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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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vintage83
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If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal? [#permalink] New post   28 Feb 2012, 21:12
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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
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D
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Bunuel
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal? [#permalink] New post   28 Feb 2012, 23:12
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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 equal? [#permalink] New post   28 May 2014, 07:21
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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 equal? [#permalink] New post   28 Feb 2012, 23:46
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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 equal? [#permalink] New post   29 Feb 2012, 03:22
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 equal? [#permalink] New post   29 Feb 2012, 07:33
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 equal? [#permalink] New post   03 Jul 2013, 01:24
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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] New post   03 Jul 2013, 02:54
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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.
D.
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal? [#permalink] New post   28 May 2014, 12:26
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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


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] New post   15 Jul 2014, 23:24
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\(x^2 - xy - 10 - 2y^2 = 0\)

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

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 equal? [#permalink] New post   22 Nov 2015, 11:19
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
now add 2 equations
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] New post   14 Dec 2015, 05:31
Expert Reply
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

The answer is (D).
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal? [#permalink] New post   14 Dec 2015, 05:59
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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] New post   20 Mar 2016, 11:54
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)

\(x-2y =5\)

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] New post   18 Feb 2017, 05:48
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] New post   19 Feb 2017, 02:21
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] New post   06 Sep 2018, 00:17
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.

Answer: D.


Hello,
In the second step, shouldn't y=-(x+2) instead of y=x-2 ?
please correct me
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Re: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal? [#permalink] New post   13 May 2019, 18:10
Here's my solution:

1) x^2 - 2y^2 = xy + 10, given that x = 2-y, then
2) x^2 - 2y^2 = 2y - y^2 + 10, now add y^2 to both sides and simplify
3) (x - y)(x + y) = 2y + 10, we know that x + y = 2, therefore
4) 2(x - y) = 2y + 10, which simplifies to
5) 2x - 4y = 10, 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] New post   25 Nov 2020, 20:01
I used substitution for this problem.

The first equation gives us the relationship between x and y.
x+y=2 --> y=2-x

Plug this into the second equation: x^2 - xy - 10 - 2y^2 = 0
x^2 - x(2-x) - 10 - 2(2-x)^2 = 0

Open this up and solve.
x = 3, y = -1
x-2y = 3 + 2 = 5
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If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x 2y equal? [#permalink] New post   15 Dec 2022, 22:32
Asked: If x + y = 2 and x^2 - xy - 10 - 2y^2 = 0, what does x − 2y equal?
x + y = 2
x = 2- y

x^2 - xy - 10 - 2y^2 = (2-y)ˆ2 - (2-y)y - 10 - 2yˆ2 = 4 - 4y + yˆ2 - 2y + yˆ2 - 10 - 2yˆ2 = -6 - 6y = 0
y = -1
x = 2- y = 2- (-1) = 3
x - 2y = 3 - 2*(-1) = 5

IMO D
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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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