is it possible to find $x$ where $y$ is equal to a whole number in a non iterative fashion

Question

Given the equation

$$\frac{635x+326}{637+x} = y$$

where $$x>0$$

Is it possible to find all positive values of $x$ (there is only one) where $x$ is positive and $y$ is a whole number.

While I can iterate all positive values of $x$, I was hoping to shortcut that as the values of x are expected to be huge.

The answer for the above equation is $x=40$ .: $y=38$

Answer 1

Here's a link to a similar question. And I don't think you can avoid iterating to find a solution.

How to find positive integer solution of bilinear transformation?