Solve the equation 2 – x = 2b – 2ax, where a and b are real parameters. Find for which values of a the equation has for solution a natural number, if b = 7	Choose the correct option:
A) 0 B) 1 C) 2 D) 3 E) 4
Solution:   We write the given equation like this (2a - 1)x = 2(b - 1)   The following cases are possible:   If 2a - 1 ≠ 0, i.e. a ≠ 1/2, the equation has single solution   x = 2(b - 1)/(2a - 1)   If a = 1/2 and b = 1, the equation gets the kind 0.x = 0 and every x is solution   If a = 1/2 and b ≠ 1, we get 0.x = 2(b - 1), where 2(b - 1) ≠ 0   In this case the equation has no solution.   If b = 7 and a ≠ 1/2 the single solution is   x = 2(7 - 1)/(2a - 1) = 12/(2a - 1)   If a is whole number, then 2a - 1 is also whole number and the solution   x = 12/(2a - 1) is natural number when   2a - 1 is positive divisor of 12.   To have a as whole number, the divisor of 12 mu be odd.But the only whole positive odd numbers divisible by 12 are 1 and 3.   Therefore 2a - 1 = 3 &lt;=&gt; a = 2 or 2a - 1 = 1 &lt; 2  - 1= 
<gadget id="calculator"> 2  - 1</gadget>
<output>1</output>
1 &lt;=&gt;   a = 1 a = 2 or 2a - 1 = 1 &lt; 2  - 1= 
<gadget id="calculator"> 2  - 1</gadget>
<output>1</output>
1 &lt;=&gt; a = 1   Answer 1
<result>B</result>
Solve the equation 2 – x = 2b – 2ax, where a and b are real parameters. Find for which values of a the equation has for solution a natural number, if b = 7