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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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by Max@Math Revolution » Tue Jan 01, 2019 5:40 am

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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. 1/8
B. 1/2
C. 1
D. 2
E. 4
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If 2x+y=3 and 4x^2-4xy+y^2=1, what is the value of xy?

by Brent@GMATPrepNow » Tue Jan 01, 2019 8:18 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

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

A. 1/8
B. 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,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by fskilnik@GMATH » Wed Jan 02, 2019 2:21 am
Max@Math Revolution 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. 1/8
B. 1/2
C. 1
D. 2
E. 4
$$? = xy$$
$${\left( {2x - y} \right)^2} = 4{x^2} - 4xy + {y^2} = 1\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,2x - y = \pm 1$$
$$\left\{ \matrix{
\,2x + y = 3 \hfill \cr
\,2x - y = 1 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,x = 1\,\,\,\,\, \Rightarrow \,\,\,\,\,y = 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = xy = 1$$
$$\left[ {\,{\rm{or}}\,\,\left\{ \matrix{
\,2x + y = 3 \hfill \cr
\,2x - y = - 1 \hfill \cr} \right.\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,x = {1 \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,\,y = 2\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = xy = 1\,} \right]$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Max@Math Revolution » Wed Jan 02, 2019 11:13 pm
=>

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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