =  y"  +  i-  1  +y'*x  -x;  or  i/^  +  ^x  -  2!/"x  +  x  =  y"*i-  1 

•'-  ^  ~   ,7n-i ir~n T-     Now  in  order  that  x  may  be  a  whole 

y       —  ^y   -  I 

number,  let  i/"  +  i  -  2?/"  =  0,  or  y  =  2  ;  then  x  =  2"'*i  -  1.     Also 

let  n  be  so  assumed  that  2"  +  ^  -  1  may  be  a  prime  number;  then 

it  will  be  found  that  i/"x  =  2"  x  (2"  +  ^-  1)  will  be  a  perfect 

number.     Thus  if  n  =  2,  we  have  2^  x  (2^  -  1)  =  4  x  (8  -  1) 

=  4x7  =  28  =14 +7  +  4  +  2+1=  sum  of  all  the  divisors  of  28. 

28.  Let  the  number  =  lOx  +  7  =  12i/  +  9  =  142  +  11  ;    then 

lOx  -  12y  =  2,  or  5x  -  6y  =  1,  whence  x  =  6f  -  1,  and  y  =  5f  -  1. 

Also  lOx  -  14i  =  4,  or  30t  -1z=1,  whence  t  =  It',  and  z  =  30t'  -  1. 

Then  X  =  6/  -  1  ^  42^  _  1  ;  y  =  5<  -  1  =  35^  -  1,  and  z  =  30t'  -  1. 

Now  assuming  t'  =  1,  we  have  x  =  41  ;    y  =  34;    and  z  -  29. 

Hence  the  least  odd  integer  =  lOx  +  7  =  410  +  7  =  417. 

'    29.  Let  X,  y  and  z  represent  the  numbers  respectively  ;  then 

x  +  y  +  z=  100  (i),  and  50x  +  30y  +  2:  =  500  (ii).     Dividing  (ii) 

by  2,  and  from  the  result  subtracting  (i),  we  have  24x  +  14y  =  150, 

T.-hence  x  =  7<  +  1,  and  y  =  9  -  12i .-.  2  =  100  -  (7/  +  1)  -  (9  -  12/) 

-  90  +  \2t.     Now  t  must  be  <  1  •.•  y  =  9  -  \2t  must  be  positive  ; 

also  t  must  be  >  -  1  because  x  =  7f  +  1  must  be  positive,  and 

since  x,  y  and  z  must  be  integral,  t  can  only  =  0,     Hence,  when 

f  =  0,  we  have  x  =  1,  y  =  9,  and  z  =  90, 


IcO  KEY  TO  [Mis.  Ex.  1- 


MlSCELLAMEOUS   EXERCTSKS. 

1        a       a       2       17  -  21a 

1.  i(l_«)_^{3a-2)  =  ----y4--=— ^^ 

2.  {(x2_X-^)+  l}2_{(x2-X-2)-  If 

=  (x2-  x-^/  +  2(r2-  x-2)+  l-{(x2-  x-^/-2(x2-  x-2)+  1} 
=  4(x2-x-2) 

3.  The  G.C.M.  of  Ihe  first  three  quantities  is  evidently  a  +  6, 
and  as  it  is  also  a  measure  of  the  remaining  quantity,  it  is 
their  G.C.M. 

b'  b^  b^-b^  +  ab         ab 

4.  Since  x  =  ^      ^,  x  -  6  =  ^      ^  -  6 

and  X  -  a  = 


b  -  0*  b  -  a  b  -  a        ~  b  -  a^ 

h'  b'  -ab^-a} 


b  —  a  b  —  a 

X  -  ft     X  -  a         b         b^-ab  +  a^     b^-b'  +  ab-a'     ab-a"* 


''a  ft         ft-  a         (6  -  G)ft     ~         (b  -  a)b  b(^b-a) 

a(b  —  a)        a 
""  b{b  -  a)  ^  V 

5.  X  +  y  +  2  =  15  (i),  X  -  y  +  2  =  5  (n),  -x-y  +  2  =  3  (iii) 
Adding  (i)  to  (in),  we  have  2s  =  18  .-.  2  =  9 

Adding  (i)  to  (11),  -we  have  2x  +  2;  =  20  .-.  2x  =  2,  and  x  -  1 
Hence  x  +  ij  +  z  =  l+y  +  9=  15.-.  y=5 

6.  5^27  X  5  -  3^x5  +  2^125  x  5  -  4^64  x  5  =   15.^5  -  6^'5 
+  10^5  -  16^5  =  (15  -  6  +  10  -  16)(/5  =  3-^5 

1 

7.  X*  +  1  =  0.     Divide  each  side  by  x-;    then  x-  +—5  =  0 

I  1 

.-.  X-  +  2  +  -^  -  2  .-.  X  +  —  =  i  V2 ;    clearing  of  fractions 


X*  f  xV2  =  -  1 ;  x»  T  xV2  +  ( Y  j   =  J  -  1  =  -  1  .-.  J  + 

Hence  x  = jz 

y2 


V2^      V2 


ilie.  Ex.  8-11.]  ALGEBKA.  151 

a         b         c  a  +  h       b  -i-  c         a  +  b        b 

'    b    ~    c    ~    d  '         °      b       ~       c       '  '  b  +  c  ~    c 

b+c       c+d        b+c        c  be         a-\-b       b  +  c 

Also =  — T-  .•. .  =  -r.     But  —  =  -r  .*.  I      ■  =  , 

c  d  c+d        d  c         d         b  ■{■  c       c  -i-d 

Hence  a  +  6  :  b  +  c  ::  b  +  c  :  c  +  d 

9.  a  :  c  ::  2a  -  b  :   2b  -  c   .-.   a  :   2a  -  b  ::  c  :   2b  -  c 
a  c  2a  2c  2a         2c 

•■•  2irr-6  =  2b^rc  or  ^^m  =  26^7^  •*•  T  =  sTTiiCArt.  106) 

a  c 

~b   ~  3c  -  2b  ■  ■  ^'^  ~  ^^'^  ~  ■^"^  '    "''  ^"^  =  6c  +  2a6  -  6(c  +  2a) 

b  ac  «   X  -2        26       2a  X  |-  26 

•■•  ¥  =  ^T^  = 7  •■•  T  = J  •'  t^^t  '^  y  'S  ^^^  '^• 

a  +  -  a  4-    2 

c  26  c 

mean  between  a  and  —  .*.  «,  —  and  —  are  in  II. P. 

10.  Sum  to  7i  terms  when  r  is  a  proper  fraction 

a|i  -  (i  -  r\\ 

a(l    -  ^")  I  V  Pji  <^ 


1  -r 


-(■-7)^     "K'-  0-7)1 


-('-7)"   H'-O-))"} 


Sura  to  cc  - = -     

I  -  r 

1 
=  P  times  the  sum  to  n  terms. 

11.    X2      -X        2/X2      -X        2Vx'*+      l+x"" 

x^   —  xT 

»  in 

xT  -  X  "  T 

"  •« 


lo2  KEY   TO  [Mis.  Ex.  11-17. 

4  1  '    ' 

x^  +  a'x  +  a'x' 

-  ax  +  a^ 

-  a-'x  -  a^x'  -  flx' 


a^x^  +  ax^  +  o"* 
a^x^   +  flx'  +  a* 

1:j.  !'■«  trial  diV.    =  I47x* 

1"'  comp.  div.  =  147x*  -  63xhj  +  dxhj^ 
2'»d  trial  div.  =  I4lx*  -  UGx^y  +  27xy 
2"«  comp.  div.  =  147x<  -  126x*y  +  llixy-  36xy»+  IGy* 

13     x'''''"'*'''^''"^"''^^"''*-    X™  +  •*  +  P  ■ X    tP-!*8-''+''-P 

'     abc 

=  abc  X  x"  =:  a6c  x  1  -  abc 

14.  {(2x-^  +  Jx  -  V)  +  ;,]{(2x^  +  Jx "  Y)  -  ,/}  =  (2x^  +  ix "  yf-y^ 
=  4x*  +  2y^  +  ix-*y*  -y^=  4x<  +  y*  +  Jx ' ■•y* 
j(x2  +  6^)  +  ax}{(x^+  6')  -  flx}  =:  (x^+  h^^-  c^x^  =  x*  +  6*  +  262x--  <i-x' 
(x"  +  yP)(x"  +  y')  =  x*"  *  "  +  x"!/'  +  x''yi'  +  yP  ♦  « 

(Sys   -.    2V3)''       (2V5    -   3V3)(3V5    +    2V3) 
^^-  33  ■*"  33 

=  a',(45-  12715  +  12  +  30  -  9VT5  +  4VT5  -  18)  = -^'^ (69  -  17^/13) 

1  1 1  12x^+1 

1         ~  4x        ~    24x^+  6x    ~    24x»  +  6a 

2x  +   ,„   ,  .   .        2x  + 


16. 


12x='+l  "^I2x''+1  12x^+1 

"~4x~ 

2x   +   1   -  (2x  -   1)  2x  +  I 

17.   TTT-r^c + 


4(4x^-  1)  "^  2(2x  -   l)(4x^  +   1)  ~   2(4x^  -  1) 

2x    +    1  (4x»    +    1)    +    (2x    +    lX2x    +    1) 

2(2r    -    l)(4x''    +    1)    ^  2(4x^   -    l)(4x»  +    1) 

4x*  +  1  +  4X''  +  4x  +  1        4x2  +  2x  +  1 
T(4i-'-  l)(4x^+  1)     ^       16x«  -  \ 


Mis.  Ex.  18-20.]  ALGEBRA.  153 

x(a  -  c)  a  —  c  X  1 

^®*  ('^  (7+a)(x  +  t)  "^  xVa^~c  ■'■  (X  +  a)(x  +  c)  "  x  +  a  -  c 

a 
:.  x^  ■>r  ax  -  ex  -  x^  -ir  ax  -^^  ex  ^  ac  \  2cx  =  -  ac  .-.  x  =  -  — 


(II)  V(-c  -   IX-c   -  2)   -   2   =  V(2^  -  3)(x   -   4)  ;    squaring 

(X  -   l)(x  -   2)  +  4  -  4V(a:  -  l)(x  -~2)  =  (x  -  3)(x  -  4); 

.-.  2^-1;^ -  3x  +  2  =  2x  -  3  .-.  4x2-  i2x  +  8  =  4x^-  12x  +  9  .-.  8  =  9 

which  is  absurd  .-.  the  equation  has  no  possible  roots. 

1  1  _  1 

('"^  (x  +  3)(x-5)  ■*■  (X  -  5)(x  +  7)       (x  +  3)(x-16) 

.-.  (X  -  r)(x  -  16)  +  (X  +  3)(x  -  16)  -  (X  -  5)(x  +  7)  =  0 

z*  -  9x  -  112  +  x^  -  13x  -  48  -  x^  -  2x  +  35  =  0  ;   x''  -  24x  =  125, 

whence  x  =  12  +^269 

6+c     1  h-c 

19.  Since  7i  =  . ,  —  will  -  ,——  ;    then  //.  mean  between 

0  —  f    n  0  +  c' 

1  2  2  6^  -  c^ 

'^''''^Ti   ^TTc      b-e  =  WTl?  =  b^-V7^-     But  a:6::6:c 
b  -  c      b  +  c        b'^  -  c'^ 

.*•  Va  =  ~e  •*•     ■>  .  .g  =  rs"; — J  •'.  also     »  ,  ■  ,  is  the  H.  mean 
6*       c*        a*  +  b*       b^  +  c'  a^  +  b' 

1 

between  n  and  — 
n 

20.  Let  w  =  work  and  x,  y,  z  =  times  in  which  ^,  B  and  C 

can  separately  perform  it ; 

w 
Then  — -  =  ^'a  daily  work  4-  B's  daily  work  (i) 

w 
-J-  =  j9'3  "  +  C's  "  (II) 

-^  =  £'3  "  +  C's  "         (III) 

w 

T 


-f-^Fi   "■        -  C's  "         (IV) 


www 


iV  lU  LU 

Then  adding  (in)  and  (iv),  we  have =  2J'8  daily 


154  KEY  TO  [Mis.  Ex.  20-26. 

to  2  1  1  1  be  -  ac  +  ab 

work  =  2 — .    Hence  —  =  —  -"r  +  —  =    "^t;: 

y  y  a  0  c  ooc 

2abc  2abc  2abc 

and  y  =  r m. ;  similarly  x  =  — —. r,  and  z  =  ^.  ,  ^^ — r- 

^      bc-ac  +  ub'  •'         ac  +  bc-ab'  ab  +  ac-bc 

«2  62  c* 

2^-  (a-bXa-c)  +  (c-fc)(a-6j  "  (a-c)(c-6)'  ^7  changing 
signs.     Hence  /.  c.  m.  =  (a  -  6)(a  -  c)(c  -  b) 
d\c  -b)  +  b^{a  -  c)  -  c\a  -b)     aH-  d'b  +  ab"^  -  cV  -ac^  +  bc^  _ 
(a  -  fc)(a  -  c)(c  -  6)  ~  a^c  -  u-b  +  at''  -  c6'  -  ac^  +  bc^  ~ 

/d'+4b'^-9c^Y    /       a^+4P-9c^/       a'^  +  4b^-9c^ 

22-  "'-(         46         )  =  (''+   —46 j  V"--~I6— j 

a^  +  4ff6  +  46''  -  9c2       9c^  -  a^  +  4a6  -  46^ 


46  46 

(a  +  2by  -  (3c)2       (3cy  -  (a  -  2by 


46  46 

(o  +  26  +  3c)(a  +  26  -  3c)       (3c  +  a  -  26)(3r  -  a  +  26) 

(a  +  26  +  3r)(o  +  26  -  3c)(a  -  26  +  3c)(26*-  a  +  3c) 
=  r66^ 

23.  a*  +  2aW  +  6*  -  20^62  =  (a"  +  6-)*  -  (a6V2)* 
=  (a^  +  a6V2  +  b-)(a?  -  a6V2  +  6^) 

Similarly  a*  +  20^6^  +  6*  -  3u26^  ^^  (a^  +  b-f  -  (ab^j3y 
=  (a'  +  62  +  ab-slZXd'  +  6'^  -  a6V3) 
a?J/  +  y^  +  a;2 

2^-    xy  +  y''  +  x^        y 

X 

25.  G.C.yi.  of  (X  +  7y)(x  -  4y),  and  (x  +  2y)(x  -  4y),  and 
(X  -  y)(x  -  4y)  is  x  -  4y  .-.  /.  c.  m.  =  (x  -  y)(x  +  2y)(x  -  4y)(x  +  7y) 
=  X*  +  4x='y  -  27x2y2  _  34jyS  +  5Gy* 

a(r'»-l)  0(1"-!) 

26.  When  r  =  +  1,  the  formula  S  =  .     becomes  S  =    ^  _  ^ 

=  a(l"-^+  1"-^+  1"-'  +  &c.  to  n  terms)  =  a(l  +  1  +  1  +  &c.  to  n 
terms)  =  na 


Mi8.  Ex.  26-29.]  ALGEBRA.  155 

,/(r"     1)  a{(  _!)»_]{ 

■When  r  ~  -  I,   S  - ; —   becomes   S  = -. — z — 

'  r  -  I  -  1  -  1 

=  a{(-  l)"-i  +  (-  l)''-2+(_i)'»-3+&c.)} 

=  a(l  -1  +  1-14-1-1  +  «fec.  to  n  terms)  =  a  if  n  is  odd,  and 

=  tt( -1  +  1-1  +  1-1  +  &c.)  =  0  if  7j  is  even. 

27.  (j)  a,  =  a  +  (t  -  l)d;  then  by  same  notation  a„  =  a+{m-l)d, 
a„  =  a+  (n-  l)d,  aj,=  a  +  (p-  l)d,  and  a^=  a+  (q-  l}d.  Then 
(p  -  q)(m  -  n)d  =  (vi  -n)(p-  q)d . • .  (p  -  q)\a  +  (wi  -  l)d  -  a  -  (n  -  l)rf} 
=  (m  -  n)[a  +  (/>  -  \)d  -  a  -  (^q  -  l)d\;  since  we  have  merely 
added  a-  d  -a  +  d  -  0  to  each  of  the  2"*  factors. 

•■•  (P  -  9)(«™  -  «-)  =  (>"■  -  ")(«p  -  «») 

(ii)  Since  a,  =  ar*'^;  therefore  by  same  notation  a„  =  ar^'^, 
a„  -  ar^'^,  a^  =  arP  "^,  and  a,  =  ar^  '^.  Then  since  (p  -  q)(>n  -  n) 
=  (ill - n)(-p -q)-  r'P'  l'<"^-'^>  =  r'^ - n;(P -  9;  _._  (^m-  n)P -  9  =  (jP -  qyn-n 

j-m  ^p  J.        J.  - 1 

But  '•"*""  =  ^,  and  r^'2-— _  Also  since  —  =  —1  =  0,  multiplying 

/^m-i\p-9        /,.p-l\m-n  far'^''^\P'^ 

by  the  latter,  we  have       „-n  =  ( ,:^^  .-.      -^^ 


p-l\m-n  /a   \P-1 

a^^J         >  ^^^^  ''  [fj 

28.  On  1^'  morning  the  watch  is  behind  the  clock  by  lis., 
and  45  hours  afterwards  it  is  only  2s.  behind  ••.  the  watch 
gains  upon  the  clock  to  the  amount  of  93.  in  45  hours,  or  }  sec. 
in  1  hour. 

Let  X  =  gaining  rate  of  watch  per  hour  ;  then  since  the  gain- 
ing rate  of  the  clock  is  -,^^s.  in  24  h.,  or  1  s.  in  240  h.,  it  is  o-{^^  s. 
in  1  h.  .-.  X  -  jL^  is  the  gain  of  the  watch  on  the  clock  per  hour 
.-.  X  -  ^j-j  =  ^,  whence  x  =  I  +  ^i^  +  t^^^q  ;  hence  watch  gains 
per  day  -^^^j  x  24  -  |g  =  49  s. 

81(2)12-  1} 

29.  (i)  S  to  12  terms  =    '  '  _  ^       =  16{(J)i^  -  1}  =  205911^ 

(11)  3^^  term  =  a  +  2d  =  4,  and  6'"  term  =  a  +  5d  =  1} 
'.3d=  ^i-4  =  -5f ;  henced=-i^.-.  a=4-2rf=  4  +  ^^'  =  5J| 
,•,  ^  series  =  5U  +  4I<^  +  4  +  3-^,-  +  Ac. 


166 


KEY  TO 


[MiB.  Ex.  20- 


(iii)  3""^  term  =  ar*=  4,  and  6'-"  term  =  ar*  =  J^  .-.ar'^  far^=  Ji-f-l 

4 
.'.  r*  =  -^y  ;   whence  r  -  I.     And  (l=—  =  4r-i-^  =  9.-.G.  series 

=  9  +  6  +  4  +  23  +  Ac. 

