If x + 5/y = 4 and y + 5/x = -4, then what is (x + y) equal to?
Answer
The correct answer is A: 0. Adding the equations: x + y + 5/y + 5/x = 0, i.e., (x + y) + 5(x + y)/(xy) = 0, so (x + y)[1 + 5/(xy)] = 0. Thus either x + y = 0 or xy = -5.
A.0✓ Correct
B.1
C.4
D.5
Explanation
Adding the equations: x + y + 5/y + 5/x = 0, i.e., (x + y) + 5(x + y)/(xy) = 0, so (x + y)[1 + 5/(xy)] = 0. Thus either x + y = 0 or xy = -5. Subtracting: x - y + 5/y - 5/x = 8, i.e., (x - y) + 5(x - y)/(xy) = 8, so (x - y)[1 + 5/(xy)] = 8. If xy = -5, this gives 0 = 8, contradiction. Hence x + y = 0.
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