Algebra Puzzle – Challenge 54

Challenge:

If \(9a^2 – b^2 = 11\) and a and b are positive integers, what is the value of \(a – b\)?

A- \(-5\)

B- \(-3\)

C- 3

D- 5

E- 7

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Factorize \(9a^2 – b^2\) and 11
(3a – b) (3a + b) = 11 × 1
Therefore, 3a – b equals to 11 or 1 and 3a + b equals to 1 or 11.
Let’s check both:
3a – b = 1 and 3a + b = 11
Solve the above system of equation:
a = 2 and b = 5
3a – b = 11 and 3a + b = 1
a = 2 and b = – 5
Since, a and b are positive integers, then, only a = 2 and b = 5 are the solutions.
Therefore, a – b = 2 – 5 = -3

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