jmc

Simplifying, we have $12(x+y)=xy$, so $xy-12x-12y=0.$ Applying Simon's Favorite Factoring Trick by adding 144 to both sides, we get $xy-12x-12y+144=144$, so $$(x-12)(y-12)=144.$$Now we seek the minimal $x+y,$ which occurs when $x-12$ and $y-12$ are as close to each other in value as possible. The two best candidates are $(x-12,y-12)=(18,8)$ or $(16,9),$ of which $(x,y)=(28,21)$ attains the minimum sum of $\boxed{49}$.