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

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Intern
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If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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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 equal?

A. 0
B. 1
C. 2
D. 5
E. 10
Math Expert
Joined: 02 Sep 2009
Posts: 52906
Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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28 Feb 2012, 22:12
7
9
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,

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 $$2(x-2y)-10=0$$ --> $$x-2y=5$$.

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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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28 May 2014, 06:21
5
2
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.

General Discussion
Intern
Joined: 24 Feb 2012
Posts: 31
Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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28 Feb 2012, 22:46
1
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 equal?  [#permalink]

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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

BR
Mandeep
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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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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Posts: 52906
Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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03 Jul 2013, 00:24
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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03 Jul 2013, 01:54
1
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.
D.
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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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 equal?  [#permalink]

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15 Jul 2014, 22:24
3
$$x^2 - xy - 10 - 2y^2 = 0$$

$$x^2 - 2xy + xy - 2y^2 = 10$$

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

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

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

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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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22 Nov 2015, 10:19
oh man..tricky one..I couldn't see the short way to solve it:

(x+y) = 2 and (x+y)^2 = 4
multiply the second equation by 2 and get:
2x^2 -2xy-20-4y^2 = 0
2x^2 -2xy-20-4y^2 = 0+
x^2 +2xy +y^2 = 4

we get:
3x^2 -20 - 3y^2 = 4
or 3x^2 - 3y^2 = 24
divide by 3:
x^2 - y^2 = 8
rewrite the equation:
(x+y)(x-y) = 8
we have x+y = 2, that means x-y = 4
x=4+y

4+y+y = 2
2y +4 = 2
y = -1

ok, now get to x:
x-1 = 2 => x=3

x-2y = 3-(-2) = 3+2 = 5.
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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14 Dec 2015, 04:31
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.

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

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

Since x + y = 2 we have y=2-x.
So 0= x^2 - xy - 10 - 2y^2 = x^2 - x(2-x) - 10 - 2(2-x)^2 = x^2 -2x+x^2 -10-8+8x-2x^2 = x=6x-18. ---> x=3 ---> y= -1 --->x-2y=5

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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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14 Dec 2015, 04:59
1
Bunuel wrote:
If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?

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

x^2 - xy - 2y^2 = 10

by seeing the x^2 and -2^y attributes, (by trail and error method) (x-2y) (x+y) = 10

Given x+y = 2

x-2y = 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 equal?  [#permalink]

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20 Mar 2016, 10:54
I went with squaring the eq 1 and using that in eq 2

eq 1 :
$$x+y=2$$
$$(x+y)^2=4$$ --> or ( $$x^2 + 2xy + y^2 = 4$$)

now, eq 2 :
$$x^2 - xy - 2y^2 = 10$$

$$x^2 + 2xy + y^2 -2xy - y^2 -xy- 2y^2 = 10$$ --> adding and substracting (2xy + y^2)

$$(x+y)^2 -3xy -3y^2 = 10$$

$$(x+y)^2 -3y(x +y) = 10$$

$$4 -3y(2) = 10$$

$$-3y = 3$$ --> y = -1

$$y = -1$$ , and $$x = 3$$ (from x+y=2)

$$x-2y =5$$

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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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18 Feb 2017, 04:48
Hi..
i solved the question by substitution method , but i see the equation can be factorised too as per 2 nd soln.
How has it been factorised?
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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19 Feb 2017, 01:21
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

I just did the small logic that is x^2-(1)xy-2y^2=10
If I do factor then it will be like (x-2y)(x+y)=10
(x-2y)2=10 .......................As given (x+y)=2
(x-2y)=5

Ans is D

Please correct if i am wrong.
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?  [#permalink]

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05 Sep 2018, 23:17
eaze wrote:
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.

Hello,
In the second step, shouldn't y=-(x+2) instead of y=x-2 ?
Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?   [#permalink] 05 Sep 2018, 23:17
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