Question: (x-y)/xy

Comment on (x-y)/xy

But by dividing the equation by 2x+1, are we not ruling out the case that (2x+1) could equal 0 i.e. x=-1/2
gmat-admin's picture

Great question!!!
It turns out that the original equation, (x - y)/xy = 2, already rules out the possibility that x = -1/2

To see why this is, let's replace x with -1/2 to see what happens.

We get: (-1/2 - y)/(-1/2)(y) = 2

Multiply both sides by -y/2 to get: -1/2 - y = -y

Add y to both sides to get: -1/2 = 0

So, there is no valid solution in which x = -1/2 (i.e., 2x+1 = 0)

Brent,

I love your videos! I had a quick question - I incorrectly applied arithmetic here and I can't figure out why it is wrong:

1... (x-y)/xy = 2
2... x/xy - y/xy = 2
3... 1/y - 1/x = 2
4... y - x = 1/2
5... y = x + 1/2

Thanks!
D
gmat-admin's picture

Your solution is solid up until line #4.

If we have a SINGLE fraction on each side, then we can use the approach you used.
That is, if x/y = a/b, then we can also say that y/x = b/a (provided a and x don't equal zero)
For example, if 3/6 = 1/2, we can also say that 6/3 = 2/1

However, if we have TWO fractions on one side, then the approach falls apart.

For example, we know that 1/2 - 1/4 = 1/4
However, we cannot then conclude that 2/1 - 4/1 = 4/1
Likewise, if 1/y - 1/x = 2, we cannot then conclude that y/1 - x/1 = 1/2 (as you have done in line #4)

Does that help?

Cheers,
Brent

Yes! Thanks for the quick response.

Hi Brent,

Is this question a candidate for the something method? I tried the something method, got stuck early on and did not pursue it further. But out of curiosity I wanted to check-in with you if the something method could be used here?
gmat-admin's picture

No, it's not a candidate for the Something Method, since there isn't anything to evaluate.
For example, if it is given that 6/(7x + 1) = 3, then I know that 7x + 1 = 2 (since 6/2 = 3)

In the video question above, we can't the deduce the values of any expressions.

Hi Brent, from x-y = 2xy , x-2xy = y, x(1-2y) = y. Wondering is this calculation also correct as the answer choice is not there? Therefore it's just a matter of calculation to match the answer choice here is needed right? Thanks Brent
gmat-admin's picture

The steps you took are correct, but you now have the variable y on both sides of the equals sign.

When it comes to questions where you are asked to solve an equation for a particular variable (y in this case), that variable cannot appear on both sides of the equal sign.

Consider this analogous example:
Solve the following equation for x: 3x + 1 = 6x - 11
Subtract 1 from both sides of the equation: 3x = 6x - 12
Divide both sides of the equation by 3 to get: x = 2x - 4
While all of my steps were correct, I still haven't solved the equation for x.

Similarly, while it's true that y = x(1-2y), we still don't know what y equals, because you have the variable y on the right side of the equation.

Great explanation thanks Brent. Was concentrated on y on its own without realising that xy has y too. Now I know why and noted. Thanks Brent.

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