Answer by jason for How to create a function to solve two unknowns

This is a linear Diophantine equation. Its solution is very well known. It can be rewritten in the form ex + fy = d. Find g = gcd(e, f). If d = g then there are infinitely many integral solutions, and these can be found using the extended Euclidean algorithm. Consequently, if d is merely a multiple of g, there are infinitely many solutions as well. If d is not a multiple of g, there are no solutions.