Since $x+x y=391$, then $x(1+y)=391$. We note that $391=17 \cdot 23$. Since 17 and 23 are both prime, then if 391 is written as the product of two positive integers, it must be $1 \times 391$ or $17 \times 23$ or $23 \times 17$ or $391 \times 1$. Matching $x$ and $1+y$ to these possible factors, we obtain $(x, y)=(1,390)$ or $(17,22)$ or $(23,16)$ or $(391,0)$. Since $y$ is a positive integer, the fourth pair is not possible. Since $x>y$, the first two pairs are not possible. Therefore, $(x, y)=(23,16)$ and so $x+y=39$.