Existence of $x,y$ Satisfying Diophantine Equation Let $a,b$ be positive integers. Prove that there exist positive integers $x,y$ such that
$$ \dbinom{x+y}{2} = ax+by $$

Solutions of the equation:  $\frac{(x+y)(x+y-1)}{2}=ax+by$
Can be written without using Pell's equation:
$x=\frac{(a+b+1)(3b-a+1)+2(b-a)k-k^2}{8(b-a)}$
$y=\frac{(a+b+1)(b-3a-1)+2(b-a)k+k^2}{8(b-a)}$
$k$ - what some integer.
