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.
$\mathrm{Squaring}(2x+3y)\mathrm{and}(2x-3y)\mathrm{both}\mathrm{and}\mathrm{then}\mathrm{subtracting}\mathrm{them},\mathrm{we}\mathrm{get}:$
$$\begin{gathered} {\left(2x+3y\right)}^2-{\left(2x-3y\right)}^2=\left\{\left(2x+3y\right)+\left(2x-3y\right)\right\}\left\{\left(2x+3y\right)-\left(2x-3y\right)\right\}=4x\times 6y=24xy \\ \Rightarrow {\left(2x+3y\right)}^2-{\left(2x-3y\right)}^2=24xy \end{gathered}$$
$$\begin{gathered} \Rightarrow 24xy={\left(2x+3y\right)}^2-{\left(2x-3y\right)}^2 \\ \Rightarrow 24xy={\left(14\right)}^2-{\left(2\right)}^2 \\ \Rightarrow 24xy=\left(14+2\right)\left(14-2\right)(\because \left(a+b\right)\left(a-b\right)=a^2-b^2) \\ \Rightarrow 24xy=16\times 12 \\ \Rightarrow xy=\frac{16\times 12}{24}\left(\text{Dividing both sides by}24\right) \\ \Rightarrow xy=8 \end{gathered}$$