Key idea. Clear the fractions, then complete a product.

1. Both reciprocals are positive, so each is less than \(1/6\). Thus \(x>6\) and \(y>6\).

2. Multiply the equation by \(6xy\): \(6x+6y=xy\). Rearranging and adding 36 gives

   \[xy-6x-6y+36=36,\qquad (x-6)(y-6)=36.\]

3. Set \(u=x-6\), \(v=y-6\). These are positive integers with product 36. The positive divisors of 36 are \(1,2,3,4,6,9,12,18,36\). Each possible u determines \(v=36/u\).

4. Adding 6 to both coordinates gives

   \[\begin{gathered}(7,42),(8,24),(9,18),(10,15),(12,12),\\(15,10),(18,9),(24,8),(42,7).\end{gathered}\]

5. Every positive factor pair was included. Reversing the algebra recovers the original equation, so every listed pair works and there are no others.

Remember. Adding a constant can turn a two-variable equation into a fixed product with finitely many factor pairs.