I have an equation with this form: $ A x + B y + C z = N$
$A$, $B$, $C$, and $N$ are known.
$A$, $B$, $C$, $N$, $x$, $y$, and $z$ are positive integers $>= 0$.
I need to find out values of $x$, $y$, $z$ that satisfy the equation.
Unless $N$ is a multiple of any of the coefficients, I cannot think of a way to solve this equation without brute-force trial and error.
What strategy should I use to solve this kind of linear equation?