30.  Let  X  =  length,  then  x  -  60  =  breadth  in  yards,  and 
x(x  -  60)  =  5500  ;  that  is  x^  -  60x  =  5500,  whence  x  =  110,  and 
x-60  =  50  3 

31.  (I)  (x3  -  ff  +  (X  -  yy  =  (^^)  =  (^'  +  ^y  +  y")' 
-  X*  +  2x^u  +  3xV  +  2xv^  +  V* 


2x^1/ 
(u)  1 
+  0 
-  2 
+  1 


7  +  0+     0  +  0+     5-     4 

+     0  + 

3+9 

+  0+     0  +  0+     0+     0 

+     0 

-14+0+28-14 

-  66  + 

64 

+  7  +     0-14 

+     7  + 

33  -  32 

7  +  0-14+7  +  33-32 

-  59  + 

100  -  23 

7x'  -  14x*+  7x-  +  33x  -32 


59x2  -  lOOx  +  23 


x'+  2x  -  I 

X*"  -  X  '  "* 

(III)  r  =  ^"^'^  +  -c'"'^  +  x"*-«  +  x"*-'  +  x"»-3  +  ic 

*■       '        X  —  X     * 

We  observe  here  that  each  term  is  derived  from  that  preceding 
it  by  dividing  by  x''.  Let  us  now  assume  that  this  is  true  to  r  -  L 
terms,  and  we  have  then  left  as  remainder  j"»  J""!  _  x"*. 
Dividing  this  by  x  -  x"^,  and  we  get  as  first  term  of  the  quotient 
x'^-^'  +  M  which  will  be  the  r"»  term  of  the  quotient  of  x"»-x  "» 
+  x-x-\  But  x'"-*  +  i  =  x"»-2'-  1  *i  +  x-=  (r-l)*J>  term  -^  x^ 
.-.  if  the  law  is  true  for  r  -  1  terms,  it  is  tru*  for  r  terms. 
Now  it  evidently  holds  for  5  terms  .*.  for  6  and  .".  for  7  terms 
and  so  on,  and  .•.  it  is  generally  true,  and  since  the  first  term  is 
x™  ^,  and  each  term  is  derived  from  the  preceding  by  t  by  x^ 
.  .  the  r^'term  is  x'""^"^^''*^-'  =  x'"  ^**.  If  hi  be  an  even  number 
the  quotient  will  contain  an  even  number  of  terms,  and  will  be 
j.fli.i+  a;in-34.  j.m-8  ^  4c.  +  x^-C^-i)  +  x^-C^M  +  x"-("'*'*; 
+  &c.  +  x"»  +  i-'™  =  x"»(x-i  +  X-'  +  x-«  +  Ac.  to  xi  ") 
+  x"^+  x'^  +  x"*  +  &c.  10  x^"**  .•.  first  part  of  quotient 
T  second  part  x  x"* 


Mis.  Ex.  32-37.]  ALGEBRA.  157 

32.  (i)  Let  V37  +  20V3  =  V^  +  Vy ;  then  V37-20V3  =  '^x-*Jij 
.-.  V1369  -  1200  =  V169  =  13  =  x  -  y.  Also  37  +  20V3 
=  X  +  y  +  2Vxy  .*.  x  +  y  =  37;  hence  x  =  25,  and  y  =  12. 
Then  V-^  +  Vz/  =  V25  +  V^  =  5  +  2V3 

(ii)  Let  V4x  +  2V4x2  -  I  =  V^'  +  V^  ;  then  V4x  -  2V4x^  -  1 
=  Vx'  -  Vy  •••  V i^6x2  _  I6z2  +  4  =  V4  =  2  =  x'  - y.  Also  4x  + 2^4x2- 1 
=  X'  4-  y  +  2\/Py  .-.  x'  +  y  =  4x  .-.  2x'  =  4x  +  2,  or  x'  =  2x  +  1,  and 
2y  =  4x  -  2  .-.  y  =  2x  -  1.     Then  -Jx'  +  Vl/  =  V^a;  +  1  +  ^/W~l 

33.  (a*  -  X*)''  =  {a*(l  -  c-^x-*)}'*  =  a-'^Xl  -  a-^x'*)"^ 

3  a-4  3-4-5  3-4-5-6 

=  a-i^(l+-a-4x-*+-a-x-+^.2:3<x-^^x-x^+j-^:^^a-iS-i« 

3-4-5-6 

+  &C.)   Hence  5">  term  =  a-^^x——a"^Sx-i*'  =a"^^x  15a"^Sx"^6 

=  I5-Z83.-16 

28  X  27  X  26  X  25  X  24  X  23  X  22 
''■   ^^  =  r-2-3-4-5-67 =  ''''''' 

35.  (X*  -  4x2  +   10   -    12X-2  +   9x-*)^    =    {(x*  -   4x^  +   4) 

+  6x-2(x2-2)  +  9x-<}^  ={(x2-2)2+2  x3x-2(x2-  2)  +  (3x-2)'^|^ 
=  x2-2  +  3x-2 

36.  Letx  =  ^l;  thenx»=l  andx»-l  =  0  .-.  (x- l)(x2+x+ 1)  =  0 
.-.  X  -  1  =  0  orx=  I.  Alsox2  +  x  =  -  1,  whence  x  =  i(-  1  +^-3) 
.-.  ^1  =  1,  or  K  -  1  ±  V^^)-     Also  I'  =1,  and  {K  -  1  +  V^^)P 

=  i( - 1  +  \r^) .-.  1^  +  {i(  - 1  ±  V^)r  =  1  +  K  - 1  +  V^^) 

=  1  +  K~^±  V~3),  i.  e.  sum  of  the  cube  roots  of  unity  =  sum 
of  their  squares. 

37.  (i)  6x  +  ay  =  ai  =  ttx-a2+6y-6^;  ax  -  bx-ay  +  by  =  0"^+ b^, 

a^  +  b^       bfa'  +  b^) 
orx(a-6)-y(a-6)  =  a2+62;  x  =  y+-^-^  .-.     ^_^     +by  +  ay  =  ab 

ba^  +  b^  ba'^+b'  6Va  +  6) 

.:-^^^  +  y(a  +  b)  =  ab;   y^a  +  b)  =  ab  - -^^^^  =  -       ^_^ 

'    -  b\a  +  b)  J2  12  ^^2  ^  j^-i 

=  ~irrb —  •••  y  =  -  ^rrs  =  v:r-a'  ^^^  ^  ^  v  +  -^rv 

""6-a        a-6~         a-6         ~a-6 


158  KEY  TO  [MtB.  Ex.  S7-iii. 

(n)  If  in  these  equations  we  •write  x  for  y,  we  shall  obtain 

values  of  x  and  y,  which  will  simultaaeously  satisfy  the  given 

equations.     Thus  i^  =  6x  +  4x  ;  or  x^  -  lOx  =  0  .•.  x^  -  lOx  is  a 

factor  of  the  reduced  equation  in  x.     Now  from  first  equation, 

x^  -   6x 

;    substitute  this  for  y  in  the  second  equation. 

3(x2  -  6x) 
=  4x  +  -^^ — ■  .-.  X*  -  12x»  +  12X-'  +  80x  =  0. 

But  we  have  shown  that  x^  -  lOx  is  a  factor  of  the  left  hand 

member  of  this  .-.  (x^- I0x)(x2- 2x-8)  =  0.   Hence  x^- 10x  =  0  ; 

whence  x  =  0  or  10.     And  x^  -  2x  =  8  ;    whence  x  =  4  or  -  2. 

Then  x  =  y  =  0,  or  10,  or  -  2,  or  4 

33.  Let  X  =  j-ds  B  sold  for  $1 ;  then  x  +  J  =  yds  ^  sold  for  $1  ; 

1  1 

—  =  what  B  received  for  1  yard,  and  r  =  what  vf  received. 

X  "^        '  x  +  i 

90         40 
Then  —  +         ,•  =  42  ;    whence  21i-  -  5Sx  =  15  .-.  x  =  3,  and 
X        X  +  i  '  ' 

I-  a     7-2       5 

39.  Insert  5  Jl.  means  between  2  and  1 ;  d  = =  „-  -,  =  — . 

'  71-1      7-1       0 

Hence  ^.  series  is  2  +  2-J  +  3}  +  4J  +  5i  +  6j  +  7;  that  is 
2  +  V  +  V  +  «  +  V  +  V  +  '•  Therefore  the  H.  series  is 
i  +  1^7  +  I'r  +  5  +  -x\  +  -if  +  f 

40.  The  /.  c.  m.  of  denominators  -  (a  -  b)(x  -  a){x  -  b)  ;  then 
clearing  of  fractions  (a  +  c)(x  -  6)  -  (i  +  r )(x  -  a)  =  (x  +  r )(a  -  b)  ; 
or  ax  +  ex  —  a6  —  tc  —  6x  —  ex  +  u6  +  ac  =  flx  +  ac  -  ix  —  6f ; 
ax  -  be  -  bx  -v  ac  =  ax  -  be  -  bx  -h  ac.  Therefore  the  given 
expression  is  an  identity. 


41.  ;^x''+  1  =  1  -  V^  •••  a-*  +  1   -    1  -  4Vx  +  Cx  -  4xVx  +  x' 

...  _  4^x  +  6x  -  4xVx  =  0  .-.  2v/x(2x  -  S^x  +  2)  =  0  .-.  2Vx  =  0, 

whence  x  =  0  . — -^  . 

/3iV-7\*  — 

Also  2x  -  3Vx  =  - 2,  whence  X  =  f ^ j    =  J(l  i  Sv'-T) 

42.  at  -  (lA  +  6c  -  Sac  -  ui  +  2ac  -  2bc)  =  ab  -t- be  +  ac 


Jils.  fes.  4S-46.]  AtQEBRA.  1^9 

43.  ix2  -  Ixy  -  -,3o!/2  -  mx  +  nij  +  ix'  +  xy  -  -j^^-j/^  +  px  -  qy 
-  \W'  +  ixy  -  II  f  +  (P-  m)x  +  (71  -  q)y 

And  |a:^  -  txy  -  -^^-y'^  -  mx  +  ny  -  Jz^  -  xy  +  -^^y^  -  px  +  qy 

=  iV^^  -  3^  y  -  liViy  -  ("1  +  p)^  +  («  +  5)y 

(X  -  8)Cx  +  6)(r  +  7j(x  -  4)       x  +  7 


(X -4)(x  +  6;(x  -  8)(x  + 5)       x  +  5 

X  a  6  1 


■(x-a)(x-6)       (a-6)(x-a)       (a-6)(x-6)  a  -  6 

.-.  /.  c.  m.  df  denominators  =  (x  -  a)(x  -  6)(a  -  b).  Hence 

z(a  -  b)  -  a(x  —'b)  +  b(x  -  a)  =  (x  —  a)(x  -  6)  ;  that  is 

ax  -  bx  -^  ab  -  ax  +  bx  -  ab  =  x^  —  ax  -  bx  +  ab ;  that  is 

/a  +  iX''      /a  +  b\» 
x^  -  (a  +  6)x  =  -  aJ  .-.  x^  -  (a  +  b)x  +  (  -^  I    =  I  — —  1    -  ab 

u'^-2ab  +  b^              a^-2ab  +  b^  a+  b        a-  b 

=  T- — ab  = .-.  X  -  — r —  =  +  — - — ,  whence 

4  »  ^  i        ' 

X  =  a  or  6 

(III)  Multiplying  by  1G8 

168  +  63x  -  48x  +  8  =  186  .-.  15x  =  10  .'.  x  =  § 

46.  Multiplying  by  (x  -  y),  (y— 2)  and  (r  -  x)  respectively, 

■we  have 

a;»  -  y^  =  37(x  -  y)\ 

^  -  z"  =  28(y   -  z)  S    .-.   by  addition    18x  -  9y  -  9c  =   0 

2»  -  x»  =   I9(z  -  x)  ) 

2x  -  y  =  5 ;    substituting  this   in   third   given   equation, 

(2x-y)'*  +  x(2x-y) +x2=  19  /.  7x*-  5xy  +  y^  =  19  j   subtract 

from  this  the  first  equation,  and  we  have  6x^  -  6xy  =  -  18 

x2  +  3 
.•.  y  = ;  substitute  this  in  the  first  given  equation,  and  we 

have  J:'+  f-^j-c  +  ir^r)    =  ^'+  ^'+  3  +  i—^J    =  37 

clearing  of  fractions  ;  2x*  +  9  +  9x*  +  x*  =  37x2  .-,  3x*  -  28x2  ^  _  9^ 

^2^3       9  +  3 
■whence  x^  =  9  or  i  .-.  x  =  f  3  or  i  J  V3  i  y  =  ~_~  =  —  -3-  =  i  4, 


160  KEY  TO  [MiB.  Ex.  46-43. 

i  +  3  10  ^^         ±  10V3 


+  JV3       ±  V3       ±  V3  3 


;    i=2x-.y  =  ±6?4  =  i2,  , 


3 

or  -  +  §V3  +  'N^  =  T  ^/3 

47.  The  question  amounts  to  finding  the  least  real  ralue  of  r, 
which  will  satisfy  the  given  equation 

bV  -  2abx  =  m  -  a^6^  -  2u'^b  -  2a},  where  m  represents  the  least 

2a         m-  a^b'  -  2a^b  -  2a? 


value  which  makes  x  rational ;   x^  — r-x  = 
.      2a       a«       a^->rm-  a^i^  -  20^6  -  2a^       m  -  a^b^  -  2a^b  -  a* 


6^ 


a        \-^in-  a2(6^  +  26  +  1 )                 a  ±  V"  -  «*(''  +  I Y 
.-.  X  -  -^  = ^ .-.  X  =  ^ 

Therefore  the  least  value  of  m  that  will  render  x  rational,  is 

m  =  0^(6  +  1)'^,  and   .-.  the  least  possible  value  of  the  given 

a 
expression  is  found  when  ^  =  -r^  and  is  therefore  a*(6  +  1)* 

48.  {(x"?   +    6    +    9x  -  SP)    -    4xP(i«i'  +    3x " »'')    +    4x2?}* 

=    {(x^^    +    3x  -  'P)^    -    2    X    2xJ'(x»P    +    3x "  »^)    +    (2xP)^j* 
=    x'P    +    3x-sp    _    2xP 

49.  (I)  S  =  {2a  +  (rt  -  l)(f}  Y  =  {Gf  +  (8  -  1)^"-}?  =  (6?  +  20)4 
=  4x  26f  =  107? 

r  /256x- i6y8x -|  /  25Gx-'6y'\ 


243x1*  - 


1   +   jx-^y  Sx''  +   2y 

25Gx-^y* 

27  6561x"-  256x-=y» 


3x*  +  2y  ~  81x*  +  54y 

.     ^    o                u         243x"{l  -  (  -  U-'yr  \     243(1  -  (  -  j )''  { 
(II.)  S^  =  as  above ^p-:^^ .= 3-^-^^ 

243  X  1       243 

=  60i 


4 


Mi8.  Ex.  50-56.J  ALGEBRA. 

a(r^-  1)  ar^(r5-l) 

50. —  =  sum  of  first  three  terms,  ana  — „ — , —  -  siilU 

J-  -  1  '  r  -  1 

ra(r^-l)-]        arVrS  -  1) 
of  next  six.    Then  72.^ TIT"  f    ~  — r^ — ,  or  dividing  each 

by  '- — ~,  we  get  72  -  r\r^  +  1),  whence  r^  =  8  or  -  9  .-.  r  =  2 

.•.  any  series  having  r  =  2  will  answer, 
g  I      ^mn  -  jnp  +  np  -  mn  +  my  -  np  _  j.  o  —  ■^ 

(X*  +  2x^  +    1)  +  x(x'  +    1)    _    (^'^  +    1)'  +  x(x^  +    1) 
^^-  (X*  +  2x-2  +    1)  -  a:(z^  +    1)    ""    (x^  +    1)^  -   x(j:^  +    1) 
(a-2+ IXx'^  +  x  +  1)       x^  +  x+l 
*"  (x^+  l)(x='-x+  1)  ~  P  -  X  +  1 

53.  x''-2(a  +  6)x  =  3a^  -  10a6  +  36^;  complete  the  square 
x^  -  2(a  +  b)x  +  (a  +  b)'  =  Sa^  _  I0o6  +  36^  +  a*  +  2a6  +  b^ 
=  4a2  -  8a6  +  4b^  =  4(a  -  b)'^  .-.  x  -  (a  +  6)  =  ±  2(a  -  6)  ;  or 
X  =  a  +  b  ±  2(a  -  6)  =  3a  -  6  or  36  -  a 


54.  Because  'Jy-x  :  V^O  -x  ::  2  :  2  .-.  '^y  -x  =  V-0  -x 
.*.  1/  -  X  =  20  -  X  ;  hence  y  =  20.  Then  from  first  given  equation, 
Vj/  -  V20  -  X  =  V^O"-^  •■•  Vy  =  2V20  -  X,  and  hence  y  =  80  -  4x, 
but  y  =  20  .-.  20  =  80  -  4x,  or  4x  =  60,  and  x  =  15,  and  y  =  20 

55.  {(x*  +  1)  -  2x}{(x'*  +  1)  +  2x}  +  2(x2  +  2x  +  1) 
=  (x2+l)2-4x='+  2(x2+2x+  1)  =  x*+2x^+  1  -  4x''^  +  2x2  +  4x  +  2 
=  x*+4x  +  3; 

(X*  +   2x^y^  +  tf){x*^  -   2x2y«    +  f)   -   2i/3(x*  -   2xV    +  y^) 

=  (x'  +  y^)^{x^  -  yh^  +  sy'Cx*  -  2x7^  +  f) 

=  (X*  -  y^f  +  2x*f  -  4xy^  +  2y6 

=  X*  -  2x'*y*  +  J/^  +  2x*y'  -  4xy*  +  2y^  =  x*  -  4xy'^  +  3j/^ 

56.  Multiply  2°'^  given  equation  by  3,  then  3x^ij  +  3x^^  =  36^. 
Add  thiS  to  1*'  given  equation,  and  we  get 

x»  +    "ix'^y  +   3x!/2  +  3/»  =   a*  +   36*   .-.   x   +  y   =   ^i?'T~3P. 

6» 


But  xy(x  +  3/)  =  i'  .-.  iy(Vtt*  +  36')  =  6*  .-.  xy  = 


^a»T36^ 


1^2                                        KEY  TO  mxB.  Ex.  b&-bd. 

Then   x''  +   2xy  +  y^  =    (a?  +   36')',   and   Axy  -   -. 

.-.  x'  -  2xy  +  y^=  (a'  +  36^)' r 

(a3  + 363)4 


Tlien  X  -  y  =  +  V  (a«  +  36^)^  (l  -  ^f^a) 


3-  -  y  =  +  («'  +  3&») 

x  +  y  =  (a'+  36^)' 


Va'  +  36« 


1  ^  /,(^-    63-v 


35 

57.  Let  X  =  number  at  first,  then  —  =  what  each  had  to  pav  , 

but  two  left,  therefore  the  number  remainining  =  z  -  2,  and  con- 

35  35         35 

consequently =  what   each  paid.     Hence  ;:  =  —  +  2 

.-.  x"*  -  2x  =  35  ;  or  X-  -  2x  +  1  =  36  .-.  X  -  1  =  +  6,  and  x  =  7 

58.  (a^  +  b^)*  =  a-(l  +  a' h^)* 

T  a^{\  +  ja'h^  +  ^rrXa'h^y  +  ^0^3^"*''^)^  ■*"  **^- 

4-3-2  /    -}i\3  as  13 

Hence  4"'  term  =  a-  x  r7T77,\a    '/)'  )     =  a-  x  Aa    -b*  =  4a*6* 


59.  (  -  2a*x*)'^  =  4a*x''     -  2a=x*  x  -  3ax'  x  2  =  I2ah^ 

-  Jox*  X  iax^  X  2  =  -  Ja^x'     -   x'    x    Jax*   x   2  =  -  ax" 

ax   X  -  3ax'  x  2  =  -  6a*x»  (l^o*  -  a)-r" 

=  -  ^^d^x*         .-.  coef.  ofx"  =  (12a'-a) 

(4a*  -  \^d')x*  .-.  coef.  of  x«  =  (4<r«  -  \'a*) 


Mis.  Ex.  60-64.]  ALGEBRA. 


163 


60.  Let  X  and  y  be  the  numbers,  x  being  the  greater;    then 

X  ■  y  ::  X  +  y  :  a,  and  x  :  y  ::  x  -  y  :  b  .-.  y{x  +  y)  =  ax, 

and  y(x  —  y)  =  bx 

xy  +  y^  =  ax 

xy  -  y^  =  bx 

2xy       =  ax  -t  bx  ;  or  dividing  by  2x,  we  have  y  =  i(a  +  b) 

If        2(a  +  6)^      (g  +  fc)' 
If  ^ax-bx  --  {a  -  b)x  .-.  X  =  ^^—^  =   ^^^  _  j)    =  2(a_6)  , 

1  4x  -  3         30x  -  23 

61-  (0  2  +         ^37  =  2  ^13x-10  =  i3x-  10 

/x^+lN  /x^  +  2   +  x-^\   _    /x^  +   1\  /(x   +  X   1)^ 

("^  (^x^  -  V  V    ^'  +  ^'*    y  ~  v^' -  V  V  ^■'  +  ^'' 

x^+l\/       x  +  x"^       N  x^  +  x+x  +  x"^ 


E^    -     \j\x'   -     \     +    X-'^j  X*   -    X^   +    1    -    X-   +    1     -    X- 

x^+2x  +  x-i^  x3+2x'  +  x  x(x''*4-l)=' 


X*  -  2x^  +  2  -  X-  ^       x^  -  2x*  T  2x^  -  5  ~  xS  -  2x*  +2x^-1 

x  +  6  x-4  x  +  2 

^^-    (X  +   i){x  -  5)  ■*"  (X  +   7)(x  +  3)  ~  (X  -   5)(x  +   3) 
.•.  I.  c.  m.  of  denom.  =  (x  +  7)(x  -  5)(x  +  3)  ;    then  reducing  to 

(X  +  6)(x  +  3)  +  (X  -  4)(x  _  5)  -  (X  +  2)(x  +  7) 
common  denom.  = (x  +  7)(x  -  5)(x  +  3)  

x2+9r  +  18  +x^-9x  +  20-x2-9x-14  x^  -  9x  +  24 


x''^  5x2- 29x-  105  x='+5x''-29x- 105 

(II)  {(X*  -  2x3  +  a:2)  +  1  (a;2  _  x)  +  .j\.}^ 
=  {(X'^  -  x)2  +  2  X  J(x2  -  x)  +  (i)2}^  =  x2  -  X  +  i 

64.  (x™  -   2y")(x™  -  ]/")    =    x-"*  -   2x™j/'*  -  x™^''  +    2y2« 
=  x^  -  3x"*i/"  +  2i/'^'' ; 

jx'"^  +   (ax"»  -  i^fx""^  -   (ax''  -  6)}  =   (x^^)     -  (ax'"  -  6)* 
=  x""^  -  a^x^^  +  2af>x'"  -  6» 


164 


b:ey  to 


[Mis.  Ex.  65-71. 


