jmc

Simplifying, we have $18(x+y)=xy$, so $xy-18x-18y=0$ Applying Simon's Favorite Factoring Trick by adding 324 to both sides, we get $xy-18x-18y+324=324$, so $$(x-18)(y-18)=324.$$Now we seek the minimal $x+y,$ which occurs when $x-18$ and $y-18$ are as close to each other in value as possible. The two best candidates are $(x-18,y-18)=(12,27)$ or $(9,36),$ of which $(x,y)=(30,45)$ attains the minimum sum of $\boxed{75}$.