If $\frac{x+y}{2}=5$ and $\frac{x-y}{2}=2$, what is the value of $x^{2}-y^{2} ?$
Solution 1

Since $\frac{x+y}{2}=5$, then $x+y=2 \cdot 5=10$. Since $\frac{x-y}{2}=2$, then $x-y=2 \cdot 2=4$.

Therefore, $x^{2}-y^{2}=(x+y)(x-y)=10 \cdot 4=40$.

## Solution 2

Since $\frac{x+y}{2}=5$ and $\frac{x-y}{2}=2$, then $\frac{x+y}{2}+\frac{x-y}{2}=5+2$ which simplifies to $x=7$.

Also, $\frac{x+y}{2}-\frac{x-y}{2}=5-2$ which simplifies to $y=3$.

Therefore, $x^{2}-y^{2}=7^{2}-3^{2}=49-9=40$.

ANSWER: 40