jmc

We apply Simon's Favorite Factoring Trick and note that if we add 6 to both sides, then the left side can be factored. Thus, $$xy+3x+2y+6=(x+2)(y+3)=121.$$Since $x,y$ are positive integers, then $x+2,y+3$ must be a pair of factors of $121$, which are given by $\{x+2,y+3\}=\{1,121\},\{11,11\}$ or $\{121,1\}$. Thus, $\{x,y\}=\{-1,118\},\{9,8\}$ or $\{119,-2\}.$ Since $x$ and $y$ are positive integers, $\{x,y\}=\{9,8\},$ so $x+y=9+8=\boxed{17}$.