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

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What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 28 Nov 2019, 01:24
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Question Stats:

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[GMAT math practice question]

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

\(\frac{7}{x+y} + \frac{3}{x-y} = 1\) and \(\frac{1}{x+y} - \frac{2}{x-y} = 5\)

A. (\(\frac{3}{4}, \frac{1}{4}\))

B. (\(\frac{3}{5}, \frac{1}{4}\))

C. (\(\frac{1}{4}, \frac{3}{4}\))

D. (\(\frac{3}{5}, \frac{1}{5}\))

E. (\(\frac{3}{4}, \frac{1}{5}\))

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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 28 Nov 2019, 03:49
MathRevolution wrote:
[GMAT math practice question]

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

\(\frac{7}{x+y} + \frac{3}{x-y} = 1\) and \(\frac{1}{x+y} - \frac{2}{x-y} = 5\)

A. (\(\frac{3}{4}, \frac{1}{4}\))

B. (\(\frac{3}{5}, \frac{1}{4}\))

C. (\(\frac{1}{4}, \frac{3}{4}\))

D. (\(\frac{3}{5}, \frac{1}{5}\))

E. (\(\frac{3}{4}, \frac{1}{5}\))


ALWAYS KEEP YOUR EYE ON THE ANSWER CHOICES.

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


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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 28 Nov 2019, 10:08
Quote:
In every answer choice, x+y≤1x+y≤1, with the result that 7x+y≥77x+y≥7.


Hi can you pls explain this in detail. I did not understand how you reached x+y <= 1
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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 28 Nov 2019, 13:05
1
devavrat wrote:
Quote:
In every answer choice, x+y≤1x+y≤1, with the result that 7x+y≥77x+y≥7.


Hi can you pls explain this in detail. I did not understand how you reached x+y <= 1


The answer choices represent options for (x, y).

Every option yields a sum for x and y that is less than or equal to 1:
A --> \(\frac{3}{4} + \frac{1}{4} = 1\)
B --> \(\frac{3}{5} + \frac{1}{4} = \frac{17}{20}\)
C --> \(\frac{1}{4} + \frac{3}{4}\) = 1
D --> \(\frac{3}{5} + \frac{1}{5} = \frac{4}{5}\)
E --> \(\frac{3}{4} + \frac{1}{5} = \frac{19}{20}\)
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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 01 Dec 2019, 21:07
=>

Assume \(A = \frac{1}{(x+y)}\) and \(B = \frac{1}{(x-y)}.\)

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 \(x + y = \frac{1}{A} = \frac{1}{1} = 1\) and \(x – y = \frac{1}{B} = \frac{1}{(-2)} = \frac{-1}{2}.\)

Adding \(x + y = 1\) and \(x - y = \frac{-1}{2}\) gives us \(x + y + x - 7 = 1 - \frac{1}{2}\).

We have \(2x = \frac{1}{2}\) or \(x = \frac{1}{4}.\)

Then we have \(y = 1 – x = \frac{3}{4}.\)

Therefore, the answer is C.
Answer: C
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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 01 Dec 2019, 22:33
MathRevolution wrote:
[GMAT math practice question]

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

\(\frac{7}{x+y} + \frac{3}{x-y} = 1\) and \(\frac{1}{x+y} - \frac{2}{x-y} = 5\)

A. (\(\frac{3}{4}, \frac{1}{4}\))

B. (\(\frac{3}{5}, \frac{1}{4}\))

C. (\(\frac{1}{4}, \frac{3}{4}\))

D. (\(\frac{3}{5}, \frac{1}{5}\))

E. (\(\frac{3}{4}, \frac{1}{5}\))

1/(x+y) = a, 1/(x-y) = b
7a + 3b = a
a - 2b = 5
Solving for a and b
a = 1, b = -2

x + y = 1
x - y = 1/2

x = 1/4
y = 3/4

C is correct.
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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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New post 03 Dec 2019, 04:31
Consider x+y=u & x-y =v
Solve for => 7/u +3/v = 1 ....Eq1
1/v -2/u = 5. .....Eq2
Solve for u & v
Then solve for x & y.
Correct ans C

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Re: What is the solution (x, y) of the following system of equations?   [#permalink] 03 Dec 2019, 04:31
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