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

Question

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

Bunuel · Accepted Answer

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.

GSBae · Answer

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.

fortsill · Answer

Very tricky....I confess...I knew there was a simpler route.....but I just did not see it.

samramandy · Answer

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

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Algebra: algebra-101576.html

DS Algebra Questions to practice: search.php?search_id=tag&tag_id=29
PS Algebra Questions to practice: search.php?search_id=tag&tag_id=50

Special algebra set: new-algebra-set-149349.html

mau5 · Answer

Given that  x^2 - xy - 10 - 2y^2 = 0 	o x^2+4y^2-4xy-10-xy-2y^2=4y^2-4xy   4y^2  and  4xy  on both sides] 	o  
 (x-2y)^2 = 6y^2-3xy+10

Or  (x-2y)^2 = 3y(2y-x)+10 	o (x-2y)^2+3y(x-2y) = 10	o (x-2y)(x-2y+3y) = 10 	o (x-2y) = \frac{10}{x+y}  = 5.
D.

manpreetsingh86 · Answer

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

PareshGmat · Answer

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

mvictor · Answer

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.

MathRevolution · Answer

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).

umasarath52 · Answer

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)

UJs · Answer

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

shikha.lakhani · Answer

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?

srdsagar · Answer

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.

Shri15kumar · Answer

Hello,
In the second step, shouldn't y=-(x+2) instead of y=x-2 ?
please correct me

attempttoo · Answer

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

jerryjz2690 · Answer

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

Kinshook · Answer

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

bumpbot · Answer

Automated notice from GMAT Club BumpBot:

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards