Inspired by the identity in Bézout’s lemma, we might ask for all integer pairs $(x, y)$ that satisfy the equation $ax+by=c$ for a given integer triple $(a, b, c)$. Firstly, we address the question of when a solution exists at all, and secondly we show how to generate all solutions from one solution. This fully parametrizes all solutions to any two-variable linear Diophantine equation. To get the first solution, we may use the extended Euclidean algorithm, which we have described using matrix multiplication and the determinant in another video.