The sum of two numbers is 25 and their difference is 9. What is their product?
We can represent the given information with the following system of linear equations:

\begin{align*}
x + y &= 25, \\
x - y &= 9.
\end{align*}To find the product of $x$ and $y$, solve for each independently.

Begin by adding the two equations:  \begin{align*}
2x &= 34 \\
x &= 17
\end{align*}Substituting in for $x$ yields a value of $8$ for $y$.

Hence, $x \cdot y = 17 \cdot 8 = \boxed{136}$