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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8235
GMAT 1: 760 Q51 V42 GPA: 3.82
What is the solution (x, y) of the following system of equations?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 59% (02:07) correct 41% (02:48) wrong based on 29 sessions

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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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Joined: 04 Aug 2010
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Schools: Dartmouth College
Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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

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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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
Senior Manager  G
Joined: 04 Aug 2010
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Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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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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Math Revolution GMAT Instructor V
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GMAT 1: 760 Q51 V42 GPA: 3.82
Re: What is the solution (x, y) of the following system of equations?  [#permalink]

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=>

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 $$y = 1 – x = \frac{3}{4}.$$

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

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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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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

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