Right you are - the answer is D.
Here's my solution:
(1) The equation x+y = 0 tells us that x = -y
If x = -y, then the exponents in the numerator will be equal: -2y = 2x
So, we can replace -2y and 2x with some even number k to get x^k/y^k
We can simplify this to be (x/y)^k
If x = -y, then x/y= -1
So, (x/y)^k = (-1)^k
Since k is even, we can conclude that (-1)^k = 1
There is only one solution, so statement (1) is sufficient
(2) xy = x/y = y/x
Solve xy = x/y for y to get y=1 or y=-1
Solve xy = y/x for x to get x=1 or x=-1
We have four possible solutions (x,y). They are (1,1), (1,-1), (-1,1) and (-1,-1)
In the original rational expression, we are raising both x and y to the power of an even integer. So, the numerator and denominator will both evaluate to 1 for all four solutions.
So, we get 1/1 = 1
There is only one solution, so statement (2) is sufficient
The correct answer is D.
Brent Hanneson - Creator of GMATPrepNow.com
