Question: What values of x have a corresponding value of y that satisfies both xy > 0 and xy = x + y?

1. \(x\leq1\)
2. \(-1< x \leq 0\)
3. \(0 < x \leq 1\)
4. \(x > 1\)
5. All real numbers

Solution and Explanation:

Approach Solution (1):

Given: xy = x + y
Subtract y from both the sides to get: xy –y = x
Factor: y(x – 1) = x

Divide both the sides by x – 1 to get: \(y=\frac{x}{x-1}\)
Note that we know \(y=\frac{x}{x-1}\), we can readily see that x cannot be equal to 1, otherwise, the denominator in y is 0, which would make y undefined.

At this point, we can eliminate answer choices A, C, and E, since they all allow for x to equal 1.
Since we are down to just two answer choices, let’s just test some x- value
For example: if x = 2, then we get: \(y=\frac{x}{x-1}=\frac{2}{2-1}=2\)

So, x = 2 and y = 2 is a possible solution to the system of equations
Since it’s possible for x to equal 2, we can eliminate answer choice B
By the process of elimination, the correct option will be D

Correct Option: D

Approach Solution (2):

Given: xy = x + y
Subtract y from both the sides to get: xy –y = x
Factor: y(x – 1) = x
Divide both the sides by x – 1 to get: \(y=\frac{x}{x-1}\)
Now take the given inequality, xy > 0, and replace y with \(y=\frac{x}{x-1}\) to get: \((x)(\frac{x}{x-1})>0\)
Simplify: \((\frac{x^2}{x-1})>0\)

Since \(x^2\) is always greater or equal to 0, it must be the case that the denominator, x – 1, is positive
In other words, it must be the case that: x – 1 > 0
Add 1 to both sides of the inequality to get: x > 1

Correct Option: D

Approach Solution (3):

Since xy = x + y
xy – y = x
\(y=\frac{x}{x-1}\)
Since xy > 0
\((\frac{x^2}{x-1})>0\)
\(x\neq0\)
\(x-1>0\)
\(x>1\)

Correct Option: D

“What values of x have a corresponding value of y that satisfies both xy > 0 and xy = x + y?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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