way to complicated, try to get the answers and plug in back... (IMO)

Well, I got x/y=9 by doing the following:

We have:

sqrt(x)+sqrt(y)/x-y=2sqrt(x)+2sqrt(y)/x+2sqrt(xy)+y

We can rewrite the above equation as:

sqrt(x)+sqrt(y)/x-y=2(sqrt(x)=sqrt(y))/(sqrt(x)+sqrt(y))^(2)

Now cross multiply:

(sqrt(x)+sqrt(y))^(3)=2(sqrt(x)+sqrt(y))(x-y)

One of the sqrt(x)+sqrt(y)'s cancel and you're left with

(sqrt(x)+sqrt(y))^(2)=2x-2y

x+2sqrt(xy)+y=2x-2y

2sqrt(xy)+3y=x

x-2sqrt(xy)-3y=0

Now factor:

(sqrt(x)-3sqrt(y))(sqrt(x)+sqrt(y))=0

sqrt(x)-3sqrt(y)=0

sqrt(x)=3sqrt(y)

Square both sides:

x=9y

x/y=9

This problem was very time consuming. I really don't think a problem like this would be on the GMAT.

How do I manipulate x/y from [1/(x-y) = 2]? Thanks.

There's no way you can get x/y from 1/(x-y)=2

1=2x-2y. That's as far as you can get.

But I got [1/(x-y) = 2] just by canceling out so I didn't make any illegal operations. How can it be a dead end?

after cancellation you should get (√x+√y)^2=2(x-y). Please recheck