65.  (i)   12(x*   -   2*)    -f    3(x    -    2)    ^    4(x»   +   2x^  +   4a:    +    8) 
=  4i'+  8x2+  igj.  ^  32 

(II)  4  -  2  +  1)20  -22  +  11-3(5-3  .-.  quotient  =  ^aW  -  3ai* 
20  -  10  +    5 


-  12  +    6-3 
-12+    6-3 


d ;    then    a  :   d   :\ 


b\ 


A       a       b       c  _„^  ____ 

ei       h       r       ri '  bed        a       b       b       0      b 


66.  Let    a  :  h  ::  b  :  c  :.  c 
be  a       b 

d  ~  b  ^  c       d 
.-.  a  :  d  ::  a"  :  6* 

67.  X*  <  lOx  -  16,  or  x^  -  lOx  <  -  16,  or  x«  -  lOx  +  25  <  9, 
or  X  -  5  <  +  3,  or  X  <  8  or  >  2  .-.  values  are  3,  4,  5,  6  and  7 ;  or 
thus(-x2+10x-16)>0i  -(x2-10x  +  16)>0;  -(x-2)(x-8)>0, 
.'.  X  -  8  must  be  negative,  .*.  x  <  8  and  x  -  2  must  be  positive, 
.-.  x>  2 


26 


26 
20 


1 


Vx  -5 


68.  By   Art.    106,    (vii)  

4Vx  -  5 

.-.  5  =  i^x  -  5  ;  or  X  -  5  =  25  .-.  X  =  30 

a       b 

69.  -7-  +  —  >  2  ;   if  a''  +  6^  >  2ab,  but  a-  +  b^  is  greater  than 

2ab  by  Art.  134,  Note  2,  .-.  y  +  —  >  2 

70.  Let  71  -  I,  n  and  n  +  1  be  the  three  numbers  ; 
then  (n  -  1)*  +  n^  +  (n  +  1)'  =  the  sum  of  their  cubes 
=  n»  -  3n^  +  3rt  -  1  +  n»  +  7i«  +  3n*  +  3n  +  1  =  3n»  +  6n  =  3n(n^  +  2) 
which  is  evidently  divisible  by  3n,  i.  e.  by  three  times  the 
middle  number. 


71. 


1 

-  5 
+  4 


1  +  0-4+       7+       0 

-  5  +  25  -  125  +  690 

+     4  -     20+100 


I  -  5  +  25  -  138  +  790 

.-.  quotient  =  a*  -  5a'  +  25a*  -  138a  +  790  -  ^^V",     ^\- 
^  u*  +  oa  —  4 


5  +    6 

-  3950 

-  552  +  3160 

-  451)7  +  3166 
4507a  -  3166 


Mxs.  Ex.  n-73.]  ALGERRA.  165 

(II)  (x^  -  x-^f   ^    (X   -  x-^f   =    (x2  +    1    +   X-  ^f 
=  x*+  2x2+  3  +  2x-2  +  X-* 

72.    It    is    evident    that    the    (n    -    ly^    factor    of 
(x^  +  (/^)(x'  +  a')   &c.,   that  is  the  n"^  factor  of  the  given 
series   is   x*^'-^        +  aS^''      ,  and  that  the  term  before  this  is 
x^^^       +  a^^^      ,  and  so  on.     Hence  the  series  is  the  same  as 

(x^  +  a').     Now   the   product  of  the   first   two   terms 

=  x^''^       -  aS^''       ;  and  the  product  of  this  by  the  third  factor 

=  x^*'         -  a^^''       ,  and  so  on.     Hence  the  product  of  the 

first  71-1  factors  will  be  (,x*  -  a   ),  and  the  product  of  this 

by  the  n"'  factor  \x^  +  a^ )  will  be  x  -  a  which  .•.  -  the  pro- 
duct of  the  given  factors. 

Note. — All  this  will  be  perhaps  more  evident  to  the  student,  if  he 
takes  a  numerical  example,  and  examines  how  the  indices  are  affected  by 
rxj'iltiplying  them  as  in  the  above  question.    Thus  suppose  n  =  5. 

Then{x(^)*     '-cSir-']{x(^y'\a^'^^'-'\ 
.  (x(-i)*  -  a(^^)')(x(i)*  +  a(^)0  =  (x'V  -  J^){x^^  +  a^'O 
-(r'-a^)=x(i)'-a(^)'=   (x^^)" '^  -  a<i)' -') 

13(2x  +  3)  -  7(2x  -  3)         x-4  12x  +  60  x-4 

^■^'  12(4x--  9)  ~  4x^  +  9  ^  12(4x2^^  ~  -ix^ Td 

x  +  5  x-4         4x'+ 20x2+9x +  45 -4x'+ 16x2+ 9x- 16 

^  4x»^  ~  4x2+9  ~  16^4-81  ' 

30X-+  18x  +  9 
~        16x*-81 


166  KEY    TO  [Mis.  Ex.  74-7?. 

74.  (I)  {(ix2  +  p/)  4-  \xy\\ax^  +ly^).  ixy]  =  (ix»  +  |y^'  -  J zV 
=  A^*  +  k'i/'  +  -.W  -  h^Y  =  tV^*  +  -8*f!/* 

(")   (2xi    +    3yi){2x^   -    3j,i)  {  (4xi    +    9y0    +   exV } 
{  (4x^   +  9i/^)   -  6x^y^\ 

^  (-ix^  -  9i/0(l6x  +  I2x^y^  +  Sly  -  36x^y^) 
=  (4x^  -  9i/^)(l6x  +  36x^y^  +  8I3/) 
^  64x^  +  144x1/^   +  324x^y  -  144xy^  -  324x^y  -   Y29y^ 


a 


=  64x^  -  129y^ 

75.  Multiplying  first  equation  bj  V^,  and  second  by  ^3,  w-e 

have  2xV6  -  6j/  ^  6V2  (i) ;    3xV6  -  6y   =    5VT8   =    15V2  (11). 

9V2       9VT2 
Subtracting  (1)  from  (n)  x^S  -  !)V2  .•.  x  =  — ,  -   =   —^  -  3V3 ; 

then   2xV3   -   3i/V2    =    6V3   x   V3   -    3i/v'2    =    18   -   3yV2   ^  6 

12 
.-.  3yV2  =  12  and  .-.  y  =  ^,.j  =  V2 


3V2 


76.  .S'n  of  1  +  3  +  5,  &c.,  =  j2  +  (tt-l)2j—  =  (2  +  2n  -  2)—  =  n^ 


Jti  n 

5   to   J/i  terms  =  {2  +  (i/i  -  1)2}  ^  =  (2  +  n  -  2)—  =  in^; 

then  sum  of  last  half  of  the  series  =  n'  -  —  =  Jn*  =  3  times  in' 

a^  +  2ab  +  b'       (a  +  b)' 
77.  The  ^1.  mean  is -t =   — -r — ,  and  the  H.  mean 

a^  -  2ab  +  b^       (a  -  by  

is r =  r —  ;    then  Art.  262,  the  G.  mean  =  -J.IH 

ab  ab      '  '  ' 


-V 


(a  +  by^(a  -  b)'       (a  +  b)(a-b)       a-"  -  ft*        a         b 


a^b^  ~  ab  ab       ~    b         a 

2ac                    2ac 
78.    By  Art.   260,  it  appears  that    H  =  -    —  .-.  6  = , 

substitute  this  for  6 

a+  c               1                        1                    n  +  r  a  +  c 

Then  — -—  =    o...  +    ».,.  =  -7^ -;  + 


ac  2ac  2ac  a{c  -a)       c{a  -  c) 

a+  c  ~  a  +  c  ~  '^ 


iiis.  Ex.  78-81.1  ALGEBRA.  l*j* 


a^  —  a*  +  ac  —  c' 


+  - 


*  *  ac       a{c  -  a)       c(a  -  c)       ac(^c  -  a)(^a  -  r ) 

2ac  -  a^  -  c^ 

.-.    1  =  7 TT r   .-.    2ac  -  a?  ~  c^  -  lac  -  a^  -  c^     Now 

(c-a){a-c) 

reversing  the  steps  of  this  operation,  we  Bhall  have  proved 

the  point  required. 

X  inx 

T9.  V  =  r  -^  s  +  t,  and  s  =  m — ,  and  t  =  nxy^  .-.  v  =  j-  +  —  +  nxy^ 

0  =  r  +  771  +  7l         \ 

8  =  r  +  7?i  +  2Yra  \  From  these  equations  r  =  1,  m  =  -  |5,  and  n-  -^ 
l:=r  +  0  +  0       ) 

jnx  X 

••■  v  =  >•  +  y  +  «^2/'  -  ^  ~  '^7  "^  ^'^''^' 

80.  {(a  +  6)x  +  (a  -  6)}{(a  +  6)x  -  (a  -  6)}  =  (a  +  6)^x2  -  (a  -  6)^ 
=  4a6  .-.  («  +  6)^x2  =  Ub  +  (a  -  6)^  =  (a  +  6)^  .-.  x^  =  1  .-.  a;  =  +  1 

a^x^  -  b'^x  -  abx^  +  b'^ ;  d'x^  -  abx'^  -  b'^x 
b^  6'^ 


(") 

ax 

y 

b 

-  —  =  X  + 
a 

b 
ax 

b^; 

(a:- 

2  -  ab)x^  ■ 
¥x 

-  bH 

'  «(a  -  b) 

6* b^ 

a(a  -  b)    '^    4a2(a  -   6)^    ""    4a^(a   -  6)^ 
6*  4a62(a  -  6)       b*  +  4a%'  -  4a6' 


+ 


X  - 


a  (a 

62 

6)' 

«(a 

- 

t) 

62 

4a^(a-6)2  ^  4a2(a- 6)^  4a^(a  -  6)^        '  '  "^       2(/(a-6) 

+  6V^^+  4a*-  4«6  6(6  +  V62  +  4a^  -  4a6 

~      2a (a  -  6)  •  •  ^  -  2a(tt  -  6) 

81.  Product  of  first  two  factors  =  a'^  -  l^-  hence  product  of 
first  three  factors  =  o*  -  6*,  aud  product  of  first  four  facto-s 
=  ft*  —  6*.  Hence  it  is  evident  that  the  exponent  of  a  or  of  6  in 
first  term  is  2",  in  product  of  fiist  two  terms  2^,  iu  product  of 
first  three  terms  2*,  of  four  factors  2',  of  five  2*,  and  so  on  ;  hence 
the  exponent  in  the  product  of  first  n  factors  will  be  2""^,  ami 
of  the  scries  to  n  +  1  factors,  the  exponent  will  he  2".  Hence 
the  required  continued  product  is  a^"  _  i-n 


1G8  KEY   TO  [Mis  Ex.  Si -'7 

82.  I  I  -  (a  +  b  +  p)  +  (ap  +  bp  -  c  +  q)  -  (ug  +  bq  -  cp)  -  qt 
+  )'\    +         p  -  (ap  +  hp)  -  cp 

-  <A  -      q  +  ("?  +  ''9)       +  9c 

1  -  (a  +  6)         -         c                    ~+  0                 +10 
Hence  quotient  =  x^  -  (a  +  b)x  -  c 

83.  {{u?x^   +    2n6x*  +    ^2^.2)    +    (^2acx^   +  26c)    +    c^x " 'j^ 
=  {(«x^  +  6x)2  4-  2  X  ex  - 1(01^  +  ix)  +  (ex  "  ^)y  =  ax*  +  6x  +  ex "  ^ 

(x  -  o)(x  +  6)  +  (x  +  a)(x  -  6)     (x  -  fl)(x  -6)  +  (x  +  a)(x  +  /-) 
^^-  Jx^  -  d')(x' -  b')  "  (^2  -  a'){x^  -  b-) 

x^-ffx  +  ix— a6+x*+ax-Jx-a6  (x^  —  a*}(x*—  6^) 


(x^  -  a^){x^  -  6^)  x^-ax-6x  +  aA  +  x^+ax+6x  +  u6 

2x2  _  2ab     x'  -  ab 


~  2x2  ^  2a6  '  x'^  +  ab 

85.  G.C.M.  of  (x2  +  px  +  ;)2)(j,.»  -  px  +  j)^),  and 
(x''  4-  px  +  p^)(x^  +  px  -  p^)  is  evidently  x^  +  px  +  /)-'. 
Otherwise  by  ordinary  rule,  thus  : 

X*  +  7)2x2  +  /)*)X*  +  2/)X'  +  J)2x2  _  p*(\  x2  +  px  +  p2)x»  -p3(^x  -  p 

X'*  +  p2x2+p*  x'  +  px'  +  p^X 

2px''-p*=2p(x'*-p3)  -px'-^x-p^ 

x^~p''')x*+p''x'^+p*(^x  -px^-p^x-p^ 

x*—p^x 

p^x'+p^x  + p*  =  P'{x-  +  px+p^  .-.  G.C.M.  =  x^-hpx  Jr  p- 

86.  5(x  +  5)(x  -  4);  V(x  -  6)(x  +  5);  and  -^(x  -  6)(x  -  4V 
Hence  /.cm.  =  ^g"(x  +  5)(x  -  4)(x  -  6)  =  33*(x*  -  Bx*  -  26x  +  12r) 


87. 


2(fl6  +  1)      _    1  -  6*        3       2(a6  +  1)      /ab  +  \y 
''  ~  ~a2^ri~^  =     a2-l  ■'    ^  a2-l     "^  V"*^^^  j 


fab  +  IV 


1  -  62       a262  +  2a6  +  1  +  n»  -  fl2i^  -  1  +  6' 
■*"  Ij^^l    ^  (a2-  1)* 

a2  +  2o6  +  /)2  flft  +  1         a  +  6  aft  +  1  ±  (a  +  />) 

"~(a2^r)2~  •■•  ^  -  "u2~T  "  i  a2^rT'  ^""^  ^  ""  i»  ^1 

(ab  +  b)  +  (a+l)       6(a+ 1)  +  (fl+ 1)       (ft+I)(a+l)      6  +  1 
(a  +  l")(a  -  1)     ^  (a-l)(a+l)  "  0^1 '' 
(6  -  l)(a-  1)  _  6-1 
a2-  \  ^  (a+  l)(a-  1)  ~  a  +  l 


Mis.  Ex.  88-90.]  ALGEBRa.  169 

88.  Multiply  both  numerator  and   denominator  of  the  first 
factor  by  x ;  then 

T*4-X-2+2(x2+l)        /x2-iy  _X-''^(X6+1)  +  2(X2  4-1)        /X2-1N 

r4_a;-2_2(x^_l)  ■  \^^Hl)    ""  x-^(x6-l)-2(x=«-l)  *  \^1 
X-  2(x*  -  x2  4- 1)  +  2\  /x^-lX       /x^-  1  +X-2  +  2\  /x^-1 


(x*  +  x2  +  1)  -  2y  \^x2+  ly      \^x^  +  1  -  x-2  -  2 J  \^x^+  1 

/x2  +  1  +  X  -  ^   /X2  -     1\  /X*  +  X=«  +    1\    /x2  -     1\  x6    -    1 

'  l^x^  -  1  +  X  -  y  l^x^  +  ly  ""  (^i*  -  x^  +  1 J  (^x^  +  \j  ""  x6  +  i 

71-1 

89.  Let  ra  represent  any  square  number;    then  — r—  will  be 

n+  1 
half  the  next  lower  number,  and  — —  will  be  half  the  next  higher. 

/7i-l\=*  n^-2n+l      47i  +  n2-2n+l      n'^+2n+l 

Then  n  +  I  —  )    =n  + 

2 

90.  Let  X,  y  and  z  represent  the  number  of  hours  taken  by  A, 
B  and  C  respectively  to  fill  or  empty  the  cistern  ;  consequently, 

1  1 

in  1  hour  A  will  fill  — th  of  it,  B,  — th  of  it,  and  C  will  empty 
X  y 

1 

— th  of  it. 


/I        1\        1 
Then  3    —  -  —     +  —  =  1 
V  X        z  J       2x 


5 —     +  —  =  1  >i.e^  once  the  contents  of  the  cistern 

\y       '^  J      ^>J 

5       1 

Jx'^  2y  ~^ 

7        3  27      5  5        1 

H'^'^^^  2~^  -  T  =  ^  5  ^  -  T  =  ^'  ^^'1  3:^  +  2^  =  '  • 
Multiplying  the  &rst  of  these  by  %,  and  the  third  by  \ 
35       5  _  ^  ^ 

6x~T~  3     f  7        5        7        5       11 

35       7       2_[     '''^^~z~~2~~E~~Q 

Also  H  -  4-  =1 


•••  4^  =  il  "  V  i  whence  -,V  =  ^,  or  y  =  3 


170 

KEY   TO                             [Mir.  Ex.  90-9P 

27      5       27       5 

Then---  —  =  7^ 

4y      2       12       z 

5      27             15         1        1 

=  '--T=i2-^  =  i2°^7  =  T--^  =  '' 

7        3        7        3 

7       7            11 

91. 

2x*  +  2x4  _  5^^  +  4j'  -  9 
3 

3x*-»-3r*-10a~'-x  +  3^ 

vei"  +6x*-  15i^+  12x^-27(2x 

J 

/  6x»  +  6x*  -  20x»  -    2x»  +    6x 

5x^4- 14x^-    6X-27 
8x<  +  3x»  -  lOx'  -  X  +  3(3x  -  27 


t3?  +  14x2  _  g_c . 


15x*+  15x^- 
15x4  +  42i»- 

-  50j=«  -  5x  +  15 
-18x»-81x 

-27x»- 
5 

-  32x^  +  76x  +  15 

-  135x^ - 
-135x^- 

160j2  +  380x  +  75 
378x2+  162X+  729 

a:2  +  i-3\5x3+  14x2  -  6j  -  27(5x  +  9 
5x»  +  5x=«-  15x 

9x^+9x  -  27 
9x2+  9j;_27 


218x2+  218X-654 
=  218(x='  +  x-3) 


92.  /.   c.   m.   of  (ax  +  6)(/'x   +  9),   and   (ax  +  6)(5x   -  -p) 
=  (ox  +  6)(px  +  y)(?x  -  p) 

=  apq3?  +  (o}'*  +  ftjjg  -  ap'^)x^  -  (a/>5  +  bp'^  -  bq^)x  -  bpq 

I.  cm.  of  x(x  -  y);    (x*  +!/*)(■«:-  yVi   a°d  y(*  +  V), 

thatisofx(x*  -  »/*)(->^^  +  ^^y^  +  2/^);  (•r*  +  yO(^*-  yO, 

and    y(x*    +   2/*)(x*   -  x*y*    +    1/*)    ^    x(x    -    t/)(x    +    i/)y 
=  xy(x^  -  y^)  =  x^y  -  xf 

93.  (i)  Multiplying  by  273,  we  have 

91x-  182a  =  78x  +  234a  -  21x  -  42a;   or  34j  =  374<J,  whence 
%  =  lla 


.Mis.  Ex.  93-97.]  ALGEBRA.  171 

(ii)  Reducing  first  member,  and   also  the  second  member, 

^■"Sx  —  5  11 

—^  =  -2—  ;    or  dividing  by  r  -  5,  we  have  —  =  ^2  _  ^ 

whence  x'^  -  I  =  6,  or  x'^  =  1 ,  or  x  =  i  ^J^ 

(ill)  Squaring  each  side,  x  4-  4  +  2^2x^  +  14x  +  24  +  2a;  +  6 
=  3x  +  34  .-.  2V2x^  +  14j;  +  24  =  24  ;  or  ^2x'^  +  lix  +  34  =  12 
.-.  2x2  +  14^  +  24  =  144,  or  x^  +  ^x  =  60  ;  x^  +  1x  +  {If  =  60  +  V- 
-  2$^  .-.  X  +  5  =  ±  ^^  .-.  X  =  5  or  -  12 

(iv)  x'^y   -    x'^  +    3x^^    -    3i/    =    V-*-'^  +    3i/,    but    x^y    =    ^ 

.-.  5  -  x2+  15  -  31/  =  'J3pl~3y,  or  20  -  (x^  +  3j/)  =  ^/x^  +  3y 

.-.  (x2  +  3y)  +  Vi-^  +  31/  =  20  .-.  (x2  +  3y)  +  (x^  +  3y)^  +  i  =  ^ 

.-.  (x2+  3i/)'  -  i  I  -  ^  =  4  or-  5  ;  squaring-  these,  x-+  3y  =  16  or  25. 

