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Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8001
GMAT 1: 760 Q51 V42 GPA: 3.82
If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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Difficulty:   25% (medium)

Question Stats: 79% (02:01) correct 21% (02:34) wrong based on 84 sessions

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[Math Revolution GMAT math practice question]

If $$2x+y=3$$ and $$4x^2-4xy+y^2=1$$, what is the value of $$xy$$?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$

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NUS School Moderator V
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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1
$$2x+y = 3$$, $$y = 3-2x$$
$$4x^2-4xy+y^2=1$$...............(1)

Substituting the value of y in equation 1, we get
$$16x^2-24x+8 = 0$$
$$2x^2-3x+1 = 0$$
$$2x^2-2x-x+1 = 0$$
$$2x(x-1)-1(x-1) = 0$$
$$x = 1 or 1/2$$
Then, y becomes 1 when x is 1 or 2 when x is 1/2
xy becomes 1

C is the answer.
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Director  P
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If $$2x+y=3$$ and $$4x^2-4xy+y^2=1$$, what is the value of $$xy$$?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$

Square the first equation, we have -

$$4x^2+4xy+y^2=9$$ and $$4x^2-4xy+y^2=1$$.... Subtracting we get -

xy = 1
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CEO  V
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Posts: 3995
Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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Top Contributor
MathRevolution wrote:
[Math Revolution GMAT math practice question]

If 2x + y = 3 and 4x² - 4xy + y² = 1, what is the value of xy?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$

This question becomes a lot easier if we recognize that (2x + y)² = 4x² + 4xy + y² (which is VERY similar to 4x² - 4xy + y²)

Here's what I mean:

Take 2x + y = 3 and square both sides to get: (2x + y)² = 3²
Simplify both sides to get: 4x² + 4xy + y²= 9

We now have two (quite similar) equations:
4x² + 4xy + y²= 9
4x² - 4xy + y² = 1

Subtract the bottom equation from the top equation to get: 8xy = 8
Divide both sides by 8 to get: xy = 1

Cheers,
Brent
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If $$2x+y=3$$ and $$4x^2-4xy+y^2=1$$, what is the value of $$xy$$?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$

$$4x^2-4xy+y^2=1$$

=> (2x+y)^2 - 4xy -4xy =1
=> (2x+y)^2 - 8xy = 1

Given 2x+y = 3

=> 9 -8xy = 1
=> 8xy =8
=> xy =1

Hence ans. C)
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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MathRevolution wrote:
[Math Revolution GMAT math practice question]

If $$2x+y=3$$ and $$4x^2-4xy+y^2=1$$, what is the value of $$xy$$?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$

Given

$$4x^2-4xy+y^2=1$$,

$$(2x - y)^2 = 1$$

2x - y = 1.

2x + y = 3
2x - y = 1
......................
4x = 4

x = 1. and y = 1.

xy = 1.

C is the correct answer.
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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First substitute 4x^2-4xy+y^2=1
Since it is in the form of (a-b)^2, the above equation can be written as (4x-y)^2=1

Also, 2x+y=3, then y=3-2x. Substitute this in the above equation, we get (2x-3)^2=1

Expand it we get 2x^2-6x+9=1

2x^2-6x+8=0
By solving the equation we get (x=-4 or x=1)

Substituting x=1 in y=3-2x, we get x=1 and y=1.

Hence, xy=1, Answer is C.

Kindly let me know if my approach is wrong.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 8001
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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=>

Since $$2x+y=3, (2x+y)^2 = 4x^2+4xy+y^2 =9.$$
When we subtract $$4x^2-4xy+y^2=1$$ from $$4x^2+4xy+y^2 =9$$, we obtain $$(4x^2+4xy+y^2) - (4x^2-4xy+y^2) = 8xy = 9 - 1 = 8.$$
So, $$xy = 1.$$

Therefore, the answer is C.
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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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We can test values for this question. One can quickly spot that x = 1 and y=1 works for both equations.

Therefore xy = 1 (Answer choice C).

It took less than 40 secs to solve

MathRevolution wrote:
[Math Revolution GMAT math practice question]

If $$2x+y=3$$ and $$4x^2-4xy+y^2=1$$, what is the value of $$xy$$?

$$A. \frac{1}{8}$$
$$B. \frac{1}{2}$$
$$C. 1$$
$$D. 2$$
$$E. 4$$ Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?   [#permalink] 03 Jan 2019, 11:01
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If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?

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