Diophantine equations where all of the terms are monomials of degree zero or one. For example, finding all integers $x$ satisfying $ax = b$, finding all integers $x,y$ such that $ax + by = c$, or finding all integers $x,y,z$ such that $ax + by + cz = d$. Probably appropriate with (elementary-number-theory).

A *linear diophantine equation* is a diophantine equation (see diophantine-equations) where all of the terms are monomials of degree zero or one.

For example, some linear diophantine equation problems are:

Finding all integers $x$ satisfying $ax = b$.

Finding all integers $x,y$ such that $ax + by = c$.

Finding all integers $x,y,z$ such that $ax + by + cz = d$.

The equation $ax \equiv b \pmod{n}$ may also be thought of as a linear diophantine equation. If we like, we may write it as $ax = b + yn$.

We may also have a *system* of such equations. For example, the Chinese remainder theorem asserts a unique solution $x$ mod $mn$ to the equations $x \equiv a \pmod{m}$ and $x \equiv b \pmod{n}$ when $m$ and $n$ are relatively prime.

More generally, every system of linear Diophantine equations may be solved by computing the Smith normal form of its matrix, in a way that is similar to the use of the reduced row echelon form to solve a system of linear equations over a field.

For reference, see linear diophantine equations on Wikipedia.