5  5 

But  since  x^y  =  5,  x'^  =  —  .-.  —  +  By  =  16  or  25.     Hence  3y'-  I6y 

=  -  5  ;  or  3y'^  -  25y  =  -  5.     From  first  of  these  equat.  ?/  =  5  or  J. 
Hence  x  =  1  or  ±  V^^ 

94.  (X  -  2)(x  -  3)(x  +  2  -  V  -  3}(x  +  2  +  V  -  3) 
r  (x2  -  5x  +  6)(x2  +  4r  +  7)  =  x*  -  x^  -  Tx^  -  9x  +  42  =  0 

95.  a-a  +  7n4-a-m  +  ni-a  =  wi 

96.  a»  +  69  =  a«  +  2a*6*  +  6«  -  2a*64  =  (a*  4-  b*f  -  (a^by2y 

=  (rt*  +  £■*  +  a26V2)(a*  +  6*  -  a''by2) 

2nb 

97.  Since  Art.  260,  J.=  l{a  +  b);   H=  —-7,  and  G  -  V^f  ^9 

have  by  substituting  these  values  for  ^,  H  and  G 
2ab 


=  1  +  v«  +  ^      /  ya  +  6      y 


T^.  =  1  + 


aft 
(2«6  -  rt2  -  ab)(2ah  -  b' -  ab) 


(a  +6)''  a6(a  +  bf 

4ab       _  ab(a  +  b)^  +  (ab  -  a^^ab  -  b-) 
(a  +  by  ""  ab{a  +  6)^ 

ai(a  +  6)2  +  a(6  -  aXa  -  b)b      db(a  +  6)^  +  a6(6  -  a)(fl. -  6) 
06  ~  06 


•,  4a6  = 


172  KEY  TO  [Mis.  Ex.  97-K"l 

.-.  Aab  =  (a  +  b)'  +  (6  -  aXa  -  b)  =  a?  +  2ab  +  b' +  ab  -  b^  -  a-+ ub 
or  4ab  -  4ub 

98.  Let  X  =  minute  divisions  the  hour  hand  passes  over;  then 
12x  -  divisions  passed  over  by  minute  hand.  Also  60  +  x 
^  minute  divisions  passed  over  by  minute  hand  between  two 
successive  transits  .-.  12z  =  60  +  x ;  or  llx  =  60  .•.  x  =  5,-',- 
=  minute  divisions  passed  over  by  hour  hand,  hence  time  in 
minutes  =  5-,*,-  x  12  =  1  h.  5/i  m. 

99.  Let  X  and  y  =  sides  of  rectangle  ;  then  xy  -  srca 
(x  +  «)(!/  -  b)  =  xy  +  at/  —  bx  —  ba  =  xy 

(X  +  c)(y  -  d)  =  xy  +  cy  -  dx  -  cd  =  xy  -  e 

ay  —  bx  -  ah       "]         ady  —  bdx  =  abd 


}ady  —  bdx  - 
bey  -  bdx  = 


cy  —  dx  -  cd  —  ej        bey  —  bdx  =  bed  —  be 


(be  -  ad)y  =  bed  -  be  -  abd 


b(cd  —  e  —  ad) 
whence  y  = h    —   t1 

Also  acy  —  bcx  ==  abc 

acy  —  adx  =  acd  —  ae 


whence  x 


{be  —  ad)x  =  acd  -  ae  —  abc 
a(cd  —  e  —  be) 


be  —  ad 

If  ad  ~  be,  and  be  +  e  =  cd 

a\cd  -  (be  +  e)]       axO        0 
Then  X  =  bc^^Hd ~  ~0~  ~  "o'  ~  ^"^  value  whftTcvor 

b(cd  -  e  -  ad)      b(ed  —  e  -  be) 
Also   y  =  ^^zr^ =  b7:rid ;    S'^^e  ad  ^  be 

b\ed  -  (e  +  be)}  _  bxO  _    0_ 
~         be  -  ad         ~      0      ~    0 


a- by  3(a-b)  x-b  a-b 

100.(1-^—6]   =^-7^r2bra=^-_     a  -  h'^°' T^b '''''' ^ 

Then  (1  -]/)'=  1  -  Y^  ■    or  1  -  3y  +  3/  -  y»  =  1  -  ^^ 


Sli3.  Ex.  100-102.J 


AtGEBRA. 


173 


.-.  -  3  +  3y  -  y'  -  -  1^^  .-.  -  3  +  3]/  -  y2  -  3y  +  Zf  -  7/  =  -  3  ; 

a  -6 

or  2\/-y'^-=  0  .-.  2!/^  =  y^;  or  2  =  y  .-.    ;— .  =  2  .-.  2x-2b-a-b] 

or  22-  -  a  +  i,  whence  x  -  ^(a  +  6) 


101. 


+  2 
-  3 


5 

+ 

0 

+  0  - 

-  3  + 

0  + 

1 

+ 

10 

+  20+  10  - 

46 

-  15- 

-30  - 

15  +  69 

+   0  + 

0  + 

0  + 

0 

-  122  - 

104+  158  +  628 

+  183  +  156- 

237- 

942 

5  +   10  +    5  -  23  -  61  +  70 

1-61  -52    +    79  +  314  +  391 
61x-  70 


5x*  +  lOx^  +  5x  -  23  - 


x''  -  2a;  +  3 

or  5x»+  10x2+  5j.  _  23x"  -  61x-il-  52x-^  +79x3  +  314J--4 
+  391X-''  +  &c. 

102.   G.C.M.   of  (y  -  3)x2  +   f/  -  9)x  -  y(2y^  _  3y  -  9) 
and  (y  +  \)x'^  +  2{y  +  lYx  -  ij(3i/  +  5y  +  2) 
=  G.C.M.  of  (y-3)x=+(y  -  3)(y  +  3)x  -  y(y  -  3)(2y  +  3) 

and  (y  +  l)x^  +  2(y  +  l^x  -  y(y  +  l)(3y  +  2) 
=  G.C.M.  of  x^  +  (y  +  3)x  -  y(2y  +  3)  and  x^  +  2(y  +  ] )x  -  y(3y  +  2) 
=   G.C.M.  of  (X  -  y)(x  +  2y  +  3)  and  (x  -  ?/)(x  +  3y  +  2) 
-  X  -  ij.     See  Algebra  Art.  73 

If  the  student  does  not  clearly  understand  this  method  of 
factoring,  he  may  obtain  the  G.C.M.  by  rule. 
Thus  x2  +  (y  +  3)x  -  y(2y  +  3))x'''+  2(i/  +  l)x  -  2/(3y  +  2)(l 

x"  +     (y  +  3)x  -  y(2y  +  3) 

(.y  -  i>  -  2/(  y  -  1) 
=  (y  -  i)(a;  -  y) 

Then  x  -  y)x^  +  (y  +  3)x  -  y(2i/  +  3)  (x  +  (2t/  +  3) 
x*  -  xy 

(22/  +  3)x  -  2/(2y  +  3) 
(2y  +  3)x  -  y(2y  +  3) 

Hence  G. CM.  =  x  -y 


It4  KEY  TO  [Mis.  Ex.  io-i-lO^. 

When  y=  1,  tbe  given  quantities  become  -  2a;^  -  8x  +  10,  and 
2x-  +  8x  -  10,  of  which  the  G.C.M.  is  evidently  x^  +  4x  -  5 

103.  a  =  mb%  and  c*  =  nb^  .-.  c  =  nH^  .-,  ac  =  mnH'*  .-.  ac  cc  b^ 

104.  (a*f  +  (m*f  =  (a^  +  m^)(^a^  -  a*m*  +  m«)  =  (<r«  +  07nV2  +  m^ 
(o^-  flmV2  +  wi^)(«*  +  ahnys  +  m*)(u*  -  a^mys  +  m*).  See 
problem  23  in  Miscellaneous  Exercises. 

105    ^'  ~    ^^    "    ^^^    =    W    +    (P    -    g)H^«    -    (T'    -    9){ 
•  (OT    +    g)''  -    j)'^  {(m    +    q)    -   p}{(m    +    g)    +   pi 

(m  +p  -  q)(m.  -p  +  q)       m  +p  -  q 

(m  -  p  +  q)(m,  +  p  +  q)  ~  in  +  p  +  q 

P^-(q  -  my         (P  -  q   +  fn)(p  +   q   -   m)  p  +    q  -   m 

(m -\r  p)'^  -  (f'   ~    {m  +   p   -  q)(^ni  +  p   +  qj  ~    m   +  p   +   q 

q'^  -   (m  -  p)^         (q  -  m  +  p)(q  +  m  -  p)  q  +  m   -  p 

(P   +   qV  -  "'^    ~    (/>  +  9  +  "0(P  +  q  -  "0  ~    p   +  g   +   771 

m  -hp  —  q       p  +  q  —  m       q  +  in  -  p 

.-.  sum  of  three  fractions  =  — ; +  + 

m  +  p  +  q       7n  +  p  +  q       p  +  q  +  m 

vi+p-q  +  p+q  —  m  +  q  +  m  —  p       m  +  p  +  q 

an  +  p  +  q  ~  in+  p  +  q  ~ 

106.  2x2*+  2-'  =  80  .-.  2^'  +  2(2')  +  1  =  81  .-.  2'+  1  =  9 
2'=  8  =  2*  .-.  X  =  3 

107.  Let  x^  -  8x  +  22  =  ?n,   then  x^  -  8x  +   16  =  m  -  6  ; 


or  X  —  4  -  +  V"*  ~  '^i  ^^^  X  =  4  +  'Jm  -  6,  in  which  m  —  6  cannot 
be  negative  if  x  be  real,  that  is  m  must  not  be  less  than  6 

108.  If  a,  />,  c  had  each  oc  d;   then  a  =  md,  b  =  nd  and  c  =  pd 
.•.  abc  =  mnpd',  that  is  abc  oc  d' 
Now  by  hypothesis  a  cc  d^  .•.  a-  md^  .-.  a'  =  m^d^ 
Also  b"  cc  nd*  .*.  6'  =  nd* 

1 

And  f'  oc   -.-  ••.  c'  =  pd'^ 


Then    taking    cube   root  of   each   side,   abc  =  y/m^np  x  <f* 
.  a!)c  oc  ti'  oc  as  if  each  of  the  three,  a,  b,  c,  varied  directly  as  d 


Mis.  ICx.  109-111.]  ALGECllA.  l75 

109.  S„  of  1  +  3  +  5  +  &c.  =  71".     (See  Ex.  LIX,  Example  13). 

n 

iS^of  (2/n+  l)  +  (2/7i  +  3)  +  (27/H-5)+&c.  =  {2(2m+ 1)  +  (^-1)2}^ 

n  n 

~  (4//1  +  2  +  2/t  -  2)—  =  (4ni  +  2/i)  —  =  (2»i  +  n)n  =  2mn  +  n''', 

and  it  is  manifest  that  the  latter  sum  exceeds  the  former  by 
2mn,  i.  e.  by  twice  the  product  of  m  and  n 

P        m 

1 10.  Let  j8  and  7  be  the  roots,  then  $  :  y  ::  m  :  n  .-.  —  =  — 

b  c 

And  Art.  208  Cor.,  $  +  y  =  -  — ,  and  fiy  =  — 

b 
i3  +  7     m  +  n         ~    a      m  +  » 
7  71    '  "  '     7  n 


Then  Art.  106, 


6         n  m  771         b         n  b        m 


7  =  -  —  X  — — ,  and  B  =  —  X7  =  —  x x =  •-  — 


(    b         n    \    f    b         m    \     b"^         vm  c 

.•.^7=    -  —  X X X ]=-—x- — ; — vS.    Butfi7-=  — 

'     \     a      m  +  nj    \     a      m  +  nj    a*     (m  +  ity   "   "py      „ 

c       b^        mn  a^c         mn  b^      (m+n)^ 

a  ~  a-    (m  +  n)^  '        ab'^  ~  {ra  -n)'^  '  '  ac  ~      mn 

111.  The  denominator  =  a\b  -  c)  +  b^c  -  be'  -  b^a  +  ac^ 
=  a\b  -  c)  +  bc(b  -  c)  -  a(b'  -  c') 
=  (b-  c){a2  -  a(b  +  c)  +  be] 
=  (6  -  e)(a  -  fc)(a  -  c) 
Similarly  the  numerator  =  a\b^  -  c'O  +  b*c^  -  6V*  -  a'b*  +  a^c* 
=  a*(b'^  -  c^)  +  b^c\b^  -  c^)  -  a^(b*  -  c<) 
=  (6^  -  c^)\a*  -  aXb'^  +  c'O  +  b-e''\ 
-  (b^  -  c^Xd^  -  6^)(a2  -  c^) 
(6''  -  c^XO'"  -  ^^)(**  -  <:^) 


The 


u  — TT — t; TT, c —  =  (a  +  bXb  +  c)(a  +  c) 

(6  -  c)(a  -  6)(a  -  c)         ^         ^^         '^ 


176 


KEY  TO 


lliifc.  Ey.  Il2-m 


112.  (ii)  Every  number  is  of  the  form  of  3n  or  3n  +  I  .*.  every 
square  number  is  of  the  form  of  9n^  or  9i^  ±  6n  +  1,  the  former 
is  evidently  divisible  by  3,  and  the  latter  becomes  so  when 
increased  by  2. 

(i)  Of  any  three  consecutive  integers,  one  must  be  divisible 
by  3,  and  since  every  even  integer  is  divisible  by  2,  and  two  of 
the  given  integers  are  even,  it  is  manifest  the  latter  of  the  two 
even  integers  is  divisible  by  4,  and  hence  that  the  product  of 
the  three  given  integers  must  be  divisible  by  3  x  2  x  4  =  24 


113.   {«(n   +    1)}»  -   [n(n  -    1)}''  =  ti'Kn.  +    1)=^-   (n   -   \y\ 
-  n\n  +  1  -  n  +  l)(n  +  1  +  n  -  1)  =  »t''  x  2  x  2n  =  4n» 

1  +  a  2  2(1   +  a^) 

,   and  x'  +  1  = 


114.  X  = 


1  -  a 


X  +   1   = 


\-  a 


(1   -  a)* 

1  +  &  2  (l+(0(l+6)  2(1 +a6) 

y=iri---y+i=rz6''^"^^i/=(iT^i6)  •••^^+^=(13^-/0 


Then 


(ai  +  l)(x'^+l)     x  + 


r2(l    +    a^)")  o 


(xy  +  \)(d'  +1)     y  +  1  ~   r     2(1+06)     1  ^  ,_^  2 

|(l-a)(l-6)|(''^-l)       1-T6 
2(a6   +    1)(1    +    a") 

(1  -g)^  1    -    h       (1    -   fl)(l    -  b)  1    -    6 


2(1    +   a6)(a^  +1)        1    -    a 
"(1    -  a)(l    -    b) 
\-b       1-6 


(1  -  ay 


\    -   a 


\-a       1-a 


=  0 


115.    1[7  +  21+35  +  35  +  21  +  7 
-  1     -    7-14-21  -14-7 


7+  14  +  21  +  14+  7  =  7x*+  14r*i^+  21xV+  Uxj^+  ly*' 
1  7+  14  +  21  +  14  +  7 
-1     -7-7-7 
-1  -7-7-7 

7  +    7  +    7  =  7x*  +  7xy  +  7y* 


His.  Ex.  116-120  ]  ALGSBRA. 

Lie.    8  +  4-6-1  +  1)72  +  36-68-16+16(9 
72  +  36  -  54  -    9  +    9 
-14-    7  +     7 
-  7)    2+    1  -    1 
2+1-1)8  +  4-6-1  +  1(4  +  0-1 
e  +4-4 


177 


+0-2-1+1 

-  2  -  1  +  1  .-.  G.C.M.  =  2x2  +  a;  -  1 


117.  (2x2  +  j._  i)(9j;2  _  4)  =  (2x  -  l)(x  +  l)(3x  +  2)(.^x  -  2) 
(2x2  +  3.  _  i)(4j.2  _  1)  =  (^2x  -  l)(x  +  l)(2x  +  l)(2x  -  1) 

118.  l.c.in.    of  denominators   =    8(1    +  x')(l    +  x)(l    -  x)* 
.-.  the  given  expression  = 

6(l+x)(l+x^  +  3(l-x)<l+T)(l+x2)  +  (l-x)^(l+xV2(l-x)^(I+x) 
'       ~~  8(^1  +  xXr^^Xl  -  x)2 

8  +  8x  +  8x2  1  +  X  +  x^  1  +  X  +  x2 


8(1 -x2)(l +x^)(l -X)  ~  (l-a;*)(l-x)  ~  1  -  x  -  x*  +  x< 

5a\  f         3a 


119.  (^  +  y)(^~Y)   +   "^   =    (^  +  ^'OC-^  -  3a)  +   llj 

15o  l.'jcx 

.•.x^+2c/x--^  =  x2+2t/x-15a+  11 J  ;  2(ix- 2ax  =  15a--7- -  Hi 

45a  -  45                       45a  -  45          45a   -  45 
.-.    x(2a   -   2a)   =    j—    .-.    x  =    ^^^7-^  =   ^-~ 

.-.   it  is   an  indeterminate  equation.     If  a  =  1  it  becomes  au 
j>lentity. 

1      1  1 

120.  (i)  If  a,  6,  c  are  in  H.  Prog".,  — ,  -r  and  —  arc  in  A.  Prog. 

1111  1         1        /I        1\        /I        1\ 

h         a         c         0'  b         a       yo        c  J       \a       c   , 

It  +  c       a  +  c 
ic  ac 

1  1  /I  IN         /I  IN        a+  c       a+  b 

Also —   =—   +   —      __-4. 


6  c  J  ac  iih 

M 


178  KEY   TO  [MiB.  Ex.  12MH. 

a+c     a+b     b+c     a+c       a+b   a+c         b+c 

.■. — ,-  =  -,    -  —    -  .•.  — -,— , and  — , —  are  in  A.  P. 

ac         ub  be  ac  ao   '    ac  be 

ab        ac  be 

And  ,,   -,—  and  t are  in  If.  Prog. 

a+  b'  a+  c  b  +  c  " 

1111  ^  ^ 

(n)  Since  -r —  —  =  —  -  -7-,  we  have  multiplying  hj  a  -r  b  -  c 

a  +  b  +  c       a  +  b  +  c       a  +  b  -i-  c       a  +  b+  c 


b  a  c  0 

a+c  b+c  a+b  a+c 

.-.-,—  +  1 1  = +  1  -  — r I 

0  a  c  b 

a  +  c       b  +  c       a+b       a  +  c 

'  '      b  a  c  b 

b  +  c    a  +  e           a  +  b 
.•.  ,  — ; —  and  are  in  A.  Prog. 

a     '       b  c 

c 

121.  .d  cc  b  and  c  a:  d  .-.  a  =  mb.  and  c  =  ;«/,  .■.  d-  — 

m 
Then  ad  -  — be  .-.  ad  az  be 
n 

122.  Area  of  circle  varies  as  (radius)^  .•.  area  =  jn(radius)' 
.-.  area  of  circles  =  ni9  +  m9  +  7nl6  +  m25  +  ni36  +  >n49 
=  by  addition  to  wl44  =  jnl2*  .-.  radius  of  resulting  circle  =  12 

71  3 

123.  5=  j2a  +  (;j-iy}y  =  {22  +  (3-l)rf}y  =  sum  of  1" 3  terms 

9 

=  {22  +  (9  -  1)J]  Y  =  sum  of  1"  9  terms 

.-.  (ll+(i)3  =  (11  +  4,/)9;    11  +  d  =  33  +  12i  .-.   Ud  =  -  23 
or  d  =  -  2  .-.  series  is  11,  9,  7,  5,  &c. 

124.  Let  m  =  the  m"",  and  n  =  the  n'"  terms  of  a  G.  series  ; 
also  let  a  =  1"  term,  and  r  =  common  ratio,  then  m  =  ar^-'^'  aud 


=  /i.»-i 


n  =  ar' 

_j.r»-l-n  +  l__«n-?»      .      J.  -    I  


m       at" 

- - __r»-l-n  +  l__«n-»  ,_,  —  , 


Mi3.  Ex.  124-127.]  ALGEBRA.  179 


m-l\  m  -  n 

,  and  common  ratio 

wT-'n 

wh«»re  m  -  the  >n^^  and  n  the  n"'  terms  of  the  series. 

