algebra

We can multiply the first equation by $pq$,

$1/p + 1/q = 1$

$q+p = pq$

We know that $pq$ is 9, therefore q + p, or p + q, is 9.

it's a system of equations:

$\frac{1}{p}+\frac{1}{q}=1$ and $pq=9$

$p=\frac{9+3\sqrt{5}}{2}$

$q=\frac{9-3\sqrt{5}}{2}$

the sum is 9