For Q80 of the Official Guide, Quant. Review, I think there is a mistake. Please tell me if I'm wrong, I just can't understand how the OG is right here.

80. If xy>0, does (x-1)(y-1)=1

(1) x+y = xy
(2) x = y

Let's focus on (2)
If x=y, substituting in y for x gives you (y-1)(y-1)=1
Therefore y^2 -2y +1 = 1
Up to here me and the OG agree!

However, now I believe this can be solved. The 1's cancel out, leaving y^2 - 2y = 0
The y can be factored, giving y(y-2) = 0.

The GMAT claims this cannot be uniquely solved for y. BUT the question says that xy>0. That means that Y CANNOT BE 0. If y was 0, xy = 0! Therefore, y must equal 2. Therefore statement 2 IS sufficient, and the answer is D not A.

What are everyone's thoughts?
And is there a list somewhere of OG mistakes?

Thanks!