SFtraveler wrote:For the following problem, I got slightly different answers, depending which side I collected the b's on to solve for b.

When I collected the b's on the left side, I got to the right answer: b=(c-ad)/(1-d).

When I collected the b's on the right side, I got (ad-c)/(d-1)=b, which wasn't among the answer choices.

Did I do something wrong or why would I get two different answers depending on which side I collected the b's?

there's no problem attached, but here' the thing: the two expressions you have are the same expression.

1-d is -1*(d-1): multiply d-1 by -1 and you'll get -d -(-1), or -d+1 = 1-d
By the same token, c-ad is -1(ad-c).

Thus, the second expression (ad-c)/(d-1), is actually the first expression multipled by -1 in both top and bottom:
(ad-c)/(d-1) = -1(c-ad) / -1(1-d).
Since you multiply both the top and the bottom of the fraction by the same constant, you haven't really changed the fraction: you can then reduce the -1s and end up with
(ad-c)/(d-1) = (c-ad) / (1-d) - so the two "different expressions" are actually equal.
If you collected all the b's on the right side and didn't find the expression in the answer choices, all you needed to do is reverse both the the top and the bottom (the equivalent of multiplying and dividing by -1, so the fraction isn't changed), and you will have found your answer.