Solving Linear Diophantine Equations

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.

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