x+y=9.....54
xy=9......91
x-y=9.....90
y-x=9.....impossible
x/y=9.....91

y-x=9 is impossible because the only two digits with a difference of 9 is 9 and 0, so it would have to be that y=9 and x=0, but that would make the number xy, 09 or just 9. But 9 is a one-digit number, not a two-digit number.

Ok thanks, I was thinking that in the question it was saying xy=9

Yeah it probably could have been written more clearly.

Since xy is a positive two-digit number, then xy=9 is it true? Confused.

Yeah, the confusing thing is that they use xy to mean two different things in this problem. When they say xy=9 in the answer choices they mean x*y=9. In other words, the ten's digit times the unit's digit equals nine. This is certainly possible if x is 9 and y is 1.

On the other hand, in the question they use xy to represent a two digit integer, where if x=9 and y=1 xy=91, and xy does NOT mean to multiply x and y.

Usually xy would be understood to mean "x times y" but since they define it in a different way in the question, they should really use a more explicit notation for multiplication.

Of course, the writers of a real GMAT question would be careful to avoid this kind of ambiguity.

even i read xy as "x times y"