The system
$$
\begin{cases}
Ax+By=15\\
Ax-By=9
\end{cases}
$$
has a solution $(2,1)$. What are $A$ and $B$?

Plug in $x=2$ and $y=1$ to get
$$
\begin{cases}
2A+B=15\\
2A-B=9
\end{cases}.
$$
Then solve for $A$ and $B$. There are many ways to solve this. During class I combined the two left sides of each equation and the two right sides. (Think of two balanced hangers where you combined them into one balanced hanger.)
This gives the equation
$$
4A=24.
$$
Thus $A=6$. Plug $A=6$ into the equation $2A+B=15$ to get
$$
2(6)+B=15,
$$
so $B=3$.

In summary, $A=6$ and $B=3$.