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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
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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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
Re: If x+y=2 and x^2 - xy - 10 - 2y^2 = 0, what does x-2y =?   [#permalink] 15 Jul 2014, 22:24
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