Question

If 2x + 3y = 14 and 2x − 3y = 2, find the value of xy.
[Hint: Use (2x + 3y)² − (2x − 3y)² = 24xy]

Answer 1

We will use the identity \[\left( a + b \right)\left( a - b \right) = a^2 - b^2\] to obtain the value of xy. 

\[\text { Squaring (2x + 3y) and (2x - 3y) both and then subtracting them, we get }:\]

\[\left( 2x + 3y \right)^2 - \left( 2x - 3y \right)^2 = \left\{ \left( 2x + 3y \right) + \left( 2x - 3y \right) \right\}\left\{ \left( 2x + 3y \right) - \left( 2x - 3y \right) \right\} = 4x \times 6y = 24xy\]

\[ \Rightarrow \left( 2x + 3y \right)^2 - \left( 2x - 3y \right)^2 = 24xy\]

\[\Rightarrow 24xy = \left( 2x + 3y \right)^2 - \left( 2x - 3y \right)^2 \]

\[ \Rightarrow 24xy = \left( 14 \right)^2 - \left( 2 \right)^2 \]

\[ \Rightarrow 24xy = \left( 14 + 2 \right)\left( 14 - 2 \right) ( \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 )\]

\[ \Rightarrow 24xy = 16 \times 12\]

\[ \Rightarrow xy = \frac{16 \times 12}{24} (\text { Dividing both sides by24 })\]

\[ \Rightarrow xy = 8\]