Problem Solving

[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

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.

=>

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