implicit differentiation problem

Attachment 28405
In the problem above I get the negative of the answer if I group the y' terms on one side of the equals sign, and the correct answer if I group the y' terms on the other. I can't figure out why this would be or how I could know i was incorrect with the negative version. Click the image twice to get a full page view.

Or is it that they two terms are, in fact, equal since both numerator and denominator are multiplied by -1?

The two results are equivalent, for the reason you stated.