Divisibility of numbers

Question

Find all positive integers $x,y$ such that $2x+7y$ divides $7x+2y$.

I somehow managed to show that $x$ is greater than $y$. But couldn't proceed further.

Answer 1

The solution is all x,y such that x=y or x=4y or x=19y. So, your question has infinitely many solutions.

Answer 2

Hint: $7x+2y=(2x+7y)k\implies(7-2k)x=(7k-2)y$. If $x$, $y$, and $k$ are all positive integers, how large can $k$ be?