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What is the solution (x, y) of the following system of equat

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by Max@Math Revolution » Wed Nov 27, 2019 11:57 pm
[GMAT math practice question]

What is the solution (x, y) of the following system of equations?

7/x+y + 3/x-y = 1 and 1/x+y - 2/x-y = 5

A. (3/4, 1/4 )
B. (3/5, 1/4 )
C. (1/4, 3/4 )
D. (3/5, 1/5 )
E. (3/4, 1/5 )
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by GMATGuruNY » Thu Nov 28, 2019 2:36 am
Max@Math Revolution wrote:[GMAT math practice question]

What is the solution (x, y) of the following system of equations?

7/x+y + 3/x-y = 1 and 1/x+y - 2/x-y = 5

A. (3/4, 1/4 )
B. (3/5, 1/4 )
C. (1/4, 3/4 )
D. (3/5, 1/5 )
E. (3/4, 1/5 )
ALWAYS KEEP YOUR EYE ON THE ANSWER CHOICES.

The correct answer must satisfy the following equation:
7/(x+y) + 3/(x-y) = 1
In every answer choice, x+y ≤ 1, with the result that 7/(x+y) ≥ 7.
Implication:
For the left side of the equation to sum to 1, 3/(x-y) must be NEGATIVE.
3/(x-y) will be negative only if x < y.
Only C is viable:
1/4 < 3/4

The correct answer is C.
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by Max@Math Revolution » Sun Dec 01, 2019 7:42 pm
=>

Assume A = 1/(x+y) and B = 1/(x-y).
Then we have 7A + 3B = 1 and A - 2B = 5 or A = 2B + 5.
Substituting the second equation into the first gives us 7(2B+5)+ 3B = 1, 17B + 35 = 1, 17B = -34, and B = -2.
Substituting B = -2 into A = 2B + 5 gives us A = 2(-2) + 5, A = 1.
Then x + y = 1/A = 1/1 = 1 and x - y = 1/B = 1/(-2) = -1/2.
Adding x + y = 1 and x - y = -1/2 gives us x + y + x - 7 = 1 - 1/2.
We have 2x = 1/2 or x = 1/4.
Then we have y = 1 - x = 3/4.

Therefore, the answer is C.
Answer: C
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