Question:
Let
\[f(x) = \frac{-px - 3}{-qx + 3},\]and let $g(x)$ be the inverse of $f(x).$  If $(7,-22)$ lies on both of the graphs of $y = f(x)$ and $y = g(x),$ then find $p + q.$

Answer:
If $(7,-22)$ lies on both $y = f(x)$ and the graph of its inverse, then $f(7) = -22$ and $f(-22) = 7.$  Hence,
\begin{align*}
\frac{-7p - 3}{-7q + 3} &= -22, \\
\frac{22p - 3}{22q + 3} &= 7.
\end{align*}Then $-7p - 3 = -22(-7q + 3) = 154q - 66$ and $22p - 3 = 7(22q + 3) = 154q + 21.$
Solving, we find $p = 3$ and $q = \frac{3}{11},$ so $p + q = 3 + \frac{3}{11} = \boxed{\frac{36}{11}}.$