If  one  of  the  terms  be  taken  as  ihe  first,  the  above  becomes 

1 


125.  a=  3;  ar*  =  ^5  .-.  r*  =  i^  .-.  r  =  ±  ?.     Art.  254,  r  ^  ~ 

22^-3         -I?  2  .       .     ^       ^^   4        8       16 

=  277^  =  -W"""^  •■•   '''  ""'''"  '"  '  -  '^  y-  9-+27 

126.  5„  =  {2a  +  (u  -  l)rf}y  ;  S^n  =  {2a  +  (2/i  -  l)rf|a 
and  Sgn  =  {2a  +  (3/i  -  l)^}y     ''^^^°  latter  half  of 

S-^    =    '^^   -    S^    =    {2a   +    (2/i   -    l)r/}«    -    {2a    +    (a   -    l)r/}"- 

3  nd  nd  n 

=  an  +  —nH  -  y  =  a/i  +  (3;t  -   1)7-  -  {2a  +  (3/i  -   l)il}y 

1  3« 

=   y(2a+(3«-l).i}y 

127.  Since  S^  and  ^2  are  each  to  n  terms,  we  have 
Sy  +  8^=  1  +  5  +  9  +  &c,  to  n  terms  +  3  +  7  +  1 1  +  &c.  to  ti  terms 
=>  1  +  3  +  5  +  7  +  9  +  11  +  &c.  to  2n  terms  =  (2n)'  =  4n^. 
Also  S,  -  Si  =  {(1  -  3)  +  (5  -  7)  +  (9  -  11)  +  &c.  to  n  terms} 
.-.  (S  -  S^y^  =  {(1  -  3)  +  (5  -  7)  +  &c.  to  n  termsf  =(  -  2  -  2  -  2 
-  &c.  to  71  terms)^  =  (  -  2nf  =  in^  .-.  S,   +  S^  =  (i\   -  S.^)' 

When  S2  =  sum  to  «  -  1  terms 
iSj  +  ^2  =  1  +  3  +  5  +  7  +  &c.  to  (2«  -  1)  terms  =  (2ri  -  1)' 
(Si  -  S.^)'=  {1  +  (5  -  3)  +  (9  -  7)  +  &c.  to  (n  -  1)  termsf 
=  {1  +  2(n  -  \)f  =  (2n  -  1)^  .-.  S,  +  S,  =  (S,  -  S,y 


180  KEY   TO  [ills.  Ex.  1I2S-133. 

128.  General  term  of  (1 +X-2)    ' 

,      ,,,     7)(p  +  ?) (p  +  2?)  '  •  •  •  {p  +  (r  -  1)7}  ^ 

2-5-8 {2  +  (r-l)3{  . 

=  (  -  D'  X 1^^^^,-^^ \.-^i 

2-5-8 (3r-  1) 

=  (_  l)''x  ■. ^a -^''^ 

When  (r  +  l)'"  term  =  Y'"  term,  r-& 

2-5-81M4-1T  „         2618 

.-.    Y-    term    =    (    -    1)«    x    ^■2.3.4-5-6  x  s"^^ "  "  =    65^^  '  ^' 

WLen  (r  -  1)^''  term  -  10'"  term,  r  =  9 

2-5-8-11-14-17-20-23-26  559130 

•■•  10'"  ^^'-^  =  (  -  1)'  ^    1-2-3-4-5-6-7-8-9  x  3«  ^'"  =-1594323^"'' 

129.  (x  -  l)(.r  +  l)(x  -  2)(x  +  2)(x  -  3  -  ^~^^l){x  -  3  +  V^^) 
=  (x^-  i)(x2-4)(x--  6x+  11)  =^  x6  -  3x5  +  6x*+  SOx^  -  51x* 
-  24x  +  44  =  0 

130.  (x  +  l)(x  -  l)(x  -  1)  =  x^  -  x2  -  X  +  1  =  0  ;  then 
x'^  +  2x*  -  3x3  -  3x2  +  2x  +  1  ^  x'  -  x^  -  X  +  1  =  x^  +  3x  +  1 
.-.  x^  +  3x  +  1  =  0  ;  or  x^  +  3x  =  -  1  .-.  x^  +  3x  +  (X?  =  ♦  -  1  =  5 
.-.  X  +  3  =^  i  iV5,  and  X  =  J(  -  3  i  V5) 

131.  Let  x  =  the  quantity,  then  ,—  =  -j  .• .  ad -^ dx  =  Ah--i-  4cx; 

46c  -  ad 

or  i/x  -  4cx  =  4bc  —  lul  .-.  x  =  —7 ; — 

a  —  4c 

2  -V3       4-2V3_    /V3  -  ly 

132.  2":rv3  ""  4T2V3  ~  (^V3  +  ly 

2  _y3\  *  ^  /V3^N  *  _  (V3-1)^(V34-1)^  _  (3-1)^  _     V2_ 

2 +V3y   ~  VV3TTy   ~         V3  +  1         ~  V3  +  1  "  V3  +  1 

133.  Multiplying  the  lower  equation  by  2  and  subtracting, 

V  1        y-1 

;  have  ly  -2z=  13  (111)  .-.  3y  -  z  +  ^  =  6  +  —  .-.  — ^  is  int. 

7/-1 
2 


Mi9.  Ex.  IX- jGG.J  ALQEBB-A.  181 

in    III  '     z  =  1>j  -  13  =  I4t -hi -13=  Ut- 6  .-.  z^  It -3, 

and  y  =  2/  +  1.     Then,  i    ^  =  1,  we  have  y  =  3,  and  2=4,  and 

X  =  2 

j.n  +  i_yTn-i    ■x"  +  x""^i/  +  a;""V  +  3;""V  +  *c.to7i+l  terms 

x"  +  x"  +  x"  4-  x"  +  &c.  to  (n  +  1)  terms 
=   when   ./   =   X    to ~2n ~" 

1  +  1  +  1  -L  &c.  to  n  +  1  terms       n  +  1 


135.  Let  m   be    their    G.C.M.,    and   q  and  5^  the  quotients 

arising  iVom  dividing  them  by  this  G.C.M.    Then  mq  +  mq'  =  45 

m^  +  wq'       q   +   q'        45         15 

and    mnn'   =    168.      Therefore    7 —  =  ■ -; —  =  — r  =  — 

'^  7«9^  5y  lt;8       56 

Whence  by  solving  the  quadratic,  or  by  inspection,  q  ~  ^  and  q' 

45         45 

-  8,  and  in  = ;  =  -rr  =  3  .-.  the   numbers  are   7x3=21, 

'  9  +5'       15  ' 

and  8  X  3  =  24 

1     x2-2x-3       1     x2-2x-15       2     x2-2x-35       92 


136.  '-■  ~r-7^^ — s+" 


5     x^-2x-8       9     x^-2x-24      13    x^-2x-48~585 

1     7/ -4         1     y-16         2     1/-36        92 
Let(x-1)2.2/;  theny  ^_g  +  y  -j^^  '  T^' yZT^Q  =  5^ 

1  1  2  92 

Now  -fT  +  -^ —  7J  ~  'Krt   •'•  subtracting  corresponding  terms* 

11  2  111  1 

2/-9  "*"  y^25  ~  7/ -49  ~     '  '^'^  y - 9  ~  y"- 49  ~  2/-49  ~  y  -  25 

7/ -49 -1/4- 9  y- 25 -J/ +  49  -40  24 

"''  (y  -  9)(i/  -  49)  "  (y  -  25)0/  -  49)  '  "''  y  -  9  ""  y  -  25 
or  5y  -  125  +  3?/  -  27  =  0  ;  or  8y  =  152  .-.  y  =  19.  Then 
(x  +  1)''  =  19,  X  =  1  =  +  VT9  and  x  =  1  f  VT9 


5  2/-9  6^9  y-25  9  13  y- 49  ^13  686  5   9^  13 
1  y  -  4  -  y  +  9    1^   y  -  16  -  y  +  25   2   y  -  3C  -  y  +  49  _ 
"^5     y  -  9     "*"  9  "  '    y  -  25     ~  13     y  -  49     ~ 

""  6  ■  y  -  9  '  9  ■  y  -  25  "  13  y  -  49  ~  '  *"■ 


182  KEY  TO  [Mis.  Kx.  1^7-159. 

137.  Let  X  =  V  +  z,  and  y  =  v  -  z:  then  2xy  -  Ay'  +  x^ 
=  2(v'-z')-4(v'^-2v2  +£2)  +  i,2+2»£  +  22=4.-.  l0vz-v'-5z-=A 
Also  x^-y^  =  v^  +  2vz  +  z'-v'+  2vz  -  z'  =  Avz  =  36  .-.  vz  =  9 
.-.  lOvz  -  v'  -  52^  =  90  -  t>2  -  5i*  =  4  .-.  v^  +  5=^  =  86 
.-.  v'  +  2vz'^5  +  5s^  =  86  +  18V5.  Extracting  square  root  (right 
hand  member  by  inspection,  and  left  hand  member  by  Art.  189) 
•\ve  have  v  ±  z^5  -  9  +  V^^  •••  Art.  186,  r  =  9,  and  +  cV^  ~  1  V^ 
.-.2=1  .-.  X  =  10,  y  =  8 

138.  XT  =  ^  =  9"^  =  (10  -  1)'^     Expanding   by  binomial 
819 

theorem,  we  have  (10  -  1)-^=  10-^+  2  x  lO""  +  3-10-*  +  Ac. 
+  Y-IO   1  +  810-9  +  9-10-1"  +  10-10-"  +  &c. 
+  1710-1'  4.  18-10-1^  +  19-10-2'"'  ^  20-10-21+  &c.  to  infinity. 
Now  S-IO-*^  +  9-10-1"  +  10-10-11  +  11-10-"  +  12-10-"+  &c. 
=  810-9  +  9-10-1'^  +  1-10-1"  +(1-10-11  j^  1-10-12)  ^  (i-io-iz 

+  210-1'')  +  (1-10-13+  4(.  ) 
=  8-10- ?>  +  10-10-1"  +  10-11  +  2- 10 -12 +3- 10-"+  4e. 
=  8-10-9  +  110- 9  +  10-11  +  2-10-"+ 3-10-"  + &c. 
=  O-IO-"  +  010-1'  +  110-11  +  2-10-"+  3-10-"+  Ac. 
Similarlyfor  18-10-1''  +  19  IQ-^J  +  2010-^  + &c.  and  generally 
for  (lO/i-  2)10-i''"*i,  &(. 

.-.  (10  -  1)-'-'^  10-^+  2-10-'  +  310-*  +  &c.  +  7.10-'  +  9-10-9 
+  10-"  +  2-10-1^  +  &c.  +  7-10-1^  +  910-1^  +  10-2-1  +  2-10-21 
+  &c.  to  infinity  =  -012345679012345679,  &c.  to  infinity. 

XoTE.— The  point  in  this  operation  is  the  sign  of  multiplication,  and 
not  the  dt^cimal  point. 

139.  nx2  —  bx  =  a}x  —  ab  r.  ax^  —  arx  ~  bx  +  ab  -  0 ;  or 
(ax  -  /»)(x  -  a)  =  0.    Now  if  we  assume  ax  -  b  ^  0;  orx-a  =  0; 

b 
the  equation  will  be  satisfied  ,•.  ax  -  6  =  0 :  or  ax  =  b  .-.  x  =  — 

b  "" 

Also  X  -  rt  =  0  .*.  X  =  rt.     Therefore  the  roots  are  — ,  and  a. 

a '  ^ 

which  are  rational  if  a  and  b  are  rational. 


Mifl.  Ex.  UO-144.]  ALGEBRA.  183 

140.  a:^  +  (a  +  b)x  =  (n  -  l)ab  ; 


2a;  +  a  +  6  =  i  '\jAnab  -  4,ub  +  a^  +  2a6  +  6^  =  +  V4/»a6  +  a''  -  2«6  +  6^ 
2x  =  -  cr  -  6  +  V^aoi  +  (a  -  6)"  .-.  X  =  i { +  V4«a6  +  (a  -  6)'=  -  (a  +  i)} 

141.  6x-Vx=l;  x-iV^=tr;  ar-JV^  +  Tk-Tir +  rA  =  vA" 
.-.  V^  -  iV  =  i  I'V  •'•  V^  =  i  or  -  J  .-.  X  =  J  or  ^ 

142.  V^  =  /rT~i  -  V-r  -  1  .••  X  =  X  +  1  -  2V-'-"^  -  1  +  X  -  1 
.-.  -  X  =  -  2Vx^^~l  .-.  x^  =  4x2  _  4^  oj.  32-2  _  4 .  ^js  _  A  ...  a;  =  § ^3 

143.  x^  +  xy  +  xz  =  a^  \ 

xy  +  y''  +  yz  =  b^  >  .••  by  addition 

xz  ■{■  yz  +  z^  =  c'  ) 
x^  +  2x2/  +  2xz  +  1/2  +  2!/2  +  2^  =  flZ  .f  ji!  4-  f^^  ainj  extracting  the 
square  root  x  +  y  +  z  =  ±  V"^  +  6^  +  c* 


-   i  V"^  +  ''^  +   ^2   .-.   X  =  i 


Va'  +  62  +    c2 


62 62 

—  =   +  V"''  +   62  +  c2   .-.   y   =   + 


y         -    '  V«''  +   6^  +   c'* 


—     r:     4-     ^a2     +     62     +     c2      .-.      2     =     + 


Va2  +   b'^  +    c2 
144.  Lot  «,10"  +  a2lO"-i  +  rT.,-10"-2  +  &c.,  +ff„.jlO  +  o„  be 

any  number  =  (10"  -  l)a^  +  (10"-i  -  1)0^  +  (10''-2  _  i)^^  +  &c., 

+  (I0-  l)ff„.i  +  a,  +^2  +rt3---  +fl'„.i4-flr„ 

Now  9  =  10  -  1,  and  each  of  the  coef.  (10"-  1),  (10" -i-  1), 

(10"-2  _  i)-..(io  -  1)   is  divisible   by  (10  -  1),    i.e.  by  9 

.-.  the  number  =  9m  +  Oi  +  a^  +  ffj  + a^-i  +  fr„,  where 

n  is   the  quotient  by  dividing  (10"  -   l)a,    +  (10" -i  -  l)a2 

f  (I0"-2-  !)«.,  + (10  -  l)a„.i  by  9 

Similarly  the   number   reversed  =  9»i^  +  "n  +  On  -i  +  ^n-a 

^  ••  •  •  a.^  +  a^  +  Oj. 

ffumber  x  4  =  36/n  +  4ai  +  4a3  +  4aj  +  •••  •  4a„  .^  +  4a„ 


184 


Key  to 


tMi«.  Ex.  46-49. 


Number  reversed  x  5  =  ioni^  +  5a„  +  5a„  .^  +  5a„  .,  +  •  •  •  •  +  5a j 

+  5a2  +  5ai 

.-.  sum  =  36m  +  Aom}  +  Oa^  +  9aj  +  9aj  +   ••••   +  9fl^   j  +  9a„ 

=  9{4?n  +  5m^  +  a,  +  a^  +  a^  + +  a„.i  +  a„} 

This  statement  may  be  generalized  as  follows  : — 

General  Theorem. — Let  r  be  the  radix  of  any  system  of 
uuinbers,  then  if  any  number  in  that  system  be  multiplied  by  any 
number  n  and  the  same  number  reversed,  as  to  its  orders,  be 
mutliplied  61/  r  -  (n  +  1) ;  then  the  sum  of  the  two  products  thus 
obtained  is  divisible  by  (r  -  1). 

145.  (a.  +  b){b  +  c)  -  («  +  l)(c  +  1)  -  (a  +  c){b  -  1) 
=  ab  +  ac  +  bc+b'^-ac  —  a  —  c— I—  ab  +  a  —  bc-\-c  =  lr'— I 


-  {(?)'- 


(147) 

a^x%2abxij+h'^i/'^+c-x'^+2rdxi/-\-dh/+a'y^-2abxy+b-x-+r'''y^-2cdxy+d^x^ 
__ 

a^(x^  +  y^)    4-   b\x''  +    y^)   +    (r{x-  +    y'Q    +    d^(x^  +    y^) 
x^  +  y^  ~ 

(rt2  +  b^  +  c'  +  d2)(a;2  +  y2) 
X-  +  y- 

148.  v'{«^(-i"^  +  4x  +  4)  -  2(((x  +  2)  +  1}  =  i  {a(x  +  2)  -  1} 

149.  G.C.M.  of  a-  +  2o6  +  i-  -  c",  and  d^  -  b^  +  26c  -  r-; 
that  is  of  (a  +  6)*  -  cr',  and  a^  -  (b  -  cy ;  that  is  of  (a  +  fr—  c) 
(a  +  6  +  c),  and  («  -  6  +  c)(a  +  i  -  <■)  is  cridently  a  •)-  6  -  c 


150. 


+  0 
-2 
+  1 


4  +  01  +  5+0  +  1 
0!  +  0 
-8  +  0 
+  4  +  0 


4  + 01 -3  +  4+1  =  4x- 


3x»  -  4x  -  1 
ar"  +  2x  -  1 


Mis.  Ex.  150-1o6.] 


ALGEBRA. 


185 


1 

+  0 
-2 
+  1 


4+0+5+0+1 
+0+0+0+0+0 

-8  +  0  +  6-8-14 
+4+0-3+4+7 


151. 


4  +  0-3  +  4+7-11-10 

4x-  3X-1  +  4x--^+  'Zx-'-  Ur'-i-  lOr-3  +  &c. 

1  1  +  0  +  0+0  +  0+0+0  +  0 
+1  +1+1+0-1-1+0+1 
-1  -1-1+0+ 1+1+0 

1  +  1+0-1-1  +  0  +  1  +  1  +  &C. 
=  1  +  X  -  x'  -  z*  +  x^  +  x^  -  x9  -  i^t*  +  &c. 
/a  b   \       fa         6\  a2  +  62  a^  +  6^ 

^^^-  \^b  ■*"  7^h)  ^  \f^b  ~^^hY  (_a  +  bXa-b)  ^  (a-6)(a  +  6) 
(n2  +  62)2       ^4  ^  23252  +  6* 
(tt^  +  62)2  =  a*  -  2a^b'^b* 

c(a-b)  c(a-b)  c(a  -  b)        (a  +  c)(b+c) 

^^^'  (a  +  c)(6  +  c)  "^  (a  +  c)(6  +  c)  ^  (a  +  c)(6+c)  ^     c(a  -  6)     ""  " 

3Cx  -  2)  X  -  3  X  -  1  1 

1  KA     ^: _ _ _    

(x-l)(r-3)       (x-l)(x-3)       (x-l)(x-3)       x-2 
3(x-2)-x  +  3-x  +  l  1  x-2  1 

(x~-  l)(x  -  3)  ~  x-2  ""  (X  -  l)(x  -  3)  ~  x-2 

(x  -  2)2  -   (X  -   l)(x  -  3)       x2  -  4x  +  4  -  x2  +  4x  -  3 
(X  -  l)(x  -  3)(x~-^  ^  (X-  l)(x-2)(x  -3^ 

1 
(X  -  l)(x  -  2)(x  -  3) 

}(xy  +  1)  +  2x{[(xy  +  1)  +  2i/}  +  (X  -  y)2 

X2j2  ^   I    ^  x'i  ^  y^ 

x^y^  +  2x1/  +   1  +  2x(xi/  +    1)    +   2i/(xi/   +    1)    +    (x   +  y)^ 
X^yi  +  1  -  x-  -  2/2 

(xy  +  1)2+  2(xy  +  l)(x  +  y)  +  (x  +  y)2  _      (xy  +  1  +  x  +  y)2 
(x2y2"+~2xyn)  -  (x2  +  2xt/  +  y^)        ~  (xy  +  1)2  _  (x  +  y)^ 
(xy+x  +  y+l)2  xy  +  x  +  y+1       (x+l)(y+l) 

(xy +  x  +  ?/+ l)(xy-x -y +  1)  ~  xy-x-y+1  ~  (x-lXy-1) 


155. 


186  Key  to  [Mis.  Ex.  Isr^lJS 

156.  (i)  ajT^  +  ix  +  c  =  0") 

^Divide  by  coefficients  of  x^;  then 
a,x2  +  6, a;  +  Cj  =  Oj 

b  c 

^■2  ^  J.  ^ =  0    ^ 

a         a  I  Let  r  and  r,,  be  roots  of  first,  and 

.,     ^1        ''i      ^  (  r  and  To,  the  roots  of  2Dd  equat.: 

a:''-)-—    +—  =  0)  "  ^ 

ttj         a,  ^ 

b  c 

Then  x^  +  —x  +  —  -  (x  -  r)(x  -  r, )  =  0 

6,        r, 
And  z^  +  — x  +  —  =(x-  r)(i  -  r,)  =  0 
a,         a       ^  ^  * 

Hence  in  order  that  the  equations  may  have  a  common  rcu-t, 

they  must  have  a  common  measure. 

