GMAT Practice Question: What values of x have a corresponding value of y that satisfies both x y>0 and x y=x+y ?
Question
What values of x have a corresponding value of y that satisfies both x y>0 and x y=x+y ?
- x \leq-1
- -1<x \leq 0
- 0<x \leq 1
- x>1
- All real numbers
Topics: simultaneous equations, inequalities
Solution
Step 1
We first rewrite the given relationship to express y in terms of x by isolating y.
x \times y = x + y x \times y - y = x y \times (x - 1) = x y = \frac{x}{x - 1}
Step 2
We then substitute this expression for y into the condition that the product of x and y is positive.
x \times y > 0 x \times \frac{x}{x - 1} > 0 \frac{x^2}{x - 1} > 0
Step 3
We note that x cannot equal 0 or 1. Because the numerator of the fraction is always positive when x is not zero, the fraction will be positive exactly when the denominator is positive.
x - 1 > 0 x > 1
Step 4
We conclude that the conditions are satisfied exactly when x is greater than 1.
x > 1
Answer
x > 1
