jmc

Multiplying both sides of the equation by $7xy$ yields $7y+7x=xy$. Re-arranging and applying Simon's Favorite Factoring Trick, it follows that $$xy-7x-7y+49=(x-7)(y-7)=49.$$ Since $x$ and $y$ are positive integers, then $x-7$ is a positive integer factor of $49$. These factors are $1,7,49$, so $x=8,14,56$, and their sum is $8+14+56=\boxed{78}$.