(ii)  Having  divided  as  before  by  cocf.  of  ar^,   let  r  and  r, 

=  roots  of  one,  and  -  r  and  -  r,  the  roots  of  the  other  equation. 

b         c 
Then  a^  H x  H =  (a:  -  r)(x  -  r , )  =  i^  -  (r  +  r  [  )x  +  rr ,   -  0 

6.        r, 
And   X-  +  —    + —  =  (X  +  r)(x  +  r,)  =  x^  +  (r  +  r,)x  +  rr,   -  0 
0|        a, 

b  h,  c  c^ 

.'.  —  =-('■  +  'i)  ~  -  — ,  and  —  =  rr,  =  — .     Hence  in  order 
a  ^  "  a^  a  Oi 

that  the  roots  may  be  equal  in  magnitude  but  opposite  in  signs, 

the  coefiBcients  of  x  must  be  equal  in  magnitude  but  opposite  in 

sign,  and  the  coefficients  of  x^  and  also  of  x"  must  be  equal. 


=  -  12x  +  20  .-.  X  =  2f 

158.  (X-  l)-(x  +  4)  =  (X  +  3)'^(x  -  2)  .-.  x'^-  2x*-  7x  +  4 
=  x'  +  4x2-  3j.  _  18  .  a:2  ^  2x  =  1 1  ;  ^/I^nx~^.  -  ±  '/U  -  +  2V3 
.-.  X  =  1  ±  2V3 

1  +  2x       2  +  2x  +  2Vl  +  2x 

159.  :; r-  = —  ;    extracting   the    square 

1  -  2x       2  -  2x  +  2Vl  -  2x  *  * 


•v/l  +  2x         Vl  +  2x+  1 

root  of  each  side, =  ±     _=::^: ;  clearing  effractions, 

Vl  -  2x         t/l  -'Ix  -  I 


Mis.  Ex.  159, 160.]  ALGEBR.\.  187 


.-.  (i)  Vi    -   4r''  -  Vl   +  2x   =   Vl    -   4P  +   Vl   -  2x  ;    or 


-  Vl  +  2x  =  Vl  -  2x  .-.  1  +  2x  =  1  -  2x  .-.  2x  =  -  2x ;  or  z  =  0 


And    (ii)   Vl  -  4x"'  -  Vl   +    2a;   =   -  Vl  -  -i-^'^  -  Vl   -    2x 


.-.    2Vl  -  4x^   =   Vl  +  2x   -   Vl   -  2x 

Squaring,  4(1-4x2)  =  2  -  2Vl  -  4x'^ .-.  2(1 -4x2)  + (1 -4x2)'  ^  i 

.:  (1  -  4x2)  +  ^(1  _  4^2)^  ^  I    .    (1  _  4^2)  +  1(1  _  4^2)i  +  j^  =  ^o, 

.-.  (1  -  4x2)^  +  J  =  4-  I  ...  (1  _  4x2)^  =  1  or  -  1  .-.  1  -  4x2=  ;  or  1 
.-.  4x2  =  J  or  0 ;  2x  =  +  iV3  or  0  .-.  x  =  +  ^V^  or  0 


(n-  1)2x2-2(71-  l)2x+  (n-  l)2  +  47i 
^^^-   (7r^l)2x2  +  2(71  -  l)2x  +  («  -  1)^  +  4ft 

(71     -     1)2x2    -    2(ft    -    1)2X    4-    («    +     1)^    _    J^ 
(71    -     l)2x    +     2(71    -     1)2X     +     (71    +     1)2    ~    T 


Art.  103  (VIT) 


2(71  -  1)2x2  +  2(71  +  1)2    _    P  +  1  (ft  -  l)2x2+(ft.+  1)2    _    P+1 

-  4(ft  -  l)2x  "  Y^  '   ^^         ^r2(n^iyi  ~  P  -  I 

...  („  _  i)2(P  _  i)x2  +  (ft  +  1)2(P  _  1)  =  _  2(n  -  1)2(P  4-  l)x ; 
or  (n- 1)2(1 -P)x2_2(ft  -  1)2(1  +  r)x  =  -  (ft  +  1)2(1  -  P); 
4(„  _  1)4(1  _  p)2a;2  _  8(„  _  1)4(12  _  pz)x  +  4(ft  -  1)*(1  +  Py 
=  4(n  -  1)*(1  +  P)2  -  4(ft  -  l)2(rt  +  1)2(1  -  P)2.  Dividing  by 
4(n-  1)2,  (71-  1)2(1 -P2)x2- 2(71-  1)2(1 -P2)x  +  (7^- 1)2(1 +P)2 

=   (ft  -   1)2(1   +  P)2  -  (ft  +   1)2(1   _  P)2 

.-.  (ft- l)(l-P).t-(ft-l)(l+P)  =  iV(n- 1)2(1 +  P)2-(ft+ 1)2(1 -P)2 

.-.  (ft  -  1)(1  -  P)x  =  (ft  -  1)(1  +  P)  +  V4Ptt2  _  4p2„  _  4a  H-  4P 

.-.  (71  -  1)(1   -  P)x   =   (ft   -    1)(]    +  P)   ±   2'^(Pn  -   l)(n  -P) 

1  +  P  2  : 

•*•  ^  =  iTp  ±  (n  -  i)(-i  _p)V(^«  -  !)(«  -  P) ;   where  in  order 

that  X  may  be  real  VC-P'^  -  l)('i  -  P)  must  be  real,  that  is 

(Pft  -  l)(ft  -  P)  must  be  positive,  and  if  n  is  positive,  in  order 

that  (Pft  -  l)(n  -  P)  may  be  positive,  P  must  neither  be  >  n 

1 
nor  <  — 


188  KEY  TO  (Mis.  Ex.  1C1-1C6. 

161.  ^=U^  +  b)  =  i(i  +  i)  =  iXfi  =  II  =  l-jlj 


H  = 

2ab 

2  X  J  X 

< 

3^  _ 

2 

=  U 

a+  b 

~      i  +  5 

G  = 

V55  = 

VTxj. 

VI 

=  1 

162. 

11  = 

2ab 

6    ■••   ^^ 

+ 

Hb  = 

2ab 

a   + 

a 

b 

Ha  -  ab  ^  ab  -  lib 

H-H  +  a     H-H''> 
or  (H-  l)a  =  («  -  H)b  .:  -—^=  j^^  .:  -^^^-  =  -^^^^ 

H-H-  a       H-  H  +  b         (I£ -  a)  -  H  _  H  -  (H  -  b) 
■'•    ~H-a       ""        H-b        •■•    '~H(H-a)      ~       H(^H  -  b) 
H  -  a  11^ ___H (g  -  6) 

•■•   Tl^H   -    a)    ~   ii(H  -   a)    ^    H(fr^^    ~    H(H~^b)  ' 

And  .'.  H  —  a,  /f  and  H  —b  are  in  H.  progression;  that  is,  H'\3 
the  //  mean  between  H—  a  and  H-b 

16.'?.  The  7J*  term  =  a  +  (n  -  1)'/  .-.  37"'  term  of  the  series 
3/  +  V  +  z<  +  &c.  =  \^  +  (37  -  1)(  -  i)  =  s^fi  -  5/  =  0 
«3,  ={2  X  6 +  (31  -  1)(-  i)}^iL=  (l2+30x-J)5/=7x3^=108i 
S,2  =  {2x6  +  (42-l)(-  1){V  =  (12  +  41  x-l)21  =  (12-C;;)21 
=  5i  X  21  =  108i  .-.  5^1  +  ^42  =  108J  +  lOSi  =  217 
a(l-r")      3Ul -(')"}      ^i" 

164.  s„  =  -^^-3^'  =     rry^  =  fl^  -  (^)"i  =  "f^  -  (■>>"i 

a  1  1     _   ^ 

i*'5.  ^cc  =  r^T  =  i(--,v)  =  r+i ~ '7" 

a(l-r")      l-(--4r     1  -  (  -  •4)'*      5  f        /^_  IVI 
^n=        l_r      -l-(-.4)-  1  +  ^  U  \      ^/     J 

S^  -  -S„  =  5  -  ^{1  -  (  -  ;)"{  =  f  -  5  +  «(  -  ?)»  =  K  -  i)" 

1C6.  Of  P' series,  5„-{2  +  (n-l)l}—  =  ("+l)y,and5p=(p+l)^ 
Of  2""  series  S„  -  {4  +  (h  -  1)3} "-  -  (3ri  +  1)  y,  and  5^  -  (3p  +  1)^ 


ilis.  Ex.  166-170.]  ALGEBRA.  189 


Of  S'-d  series  S„  =  {6  +  (n  -  1)5}  ^-  =  (5«  +  l)~,  and  Sp  -  (bp  4-  V)~ 
Of  4'"  series  <$•„  =  {8  +(n  -  1)7]  2"  =  (?«  +  Oy,  and  5-^  =  (^  +  1)|- 

.-.  of  the  series  (71+  1)—  +   (3?i  +  1)—  +  (5/i  +  1)—  +  &c., 

n 
where  the  first  term  is  Qi  +  1)—  and  the  common  difference  is 

2rt  X  Y  =  ri^,  the  Sp  =  {2(7i+  l)y +  (;'-  l)n^}y  =  (i^  +  n+7'w^-w^)-;j- 
;[>  pn 

;)  p  /» 

Also  of  the  series  (p  +  1)—  +  (3/)  +  1)—  +  {op  +  1)—  +  Ac. 

P  7? 

where  the  PHerm  is  (/'+!).;  and  the  common  difference  is  2px~=p^ 


2 


2\._ 


•!^n  -  {^CP  +  1) y  +  (ra  -  l)iJ==}y  =  (i)^  +  p  +  np2  -  p^)^  =(P  +  ^P'^Yj^ 

pn 
-  (1  +  pn)-^-  .-.  Sp  of  the  former  series  =  Sn  of  the  latter  series. 

167.  {(X  +  y)  -  V^j{(T  +  y)  +  V^^J  =  (^'^  +  2x»/  +  y^)  -  xy 
=  x^  +  XT/  +  y^ 

[(x'^  +  y'^)  +  xi/\{(x'^  +  ?/2)  -  xy]  =  (x^  +  y'^'f  -  xhj^  =  x*  +  xhf  +  ?/* 

168.  8(52  -3x8)^+  5(5^+  3x8)^  =  8(25  -  24)^  +  5(25  +  24)^ 
=  8Vl  4-5V49=  8  +  35  =  43 

169.  Vl6a*-  96a^6  +  216ft^62-  216a6«  +  8I6*  -  4a2-  12a J  +  96^, 
and  V4a^-  12a6  +  96^  =  2a  -  36 

\        \        \  f  a        \  11  3c        26 

170.  --  +  Ti  =  -J  -r  +  <^  •    Also  -  TTT  -  ^  =  -  ":;^7~  -  :r^^  ; 

a      4(/     a(i\^4         y  26     3c  66c      66c  ' 

a         c  3c         26 

but  since  -7-  =  -t»  it   follows   that   6c   =   ad,   .•.   -  —7—  -  — r- 

b         d'  66c        66c 

1\         /  b         1 


3c 

26              c 

b 

~   ~  6ad 

6a<i           2a<i  ~ 

3ad   " 

1 

1         1         1 

1  /a 

ft 

26  ~  37  ■*"  4^ 

^(^4 

1  /ft 
~  ad^A 

6        c        \ 

-T-Y"-') 

ad  y         \^  3        ad 

1        _       A       A       i 
'^'   "T^ai"¥^ad 


190  KEY   TO  [Mis.  Ex.  171-1,5. 

12(x+  1)^ 
171.  Multiply  by  4(j:  +  1),  and  8x  +  12  =  4x  +  5  +      ^^  ^  — 

Reducing  and  then  clearing  of  fractions,  we  have  12i*+  25j  +  1 

=  12x^+  24x+  12  .-.  j;  =  5 

2x  +  b       2a  +  b 
^ '2-  i^Tii  =  ^^"^T^  •■•  clearing  of  fractions, 

.-.  2a2x  +  2a6x  +  d^b  +  ab^  =  2ax*  +  2a6x  +  6x^  +  6^* 
.-.  2ax^  +  6x2  _  2a^x  +  b^x  =  d^b  +  al? 
(2«  +  h)x''  -  (2a^  -  P)x  =  ab(a  +  b) 
4(2tt  +  by^x^  -  4(2a  +  b){2d^  -  b'^)  +  (2a^  -  b^f 
^  iab(a  +  b)(2a  +  6)  +  {2a?  -  b^y  =  4a*  +  8a»d  +  Sa^b^  +  4ub'  -r  u^ 
.-.  2(2a  +  b)x  -  (2a2-  b^)  =  i  V4a*  +  8c^b  +  Sa^lr'  +  4ai»  +  b* 
=  ±  (2a2  +  2ab  +  b')   .-.  2(2a  +  b)x  =  2c?  -b^\  (2o?  +  2ab  +  b^) 
=  4a^  +  2aJ,  or  =  -  2a6  -  26^  =  2a(2a  +  b),  or  =  -  2b(a  +  b) 
2a(2a  +  6)  ft(a  +  6) 


«+l  ab  +  a  a+1        o6  +  a 

173.  Since  x  =    .^,— Ti  and  y  -    cT-i  i  -"^  +  y  -    l  ,  r  +  — rTT 
ao+l'  '      ao  +  1 '  ^     ao  +  I       a6  +  1 

2«  +  aft  +  1 
"aftTl 

X  +  y_-_i 

X  +  1/  +  1 


2a  +  aft  + 

1 

- 

1 

a6  +  l 

2a  +  a6  + 

1 

+ 

1 

a6  +  1 
a 

2a  +  o6 

+  1  - 

ab 

- 

1 

ab  +  1 

2a  +  a6 

+  1  + 

ab 

+ 

1 

ab  +  1 


2a  +  2a6  +  2  ~  a  +  ab  +  I 

174.  2x«  -  2xr  -  2xy  +  2yr  +  2y^  -  2xy  -  2yz  +  2xz  4-  3c*  -  2x; 
+  2xy  -  2yz  =  2x2  ^  2^!/^  +  2;^  _  2xy  -  2xz  -  2yz 

=  (X2  _  2xy  +  y2)  +  (X2  -  2XZ  +  I*)  +  (y3  _  2i/r  +  s-") 

175.  (a  +  hf  -  c^  («=»  -  b-f  +  4(76c3^-  c* /(a  -  6)^  r  c* 

/  (a=«  -  ^2/  -  {a-_W\_ 
(a  +  6)^c2  -  c* 
(g  +  6)2c»  -  c* 


.U.S.  £x.  176,177.]  ALGEBRA.  191 

17G.  x«-l)j;i<'  +  x9  +  a;«  +  2x'  +  2a:*  +  2j;*+x2+x+  l(x=+a  +  l 


rio  _  x^ 


a;9  +  2x^ 

X^  -X 


X*  +  2x'^  +  2x 
x«  -  1 


2x^  +  2x*  +  2x*  +  2x^  +  2x  +  2 
x'  +x*  +  x'  +  x=^  +  x+  l)x8-l(x 

x"  +  x"  +  X*  +  X^  +  X^  +  X 


-  X*  -  X*  -  X*  -  X^  -  X  -  1 


ar"  -;-  a*  +  z^  •{■-x^  +  x'+  i )x'  +  x*  +  x^  +  x''  +  x  +  1  (x^  -  x 

;i;^  +  X^  +  X«  +  a*  +  X^  +  X^ 


—  x^  —  x'  +  X  +  1 

—  x^  —  x'  -  X*  —  x'  -  X^  —  X 


X*  +  x=*  +  x^  +  2x  +  1 

X*  +  x^  +  a'^  +  2x  +  l)x«  +x*  +  x^  +  x2  +  x  +  if  x 
x"  +  X*  4-  x^  +  2x^  4-  X 
-    x^+  1 
x2  -  1  ^x*  4-  x^  +  x2  +  2x  +  1  (x2  +  X  +  2 


ii;=*  +  2x» 

G.C.ilf.=x+l)x2-l(x-l 

X^-  X 

fl;2  +  x 

2x2  .^  3j. 

-x-1 

2x^-2 

-X-  1 

Sx  +  3 

3(x  +  1) 

Cix    +    3)(x    -    1) 

(X    -    7)(x    4-1)         X    4-    2 

^''-     (X    4-    6)(x2    -    1)    ' 

(X    4-    5)(x2   -    1)    ~    x^   4-     I 

2x2  4-  X  -  3  -  (x2  -  6x  -  17) 

X  4-  2            x2  -  Tx  4-  4           X  4-  2 

(x4-5)(x2-l) 

x2  4- 1   ~   (X  4-  5)  (x^  -  ]  )  ~  x2  4-  1 

(x2    4-    7x    4-    4)(x2    4-    1) 

-    (X    4-    2)(x2    -    l)(x    4-    5) 

(X4- 

5)(x*-]) 

(x^  4-  7x  4-  4)x2  4-  x2  4-  Yx  4-  4 

-  (x2  4-   Yx  4-  10)x2  4-  x2  4-   1x  +  10 

(x  + 

5)(x4  -  1) 

2x2  +  14a.  +  14  _  6x2       14^^  - 

-  4x2  +  14 

(X  4-  5)(x*  -  1)       "  (X  4- 

5)(x*-  1) 

102  KEY  TO  [Mi5.E2l.17S,  i:y. 

(178) 

(a  + V-])^-f  (a-Zyin')^     a^+  2ah'f^  -b^-ir  a?-2abf:ri^b* 
(a  -  b'j'-^lXa  +  6V  ^1)     ~  a^  -  b\-j'^y 

ia^  -  262  ^  2(a2  -  }P) 
""  'HF+'b^  ""      a^  +  6^ 

179.  Let  X  —  3y  be  the  first  of  any  four  positive  quantities  in 
^^.P.,  and  let  2i/  be  their  common  difference. 
Then  the  four  quantities  are  x  -  St/,  x  -  y,  x  +  y  and  x  +  3y. 
And  the  sum  of  the  extremes  =  x-3y  +  x  +  3y=  2x. 
Also  the  sum  of  the  means  =  x  -  y  +  x  +  y  =  2x.  And  2x  =  2x 
.•.  the  sum  of  the  eitremes  =  the  sum  of  the  means. 

X 

Again  let  -j  be  the  first  of  four  positive  quantities  Lo  (?.P.^ 

and  let  y  by  their  common  ratio. 

X       x 
Then  the  four  quantities  are  — r,  — ,  i  and  xv. 

X             x  +  xy^                                  X           x-r'jy 
Sum  of  extremes  =  — 7  +xi/==  — t~\  sum  of  means  =  — Vx- - 

X  +  xy^   >   X  +  xy  > 

Then -^ p.   — '-,  according  as  x  +  x^  p  xy  +  ly' ■ 

or  as  1  +  y^  =  y  +  J/2;    or  as  (1  +  y)(l  -  »/  +  2/^)  =  y  (1  +  >/)  ; 

or  as  1  -  y  +  y^  ^  y ;    or  as  1  +  y-  =  2y.     But  1  +  y^  >  2y  by 

X  +  xy^      X  +  x>/ 
Art.  134  Note  2,  .-.  5—  > -,  that  is  the  sum  of  the 

extremes  is  greater  than  the  sum  of  the  means. 

Lastly,  if  as  before  x  -  3y,  x  -  y,  x  +  y  and  x  +  3y  are  in  ^J.P., 

Ill  I 

their  reciprocals ^,  ,  — , —  and  — —r-  are  in  H.P. 

^  X  -  3y'  X  -  if  X  +  y  x  +  3y 

1  1  2x 

■T^^^"    rr:-3y   +   rr^    =   x»  -   9y'    =    ^""^    °f   extremes. 

I  1  2x 

^"•^     T^rj   +    — +~7   =    x^  "-   y»    =   ^""^    of  means. 


ii-..  Ex.  170-134.]  ALGEBRA.  193 

Now  whether  y  he  positive  or  negative,  y^  is  necessarily  positive, 

2x                  2x 
and  therefore  x*  -  y*  >  x*  -  9y^,  and  .*.  -j- — ^-3  >  -j j ; 

that  is  the  sum  of  the  extremes  is  greater  than  the  sum  of  the 

means. 

2/1  -  1 

180.  S^n-i   =   {2a  +    (2a  -   2)0'} — r ,    and   when   d    -    a 

2n-\  2/1-1 

S.n.\  =  {2a-  (2«-  2)a}— ;7—  =  {2a  +  2a72  -  2«}— r—  =  na(2n  -  1) 

Also  (2/1-  I)">  term  =  a  +  (2a- 2^  =  a  +  (2i-2)a  =  a+2ara-2a 
=  a(2n  —  1)  .*.  eum  of  2«  —  1  terms  =  the  (2/1-  1)*  term  xra  when 
the  series  is  ascending,  i.  e.  when  the  first  term  is  the  least  and 
the  last  term  is  the  greatest. 

