jmc

If we add $1$ to both sides of the equation, the left hand side may be factored using Simon's Favorite Factoring Trick. Thus, $$xy+x+y+1=(x+1)(y+1)=77.$$ Since $x,y$ are positive integers, then $x+1,y+1$ must be a pair of factors of $77$, which are given by $\{x+1,y+1\}=\{1,77\},\{7,11\}$. Thus, $\{x,y\}=\{0,76\},\{6,10\}$, though only the latter satisfies the given conditions. Hence, $x+y=6+10=\boxed{16}$.