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 $p + q.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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 $p + q.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{36}{11}
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 $p + q.$
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