181.  ab  +  b'/a^lc^  =  x'  .:  b^jd^  -  x'-*  =  x^  -  a6  ;  a^h'^  -  Px^ 
?  X*  -  2a6x2  +  a'b'^  .-.  i*  -  2a6x2  +  fi^x^  =  0  ;  or  x^(x^  -  2ab  +  6^)  =  0 
.•.  X  =  0  or  X  =  +  \JbCia  -  b) 

182.  3x^  +  x'':=3104;  36x*  +  12x«  +  1  =  37249  ;  6x^  +  1^  +  193; 
6x^  -  192  or-  194;  x*  =  32  or-32§  ;  x^  =  2,  hence  x  =  64; 
or  x'  =  (  -  32i)«,  whence  x  =  VP^'iP  =  32J  ^/W>1  =  '-'^  ^ 
=  ^^7=^7857 

x2+  2ax  +  x2_x2+  2ax-  a}  _  h'^ ■¥  2hx  ■>>■  x^ -  b^  +  26x-x2 

4ffx  46x  a  b 

or  6x^  +  ax'  =  a6^  +  ba'  .-.  (6  +  a)!^  =  ab(b  +a)  .-.  x^  =  ab,  whence 
I  =  -I  ^Ja5 

184.  Vx2+  V^^  +  96  =  11  -X  .-.  x'  +  ^Jx'  +  9Q  =  121  -  22x  4- x^ 
.-.  V^^  +  96  =  121  -  22x.     Again  squaring 
x2  +  96=  14641  -  5324X  +  484x2  _._  483^2  _  5324X  =  _   14545; 

5324  14545 

orx^--g3X  =  ---g3- 

^   5324   /2662Y_  7086244   14545   708C244- 7025235 
■'■  ^  ~  483  ^  ■*"  I^TSS  y  "  23328'9'  ~  "483"  ~     233289 

N 


194  KEY  TO  [MiB.  Ei.184-186. 


2662  /  61009  247 


3   ~  -  V  23: 


=  ± 


483       -    »   233289     -^483 
2662  +  247     2909  2415 

••■  ^  =  ~l8l~  =  483  ■'   ""'  433"  =  ^-^^  °^  ^ 

185.  Let  X  -  the  left  hand  digit,  and  y  =  the  right  hand  digii ; 

then  the  number  is  lOx  +  y 

\Ox  +  y 
.•. -^  =  21,  whence  lOx  +  y  =  21x  -  21y,  or  22i/  =  l\x,  or  x  =  "]/ 

•*■  ~y 

lOx  +  w 
Also  — ^7 —  +  17  =  lOy  +  X,  whence  27x  +  IBj/  =  llxy  +  10.'/-+  x' 

But  X  =  2y ;  substituting  this  in  the  last  equation,  we  have 
547/  +  182/  =  22i/^  +  10^=*  +  4y^  .-.  72  =  36i/,  whence  y  =  2 
And  X  =  2y  =  4  .-.  the  required  number  is  42 

186.  Let  X  =  minutes   per  mile   taken   tj    B,    then  x   -t-   1 

60 
=  minutes  per  mile  taken  by  ^i? ;    —  =  miles  per  hour  of  B,  and 

60 
— — r  =  miles  per  hour  taken  by  ^i. 

The  second   time   round   the   rate  per  hour   of  B  -  ■ —  -  3 

'  X 

60    -   2x  60  62  +    2x 

,  and  rate  per  hour  of  .9  = t—t-  +  2  = 1- 

X        '  '  X  +  I  X   +    1 

And  since  the  course  is  2  miles  long,  the  time  in  hours  taken  by 
B   to   go  round   =   -^^^  =  ■sTTT  =  sTT^   •'•    *'°^«  '° 

X+1  X  +    I 

60x  +  60 
minutes  required  to  go  round  =  ^oT"! —  •'•  minutes  per  mile 

30x  +  30 
taken  by  .^  =  -^.  ;  similarly  minutes  per  mile  in  2'"'  round 

30x 
required  hj  B  =  vcTp^ — ,  and  since  ^5  does  the  two  miles  in  two 

minutes  less  than  B,  his  time  per  mile  will  be  one  minute  les? 

30x   +   30  30  r 

thnn   B  .-.   -y .  +    1    =   ^o~^~x'   ^^*°"'®  ^7  reduction 


Mis.  Ex.  186-190.]  ALGEBRA.  195 

3')j;  +  30  +  31 +x     31J  +  61        30x 

^,— =  -XT-; —  =  -T .-.  930x  +  1830  -  Slx^  -  61z 

31  +  X  31+x        30-x 

5  930x  +  30x2;    or-61x2-61x  =  -  1830,  whence  x^  +  x  •=  30 ; 

x^  +  x  +  i  =  -ijJ-  .-.  x+  i  =  i  Vj  and  x  =  5 

.-.  jI's  rate  P'  round  =  5  +  1  =  G  min.  per  mile  ^  10  miles  per  hour 
B's  rate  1"'  round  =  5  minutes  per  mile,  or  13  miles  per  hour 
Jl's  rate  2°''  time  round  =  10  +  2  =  12  m'les  per  hour 
.B's  rate  2°''  time  round  =  12  -  2  =  10  miles  per  hour 

■^fhole  time    of  B  for  both  rounds   =   10  +    12   =   22   minutes 

Whole  time   of  j1  for  both  rounds  ==    12  +   10  =   22  minutes 

.-.  neither  horse  wins. 

187.  Let  X,  X  +  1,  X  +  2,  X  +  3  and  x  +  4  be  any  five  conse- 
cutive integers;  then  x(x  +  2)(x  +  4)  +  (x  +  1)^  +  (x  +  3)^ 
=  (X  +  2)(x2  4-  4x)  +  x^  +  3x2  +  3x  4-  1  +  ^^.s  ^  Q^-i  +  2lx  +  27 
=  (X  +  2)(x2+4x)  +  (x^+4x'+  5x  +  2)  +  (4x  +  8)  4-  (x='+  8x^4-  2 Ix  4- 1 8) 
=^  (X  4-  2)(x'''4-  4x)  4-  (X  4-  2)(x2+  2x  4- 1)  4-  (x  4-  2)4  4-  (x  4-  2)(x2  4-  6x4-9) 
■■--  (X  4-  2){(x2  4-  4x)  4-  (x^  4-  2x  4-  1)  4-  4  4-  (x-  4-  6x  4-  9){ 
=  (X  4-  2){(X''  +•  2x  4-  1)  4-  (x2  4-  4x  4-  4)  4-  (x^  4-  6x  4-  9)} 
:r  (X  4-  2){(x  4-  \y  4-  (X  4-  2)2  4-  (X  4-  3)-'}  =  product  of  middle 
number  by  the  sum  of  the  squares  of  the  middle  three. 

188.  X*  4-  J*  4-  X*  4-  4x^y  4-  6xV  4-  4xy^  +  y* 

=  2x*  4-  ix^y  +  6x^y"  +  ixif  +  2^*  =  2(x*  4-  2xhj  +  3xh/^  2xy^  +  y*) 
=  2(x2  4-  xy  4-  y^f 

189.  (x^  +  y^  +  xhj  +  xy^)(  x^  -  y^  -  x'^y  4-  xi/^) 
=  {(x"  +  xy^  4-  (/  +  x2i/)}{(x«  4-  xf)  -  (y«  4-  x'^y)] 

-.  (x^  4-  xy'^f  -if  +  x^yy  =  xS  4-  2xV  +  ^^i/*  -  (y*  +  2xV  4-  xh/) 
=  x^  4-  x*i/2  —  x'^y^  —  y^ 

190.  x^  =  ('JoTb  ±  -{a'^f  =  a  +  b  ±  2\J'aF^'l?'  +  a  -  b 
=  2a  +  2Vtt^  -  b'^  ; 

ax^-  ix4  =  x\a  -  ix2)  =  {2a  4  2V5^^{a  -  (ia  ±  W^'  -  6')1 
=  {2a  +  2Va^  -  b']{ia  :f  i^/d^  -  6^j  =  {a  +  Va^  -  b'Xa  :f  ■^fF^'F) 
=  a^  _  (a2  _  J2)  r-  62 


196 


KEY    TO 


[Mis.  Ei.  191-i:9. 


101.   ax'  +  (ay  +  az  +  2ry)x'  +  (by- +  2cy' +  2  ij2)x  +  (by*  +  btfz) 
r-x  +  (y  +  z) 

a    +(ay  +  az  +  2aj)     +  (by^  +  2rr/'  4  2ajz)  +  (by^  +  bt/z) 
-  (ay  +  az)  -  (2cy-  +  2cyz)  -  (bi/  +  by^z) 


1 

-  (i/  +  =) 

a  +  2ry  +  by^ 

.-.  quotient  =  ax^  +  2cyx  +  by'^ 

192.  (x"/  -  (1")^  ^  (x"  -  1")  =  J."  +  1"  ^  j:"  +  1 

193.  l-l  +  l-a;  +  2x-3  +  5x+24-4-5x-4  +  JC 

194.  a(b'''  +  26c  +  c^)  +  b(c^  +  2ca  +  a')  +  c(a^  +  lab  +  b*) 

-  j((i^  -  ab  -  ac  +  bcXb  +  c)  +  (6'  -  be  -  ab  +  ac)(a  +  c) 
+  (c^ -  ac  -  be  +  ab)(a  +  b)\  -  ab^  +  ac^  +  be^  +  ba-  +  ca-  +  c6^  +  6a6c 

-  u'^b  -  a^c  -  i'-^rt  -  b~c  -  c^a  -  c^b  +  6abe  =  I2abe 

195.  {(b  +  e  -a)  +  (e  +  a-b)  +  (a  +  b  -  e)]x  +  {(e  +  a  -  b) 
+  {a  +  b-c)  +  (b  +  c-  a)}y  +  \(a  +  b  -  e)  +  (b  +  c  -  a)  +  (e  + a-  b)\z 
:^  (0  +  b  +  c)x  +  (a+  b  +  c)y  +  (a  +  b  +c)z  =  {a-^b  +  c)(x  +  y  +  z) 

196.  (x  +  2y)^x(x-2yy  =  (x-- 4y=/  =  x^  -  12xV+  48xV - 64.y« 
(u  +  fcV^~0  ("  -  W^)  =  «'  -  ^"(  -  1)  =  a*  +  i"'^ 

197.  j((X  +   6   +   cX^   +   b   -   e)\{(c   -   a  +  6)(c  +   0-6)} 

-  {(a  +  6)'' -  c-^jfc^ -  (a  -  6)^}  -  c=={(a  +  b)-  +  (a- b)-]  -  (a=« - b'f  -  e* 
=  2(ih^  +  2b^e^ -  «*  +  2a^/)^  -  b*  -e*=  2a!'b'  +  2oV  +  26*c»-  a*  -  i*  -  c* 
(X  +  1  +  x-i)(x  -  1  +  x-1)  =  j(x  +  z-1)  +  lj{(x  +  i-i)  -  IJ 
=  (x  +  x-1)^-  1  =x-+  2  +x-2_  1  -  x2+  1  +X-* 

198.  (2x*  -  SxV  +  4xV  -  Sxy*  +  6y*)  4-  6xV 
=  JxV-'-ixj/-'  +  ?  -Jx-iy  +  x-V 

(x2+2i+l)(x=-2x  +  3) 


(X*  +  4x  +  3)  ■;•  (xH  2x  +  1)  - 


X-  +  2x  +  1 


x'-2x  +  3 


,»,.(,.-,.,.(.'-f)  =  {a(x-f)'}.p-|) 

=  8(x'  +  Jx*y  +  ii/^)  =  8x^  +  4x*y  +  2]^* 


Mis.  Ex.  199-204.]  ALGEBRA.  19"? 

(x«  -  apx-  +  "'^px  -  u^)  ^  (x  -  a)  -  {(x^  -  a*)  -  apx(x  -  a)\  v  (r  -  a) 
=  (x-  +  ax  +  a-)  -  apx  =  x'^  +  (I  -  p)ax  +  a^ 

200.  (i)  (j2  -  3x  -  4)  =  (X  +  l)(x  -  4),  and  (x-  -  2x  -  8) 
=  (X  +  2)(x  -  4)  ;  (x2+  X  -  20)  =  (x  -4)(x  +  5)  .-.  G.C.M.  =  x  -  4 

(u)  3x'+  4x2-  3x  -  4  =  ^2(3_r  +  4)  _  (3x  +  4)  =  (x^ -  l)(3x  +  4) 
2x*  -  'ix'^  +  5  =  (2X''  -  5)x-  -  (Jx^  -  5)  =  (2x2  _  ^^(^^.2  _  j) 
.-.  G.C.M.  =x^-  1 

(ru)  Let  p  be  the  G.C.M.  of  m  and  n;  then  the  G.C.M.  of 
(a-"  +  a""'),  and  (x"  +  a")  =  x?  +  a?,  and  of  (x™  -  a"*),  and  x"  -  «" 
=  xP  -  flP  .-.  required  CCilf.  =  (x^  +  aP)(xP  -  a?)  =  x^p  -  a^P 

201.  (i)  I. cm.  of  (x  -  2a)(x  +  a),  x2(x  +  a),  and  a(x  +  a)(x  -  a) 
=  0x^(3;  -  2(r)(x  +  ff)(x  -  a)  =  ax^  -  2a^x*  -  aV  +  2a*x^ 

(II)  x^  -  x't/  -  a-x  +  try  =  (x-  -  a'^){x  -  y);  x^  +  ax''  -  xy^  -  ay' 

=  (x  +  «)(x2 -y-)  .-.  l.c.m.  =  (x^  -  a-)(^x^ - 1/^)  =  x* -  x^  _  a^x'^  +  a^y^ 

(a-hb  -c  +  d)(a  +  b-c-d)       (b+c-a  +  <r)(b  +  c-a-d) 

0A9     ! ^ 4.  ^^ 2i 

(a  +  6-c-d)(a  +  6  +  c  +  rf)       (6 +0  4  a  +  dX*  +  c-a-(/) 

(c  +  a  -  A  +  d)(c  +  a  -  6  -  d) 
■^  {c  +  a  +  b +d)(c  +  a- b -dj 

a  +  b  -  c  +  d       b  +  c  —  a  +  d       a  +  c  —  b  +  d 
a  +  b  +  c  +  d  "*"  a+  b  +  c  +  d  "*"  a  +  6  +  c  +  d 
a+6-c  +  rf+6  +  c-a  +  <i  +  a  +  c-6  +  i       a4-6  +  c  +  3rf 
a  +  6  +  c  +  d  ~    a  +  6  +  c+d 

2(f 
"*"  a+  b  +  c  +  d 

x^  +  2xy  +  j/2  -  z^       (x  +  y)2  -  z''  _  (I  +  y  +  2)(x  +  y  -  2) 
^''^-  x^  -  y^  +  2yz  -  z^  ~    x^  -  (y  -  2)^   ~  (x  +  y  -  2)(x  -  y  -I- 1) 
X  +  y  +  2 
~  x-y  +  s 

a\a  +  b)     a(a  -  b)  2ab         a»  +  a^b  -  a(a  -  b)^  -  2ab'^ 

^^*-  b(a^-b^)  "  6(a  +  t)  ~  a"^^^  =  6(0^  -  ft^ 

c^+a^b-a^+  2a:^b  -  ab^  -  2ab^  _  Za'b  -  3ab^  _      3ab(a  -  6) 
""  6(a2  -  6^  "^    6<a^  -  i^^    ^  i(a  -  6)(a  +  A) 

3a 
~  a  +  6 


198                                           KEY  TO  CMis  Ex.  205-;il0. 

''a^  -  ox  +  ax       a^  +  ax  -  ax\  (a  +  z)*  +  (a  -  x)^ 

—  X I    4- 

a  +  x        J 

2a^  +  2x2           g*  q2  _  3.2 


20-^-    1  --^^r-  ^  a  +  x  -  a'-x' 


a-x       a+xj   ■     a^-x-=         a«-x''    '    2{a^  +  x*)       2(a^  +  x^) 
4a&  4(76 

a  +  b'^^'^  a  +  b  '^  "^        Gab  +  2a*       Gab  +  26« 

^^''"       4«6  "^      4u6  ""   2a6  -  2a«  ■*"  2fl6  -  2i«  ' 

~TT  -  2«  — T-i:  -  26 

a  +  0  a  +  0 

dividing  numerator  and  denominator  of  l*'by  2a.  andof  2'"'by  26, 

36  +  fl        3«  +  6        36  +  a        3a  +  6        26  -  2a 

we    get  —r- +    r    =    -. -r =    -r =    2 

°        0  —  a  a  —  0  b  —  a  0  —  a  0  —  a 


207.  (i)  V^^*  -  4x«  +  4x^  -  4X''  +  8x  +  4 
=  V(x*  -  4x=*  +  4x2)  _  4(x^  _  2x)  +  4  =  x*  -  2x  - 


(u)    V--{4-44-(y)j  = 

/a«      2a6      2af      6'-      26c      c^\^      a       b 


2  -  —     =  2x2** 

3  /  3 


208.  The  square  of  which  «V  +  6x  are  the  first  and  second 

62 
terms,  is  a-x-  +  6x  +  -r-^  .-.  in  order  that  a^x'  +  bx  +  Ic  +  b^  raav 
'  4a^ 

6«  c  1 

be  a  perfect  square,  we  must  have  6c  s-  b'^  -     -7  .-.  -r  +  I  =   .  n 

1        c 

and  .•.  T  ^  — r  =  1 
4a'       6 

209.  (1)  mnx  +  ainn  =  »rx  +  am^  .-.  7?i;jx  —  ii^x  =  am*  ~  amn, 

<jm(m  -  n)        am 

tliat  is  (mv  —  7r)x  -  am^  -  amn  .-.  x  =  — ; r  =  

^  '  n(m  —  n)         n 

(11)  2x2  _  i3j-  -  _  6,  whence  x  =  6  or  J 

7x  +  1  400  /x  -  J\         400x  -  200 


7x+  1  400 /x-  J\         400x- 

2''^-  (')  ,3-^rii^  =  v(^tJ  =  -3TT- 


2 

whence  242 Ix-  -  G411x  =  -  25&S,  or  807x*  -  21.37x  ^  -  866 ; 
2i:;7     /2137\»    45667G9  -  2795448     1771321 


X'  - 


IGU;  (1614)*       ~   (1614)^ 


Mis.  Ex  210-214.]  ALGEBRA.  199 

2137  _     V1771321     ±  1330-9         _2137  +  1330-9  _ 
•"■  ^~  1G14~-      1614      ~      1614     •'•^~  1614  =  +  2'14 

or  -  0-49 

(II)  x2  -  2(tf  +  b)x  +  (a  +  6)2  =  4(a2  -  2ab  +  6^)  =  4(a  -  i)^ 
.-.  X  -  (a  +  6)  =  ±  2(a  -  6),  whence  x  =  3a  -  6,  or  36  -  a 

211.  ex    -   acy   =    abx   +    by    .•.    x(c    -   ab)    =   y{ac   +    b) 

x(c  -   a6)  ax(^c  —  a6) 

.•.   y  = — r — ■.     But  X  -  au  =  b    .•.   x  —  — r — ■  =   b 

■'  ac  +  b  ^  ac  -^  b 

acx   +   bx   ~   OCX  +  a^bx         bx   +   d^bx  b   +   a^b 

ac  +  b  ~        ac  +  b        ~  ac  +  b     ~ 

ac  +  b  c  —  ab 


and  y  = 


I   +  a^  "       1   +  a-' 

(n)  x'^  +  2xy  -i-  y^  =  49,  and  x^  +  xy  ■{■  y^  =  37  .-.  xy  =  12 

x^  -  2xy  +  y^  =  1 
.-.  X  +  y  =  +  7,  and  x  -y  =  ±1  .'.  2x  =  +  8,  and  x  =  ±  4  or  +  3, 
and  y  =  +  3  or  +  4 

212.  Subtracting  the  second  of  the  given  equations  from  the 
first,  we  have  3/(2  —  x)  =  a^  —  c^;  to  which  adding  the  third 
equation,  we  have  2yz  =  2a^  .•.  yz  =  0?  .•.  xz  =  6^,  and  xy  =  c* 


xz 

y      a* 
X  ""  6^  • 

a«x 

ac 

a^x 
.xy-x-  ^,  =c^. 

ab 

b'c' 

.x'  =  —rr. 

a-* 

6c 
•.x  =  ±- 

whence  also  y  -  +  -j-,  and  z  =  +  — 
^     -  b  '  -  c 

213.  Let  X  -  ^'s  age,  y  =  B'a  age,  and  z  -  Ca  age;    then 

32  4 

y  -  X  =  2(2  -  y);    x  +  y  •=  y,  and  x  +  y  -  12  =  -^(2  -  G)  ; 

3t/  -  /  =  22 ;    2x  +  2y  =  32 ;    3x  +  3y  -  42  =  12  ;    Cx  +  6y  -  9:  -  0, 
and  Gx  +  6y  -  82  =  24  .-.  2  =  24  ;  y=  21,  and  x  =  15 

214.  S,j  =  {2a  +  (n-  l)rf} y  =  {3  +  (12  -  1)^}G  =  (3+  11  x  3)6 
=  (3+ V)'<6=  117 

i»'n  01  IS  +  25  -f  3-,^-  =  S„  of  1  +  2  +  3  +  &c +  /Sn  of  ?  +  3  +  ^ 


200  KEY   to  [Mi8.  Ex.  21i-2n 

n  n        n(n  +  1)       §{1  -  (O"! 

