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

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New post 01 Jan 2019, 06:52
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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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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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New post 01 Jan 2019, 07:01
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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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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New post 01 Jan 2019, 07:07
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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Re: If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?  [#permalink]

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New post 01 Jan 2019, 09:37
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

Answer: C

Cheers,
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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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New post 01 Jan 2019, 10:23
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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New post 01 Jan 2019, 11:29
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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New post 01 Jan 2019, 11:54
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.
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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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New post 03 Jan 2019, 00:27
=>

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.
Answer: 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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New post 03 Jan 2019, 11:01
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 \(xy\)?

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