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If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

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If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 28 Feb 2012, 20:12
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If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

A. 0
B. 1
C. 2
D. 5
E. 10
[Reveal] Spoiler: OA
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 28 Feb 2012, 22:12
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vintage83 wrote:
If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

A. 0
B. 1
C. 2
D. 5
E. 10

Hi,
Can you please help me solve the equation above

Thanks much.


Welcome to GMAT Club. Below is a solution to your question.

Given: \(x^2-xy-10-2y^2=0\);

Rearrange \(x^2-y^2-y^2-xy-10=0\);

Apply \(a^2-b^2=(a-b)(a+b)\): \((x-y)(x+y)-y^2-xy-10=0\) --> \((x-y)(x+y)-y(y+x)-10=0\);

Factor out \(x+y\): \((x+y)(x-y-y)-10=0\);

Since given that \(x+y=2\), then we have that \(2(x-2y)-10=0\) --> \(x-2y=5\).

Answer: D.
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 28 Feb 2012, 22:46
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Very tricky....I confess...I knew there was a simpler route.....but I just did not see it.
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 29 Feb 2012, 02:22
Square the first equation and subtract from the second and there you are.

You would be able to factor out and find the values of x and y. x = 3 and y = -1

x - 2y => 3 - 2(-1) = 5

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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 29 Feb 2012, 06:33
Thank you very much. I get it now! I think I got confused at the last step and was stuck for a while.
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 03 Jul 2013, 00:24
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 03 Jul 2013, 01:54
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vintage83 wrote:
If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

A. 0
B. 1
C. 2
D. 5
E. 10


Given that \(x^2 - xy - 10 - 2y^2 = 0 \to x^2+4y^2-4xy-10-xy-2y^2=4y^2-4xy\) [By adding \(4y^2\) and \(4xy\) on both sides]\(\to\)
\((x-2y)^2 = 6y^2-3xy+10\)

Or \((x-2y)^2 = 3y(2y-x)+10 \to (x-2y)^2+3y(x-2y) = 10\to (x-2y)(x-2y+3y) = 10 \to (x-2y) = \frac{10}{x+y}\) = 5.
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 28 May 2014, 06:21
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vintage83 wrote:
If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

A. 0
B. 1
C. 2
D. 5
E. 10


This problem can still be easily solved without utilizing a "trick" in the algebra; you have two equations, just solve for x & y.

\(x + y = 2\)
\(y = x -2\)
\(x^2 - xy - 10 - 2y^2 = 0\)
\(x^2 - x(2-x) - 10 - 2(x^2 - 4x +4)\)
\(x^2 - 2x +x^2 - 10 - 2x^2 + 8x -8\)

Here, the quadratic conveniently cancels out, so we're left with a linear equation.

\(6x - 18 = 0,\)
\(x = 3.\)

Using x = 3 and our first equation, we have that y = -1, and thus that x - 2y = 3 - 2(-1) = 3+2 = 5.

Answer: D.
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 28 May 2014, 11:26
vintage83 wrote:
If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?

A. 0
B. 1
C. 2
D. 5
E. 10


x+y=2 ----------1); squaring both sides we have;
(x+y)^2=4 ------------2);
Now, x^2=10+2y^2+xy; adding y^2 and 2xy on both sides we have;

x^2+y^2+2xy = 10+ 3y^2+3xy;
(x+y)^2= 10+ 3y^2+3xy; from 2) we have
4=10+ 3y^2+3xy;
-6=3y(x+y);
from 1) x+y=2
-6=6y; y= -1;
therefore x=3 from 1)
thus x-2y= 3-(-2)=5
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 15 Jul 2014, 22:24
\(x^2 - xy - 10 - 2y^2 = 0\)

\(x^2 - 2xy + xy - 2y^2 = 10\)

x(x-2y) + y(x-2y) = 10

(x+y)(x-2y) = 10

\(x-2y = \frac{10}{x+y} = \frac{10}{2} = 5\)

Answer = D
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =? [#permalink] New post 18 Aug 2015, 17:43
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Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?   [#permalink] 18 Aug 2015, 17:43
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