=  {2  +  (n  -  1)1{-  =  (2  +  n  -  D-  =  ^—  +        l-i 

n(n  +  1) 
=   -^—  +2-2(3r=i{n(;i+l)  +  4-4(D"} 

.•J^  of V2  +  5 V3  +  W2  =  y-TTI  =      i_|      =  ^^-^-  =  3V2  +  2V3 

215.  a,  -ftj  -a.,  •  •  •  -a,  =  a''  .-.  a,  -a^  -Aj  •  •  •  -(Tp.j  =  a,  P"^  ' 
a,  ■(/,«,  ••••flp  =  a.p'  .-.  aj,  =  a,?"  ■=-  a/P-i  '  =  a.'P-i  .-.  a,  is 
formed  from  a^.^  by  mnltipljing  it  by  a^^  .-.a,  +  a.,  +  Cj  +  Ac, 
is  a  Geom.  series  having  a,  for  first  term,  and  «,^  for  commou 

ratio.     Then  .S'„  =  «r— z-37-   =  "' "o^'^ri 

216.  Squaring  each  side  and  transposing,  we  get 

X*  -  20x^  +  94x^  +  60x  +  9  =  0;  extracting  the  square  root  of 
each  side,  we  have  x^  -  \0x  -  3  -  0  .-.  x^  -  lOx  +  25  =  28: 
X-  5  =  ±  2V7  •••  a:  =  5  +  2V7 

217.  Multiplying,  we  have  -  30x*  +  46a*  +  7x*-23x  +  4  =  4 
.-.  x(30x^  -  4Gx-  -  7x  +  23)  =  0 

.-.  x{30a=*  -  30X''  -  lOx^  +  16x  -  23x  +  23}  =  0 

.-.  x{30x'''(x  -  1)  -  16.t(x  -  1)  -  23(x  -  1)}  =  0 

.-.  x(x  -  l)(30x2  _  iQj.  _  23)  =  0  .-,  X  =  0.    Also  x  -  1  =  0  .-.  x  =  I 

Also    30x-  -    16x    =    23,    whence    x*  -     *,x    4-    jVV    =    III 

.-.  X  =  -^\(i  f  iV^54) 

218.  The  given  series  is  double,   i.  e.  is  equal  to  the  .i  series 

1+2  +  3  +  4  +  5  +  &c.,   +  the  G  series  1  -  2  +  4  -  8  +  16  -  (to. 

Then  sum  of  .^  series  as  follows  : — 

4;i 
S,„  =  {2  +  (4,. -l)}y  =  (4/i+I)2/. 

4n  +  1 

S^n.x  -  {2  ^  (4'i  +  1  -  Dl—^T"  "  (-'*  ^  '^(^"  ^  ^> 

*\n..  -  {2  +  (4«  +  2  -  1)}^^  =  (4n  +  3)(2«  +  1) 
S^n*s  =  {2  +  (4;»  +  3  -  1)}^^-J~  =  ^<"  ^  ^^<*"  ^  ^^ 


r.lis.  l^x.a3.219.J  ALQiiBllA.  201 

Also  Slim  of  G  soiie.^  as  follows : — 

(  -2)-"-  I        10"-  1         1 

■  =  -:-(!  -  16") 


'4n  +  3   -  _  3  -     3 

.-.  of  given  series  5,„  =  2n(4ft  +  1)  +  J(l  -  16") 

-^4™.!  =  (2«  +  l)(4n  +  1)  +  J{1  -  (  -  2)*"'M 
«4n^.  =  (4«  +  3)(2n  +  1)  +  Kl  -  4-"^^) 
5^4™^.  =  2(n  +  1)(4«  +  3)  +  i{  I  -  (  -  2)*"^'} 
2 IP.  Let  X  =  nnmber  in  width,  and  y  -  number  in  the  length  ; 
then  xy  =  whole  number  in  the  bunch.     Also,  since  y  >  10  but 
<  20,  y  =  a  number  of  two  digits  .-.  when  x  is  written  to  the  left 
of  y  it  must  occupy  the  third  or  hundreds  place  .-.    IOOj:  +  y 
=  the  number  in  scale  of  10. 

Also  since  x  <  10,  it  consists  of  but  one  digit,  therefore  when 
written  to  the  left  of  y,  the  number  will  be  represented  by 
lOy  +  X  which  .-.  =  number  in  scale  of  10 

Again  in  similar  rectangles  the  perimeters  are  as  the  cor- 
responding sides,  and  whole  perimeter  of  first  bunch  =  2(x  +  y), 

x^y 
and  of  second  bunch  xy  .-.  2(x  +  y)  :  xy  ::  x  :  — — y — r  =  width 

xy'' 
of  2nd  bunch,  and  2(x  +  y)  :  xy  ::  y  :  — — -— r  -  length  of  2nd 

bunch    .•.    whole    number   of    matches    in    the    second    buncb 
x'-y  xy'^  a'y* 


2(x  +  y)       2(x  +  y)       4(x  +  yf 
Then  from  first  condition  lOOx  +  y  :  xy  :;  a  :  2  (i) 

"         second     "  \0y  +  x  :  xy  ::  a  -  10  :  4  (ii) 

"     "-^''^    "      4(^i;y^  =  4^^  <"'> 


202  KEY  TO  [Mis.  Ex.  219. 220. 

From  (ii)  20y  +  2x  :  xy  ::  a-  10  :  2  .-.  20y  +  2x  +  5y  :  xy  y  a  •  " 
.-.  20y  +  2x  +  5xy  :  xy  ::   lOOx  +  y  :  xy 

.-.  20y  +2x4-  5xy  =  lOOx  +  y  .-.  5xy  =  98x  -  19y  (iv) 
Also  from  (iii)  x V  =  l'5(x  +  y)^  .-.  xy  =  4(x  +  y)  .--  5xy  =  20(x  -t-  y) 
Substituting  tliis  in  (iv),  we  have  20x  +  2y  =  93x  -  19y,  whence 
2x  =  y. 

Again  substituting  this  in  (iii),  we  have  x^  =  16(x  +  y)^, 
that  is  4x*  =  lG(x  +  2x)^  .-.  2x''  =  4  x  (3x)  .-.  2x^  =  12x,  or  x  ^  G 
.•.  y  =  12  ;  and  xy  =  6  x  12  =  72  =  number  of  matches  in  llie 
bunch. 

220.  Since  the  conditions  giving  the  equations  (i)  and  (n) 
remain  the  same,  these  equations  and  .•.  also  (iv)  which  is  derived 
from  them  independently  of  (iii),  remain  the  same. 

•.  we  have  but  to  solve  in  positive*  integers  the   equation 
5xy  =  98x  -  19y,  remembering  that  x  <  10,  and  y  >  10  but  <  20 
5xy  =  98x  -  19y 
.-.  5xy  +  19y  =  98x 


1862 


1862 
Now  since  y  is  an  integer,  ^"7~,^  is  also  an  integer. 

And  .since  x  is  integral,  5.r  +  19  must  equal  an  integral  divi.^or 
of  1862,  and  further  since  x  is  finite,  positive  and  less  than  10, 
5x  +  19  will  be  >  19  but  <  69  and  will  end  in  9  or  4  according 
as  X  is  even  or  odd. 

Now  the  only  divisor  of  1862  fulfilling  these  conditions  is  49 
.-.  5x  +  19  =  49 

.-.  y  =  6 

98x 

=  12 


*'      5X+19 
XV  =  72 


•They  must  be  positive  IVom  the  nature  of  the  problem. 


Mis.  Ex.221.]  ALGEBRA.  203 

221.  Let  X  =  rate  per  hour  of  the  express  do-wn ;   y  =  rate  per 

hour  of  accommodation  down,  and  d  =  distance  from  Stratford 

d  ,    d 

to  Toronto.     Then  —  hours  =  time  down  by  express,  and  — 

X  y 

d 
=  time  down  by  accommodation.     Also  —  =  cents  per  mile  in 

d  d'^ 

express    fare,    and   —   x   d   -   —   =   whole    fare    by    express. 

d                                                                  d 
X  -  —  =  rate  of  expressing  going  up  .-. r  =  hours  on  road 

X 

going  back  ^' 

But  if  the  fares  had  varied  as  the  velocities  ;    then  fare  at 
X  :  fare  at  y  : :  r  :  y  .-.  fiire  at  x  -  fare  at  y  :  fare  at  x  :;  x  -  1/  :  x 

But  in  this  case,  fare  at  x  -  fare  a,t  y  =  d  cents,  and  since  fare 

d' 
ny  express  to  Toronto  remains  the  same,  d  :  —  ::  x  — y  :  x  (i) 

d\  d 

Also  fare  at  x  :  fare  at     x ::   x  :  x  -  —  .-.  fare  at 


d\  d 

X  —  fare  at  (  x  -  —  )   ;  fare  at  x  : :  —  :  x 


f  d\  d 

But    fare    at    x    -    fare    atlx-  —  )    -    x    -    —    cents 

\  X  X 

d      d^        d  ^  ^ 

.•.  X :  —  : :  —  :  x  (u) 

XX  X  ^ 

Using  the  formulas  now  found  in  -expressing  the  remaining 

d       1  /dy 

statements   in   the   problem,   we   obtain    —  ~  "Tl  —  I      ('")  > 

Then  from  (m)  dy  =  2x^  (iv) 

from    (i)  x^  =  rf(x  -y)  =  dx  -  2x^,  by  (iv)  .-.  d  =  3x  (v) 
from  (11)  x^(x2  -d)  =  d^  .-.hy  (v)  xXx^  -  3x)  =  27x3 
•.  x^  -  3x  =  2Tx  .-.  X  -  3  =  27  .-.  x  =  30  .-.  d  -^  3x  =  90  miles 

d^       90  X  90 

«  distance    from    Toronto    to    Stratford ;    and    —  =  — :^ — 

'  x  30 

=  270  cents  -  $2-70  =  fare  from  Toronto  to  Stratford. 


204  KEY  TO  (Mra.  Ex.  222-224. 

222.  xyxHTbix^  +  9)(VPT25  -  1)  -  ^b-Jx^  +  25  =  5(x*  +  45) 
.-.  (i)  x^^p^Tb{x^  +  9)(VP+^  -  1)  =  5(x2  +  25  +  9VF+T5  +  20) 
=  5{(-/F+25)(VJ^TT5)  +  9V^?T25  +  20} 
=  5{(Vx^+25)(Vj^"25)  +  5Va:^  +  25  +  4Vx^+25  +  20} 
r  5(/F+T5  +  5)(VF+T5  +  4)  (II) 


But  (/FT25  +  5)(V:F+  25  -  5)  -  x''        ^ 

J.   .-.  from  (i)  and  (ii) 

And  (y/x''+20  +  4)(V^^-  25  -  4)  =  x^  +  9j 

(V^M^23  +  5)(Vx^+25  -  5)(Vx^TT5)(Vx^+25  +  4)(V-c^+25  -  4) 

(Vx^'iS  -  1)  =  5(Vj^T5  +  5)(V-r-  ■(-  2"5  +  4) 

.-.  (^/Jiy^TTS  -  5)(Vi^+25)(Vi^+2~5  -  4)(^y/FT^2~5  -  1)  =  5  (ill) 

.-.  (x^  +  25  -  5'/x^T25)(x^  +  25  -  5yfPT25  +  4)  =  5 

.-.  (x^  +  25  -  5Vj-n^25)2  +  4  (x^  +  25  -  5'/x^TT5)  =  5 

.-.  (x2  +  25  -  5V?+T5)2  +  4(x^  +  25  -  Vx^+25)  +  4  =  9 

.-.  x2  +  25  -  5Vx^T25  =  -  5  or  1 

.-.  (x2  +  25)  -  5V;?+1:"5  +  2/  =  f  or  2^^ 

5  +  V5  5  4  V^ 

.-.  V^"  +  '-S  =  -    .^   —  or '^ — 

.-.  x2  +  25  =  1(30  4  10V5)  or  i(54  i  10V29) 


\7hence  x  =  KV  +  lOV^  -  70)  or  i(V  +  10V29  -  46 
Also  -^x'+Jb  +5  =  0,  whence  'Jx'^+25  =  -  5,  or  x*  +  25  =  25  .-.  x  =  0 
VPT^S  +  4  =  0,  whence  VPTUB  =  -  4,  or  x-  +  25=  16  .-.  x^--9, 
or  X  =  +  3-f^n. 


223.  Let  X  =  number  of  yards  dug  at  $1-25;  then  100  -  j 
-  number  of  yards  dug  at  $0-75  .-.  l-25x  =  50  =  -75(100 -x) 
Therefore,  we  have  two  independent  equations  containing  onlj 
one  unkuown  quantity,  and  any  solution  obtained  from  on<i 
equation  is  inconsistent  with  the  other  ;  consequently  thn 
problem  is  impossible. 

224.  Let  X  =  length  of  one  side  of  rectangle  and  y  =  other  ; 
then  xy  =  area,  and  2(x  +  y)  =  perimeter  of  the  r&ctaagle ; 
and  xy  =  area  and  4'^xj/  =  perimeter  of  the  square. 


Mis.  Ex.  224. 225  ]  ALGEBRA.  205 

(i)  .-.  xy  =  Am^lyy  (i),  and  xy  -  2n(x  +  y)  (ii).  From  (i)  Viy 
=  4/;t  .-.  xy  -  16//!^  (m),  substitute  this  in  (ii),  and  we  get  \^rr? 
=  2n(x  +  y)  .-.  8;?i^  =  n(x  +  y)  (iv).  Squaring  (iv),  we  have 
64«i*  =  n'{x  +  yY ;  multiplying  (in)  by  4/i^,  we  have  64/tt^i^  =  4n''xy 
•-  by  subtraction  64//i^(7n.^  -  n^)  =  n^(x  -  y)'^  .-.  ±  8m^w^  -  ?i^ 

=  n(x-y)  (v) 

4))i 


Adding  (iv)  and  (v)  and  reducing,  we  get  x  = (m  ±  ^iri''  —  ri^) 


Am 


Taking  (v)  from  (iv)  and  reducing,  we  get  y (m  :f  ^/n^  -n') 

(ii)  When  the  perimeters  are  equal;  then  takingxandyas  before, 

(x  +  yy 
we  have  2(x  +  y),=  perimeter  of  the  square,  and  — - —  -  its  area ; 

(X  +  y)'^ 

=  2/rt(x4-?/)  (r)  ;  xy  ^  2n{x  +  y)  (ii).     From  (i)  x  +  y 

=  8hi  (hi),  substitute  this  in  (ii),  and  xy  =  IGmn  (iv) 
Square  (m),  subtract  4  times  (iv)  and  then  take  the 
square  root,  and  we  have  x  -  y  =  ±  i'^Jm^  -  mn  (v) 
Adding  (in)  and  (v)  and  reducing,  x  =  4(m  ±  ^/n^^^^lflli) 
Subtracting  (v)  from  (ni)  and  reducing,  y  -  A{^ii\   '\  ^iti^  -   mn) 

225.  Let  X  =  age  of  younger  at  first  trial,  and  y  -  age  of  elder. 

Let  r  =  ratio  of  throw  to  age  at  first  trial,  and  r  =  ratio  of  gain 

of  one  to  age  of  the  other  at  second  throw  .-.  first  throw  of 

younger  =  rx,  and  first  throw  of  elder  =  ry;  gain  of  younger 

=  r  (y  +  1)  ;   gain  of  elder  =  r  (x  +  1)  ;   second  throw  of  younger 

=  rx  +  r  (y  +  1)  ;    second  throw  of  elder  =  ry  -^  r  (x  +   1). 

2(x4- l)(y  4- 1) 

Also  H.  mean  of  their  ages  at  latter  trial  = r ; 

"  X  4-  y  4-  2        * 

r(x  4-  y) 
^.  mean  of  first  throws  =   — -z — ,  and  ^.  mean  of  2nd  throws 

r(x  4-  y)  4-  r,(x  4-  y  4-  2)                                                 r,(x  4- y  4- 2) 
=  ^ .•.  difference  of  ./I.  means  = ^ 

Longest  throw  =  second  throw  of  the  elder  =  ry  4- r  (x  4-  1); 
value  of  ratios  compounded  of  ratio  of  throw  to  age  and  gain 


206  KEY   TO  [Mis.  Ex.  226 

to  age  of  other  =  rr  ;  value  of  ratio  formed  by  multiplying 
antecedent  of  this  eompound  ratio  by  J  product  of  ages  at 
second  trial  =  irr^(x  +  l)(y  +  1);   value  of  the  ratio  of  which 


this  is  the  duplicate  =  iV",(^  +  l)(y  +  1)  ;    value  of  the  ratio 
compounded  of  the  ratio  of  throw  to  age  of  one  with  gain  of  one 

r 
to  age  of  other  =  — 
I 
Then  using  the  values  thus  expressed  in  stating  the  problem, 
we  have  the  four  equations : — 
ry  -  7x  =  24  ;  or  r(y  -  x)  =  24  (i) 

\ry  +  r  (X  +  1)}  -  {rx  +  r^y  +  1)}  =  25  ;  or  (r  -  r  )(y  -  x)  =  22  (u) 
ry  +  r,(x  +    1)       2(x   +    l)(y  +   1) 
r,(x  +  y+2)    =   ^T7^^2 ',    or   r   =    r,(x  +    1)   (iii) 


W'-,(x   +    l)(y   +    1)   =   — ;    or   (X   +    l)(y   +    l)r»   =    4r    (iv) 

t 
Then  (i)  -  (ii)  gives  r  (y  -  x)  =  2  (v)  ;  substituting  (iii)  in  (ii) 
rx(y  -  x)  =  22  (vi)  ;  dividing  (iv)  by  (v),  we  have  x  =  II 
=  age  of  younger  at  first  throw  (vii).  Substituting  (vii)  in  (iii) 
r  =  12r  ,  and  in  (v)  r  (y  -  11)  =  2  ;  also  substituting  (iii)  in  (iv) 
and  reducing,  r*(i/  +  1)  =  4  (vm).  But  r(y  -  11)  =  ? 
.".  r^(y  -  11)  =  2r  ;  subtracting  this  from  (vui),  we  have 
12;f  -  4  -  2r  .-.  6r-'+  r  =  2  .-.  r »  +  ^r  +  -rir  =  i  +  r\i  =  T^\ 
■■•  '■j  =  i  iV  -  I'i  =  i-  S"<^  ^  =  ^2r  .-.  r  =  6,  and  since 
r(y-  11)  =  2;  i(y  -  11)  =  2  .-.  y  -  11  =  4,  or  y  =  15  =  age  of 
elder  .-.  throws  at  first  trial  =  11  x  6  =  66,  and  15  x  6  =  90  ;  and 
throws  at  second  trial  =  66  +  K15  +  1)  =  74,  and  90  +  J  (11  +  1) 
-96 

TBS  BKD. 


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tin  EaMy  Mode  of  Teacliinfc  tlio  Riidioientii  of  Latin 
(iranmmar  to  Beginners.  By  Tuohas  Javfrsx  &o- 
T-.y.nTSOV,  Esq.,  M.A , i 

Tro:  jise  on  FrencU  Pronunciation  and  Gonder*. 
l^y  J.  B.  Anoklvy  Lafont,  K.-q.,  French  Ma*  r,  Gnunmar 
and  Central  Schools,  Hamilton.  Out 

f'iiinock'w  improved  Edllfo-  ^r  <;o!d«n:  rZt's  His- 
tory of  England.  {Sec  niEditiOv.)  By  W. 
C.Taylor,  LL.D,  T.C.D.  ..:. - 

Element!*  of  Elocution.    By.,.  ..  >„....u,  M.U.C.S ' 

itluHhio  Header.    By  Kev.  Wair,  Wamiltos,  D.Dj»|^t .(' 

Uriti-<!i  American  Reader,    ^iy  Key.  .1.  O.  ^aWhwicK... 

OutlJuoM  of  tthrouolosy.    By  Mrs.  Go;  "—^       * 

<'la«<sical  EnjrllMli  SpeJIius  Pook.    I 

A  Couiprchenxive  System  ol  iSook-I 
sle  and  Double  Entry.  By  Th 
Accountant 

'f.ie  A-B-C  Simplified,  and  Reading  M!;> 

oity  ofl  ittlt  Children.    By  G.  G.  Vaskv 

JOHN  LOVELL,  rrviler  an.: 
.  July,  1871. 


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