What's the most straightforward way to ask Mathematica to find all solutions of an equation like

3x + 2y + 4z == 0 (mod 11)

(for instance), where either x, y, z can be considered to be integers in the range -5...5, or equivalently they belong to the ring Z/(11Z) of integers modulo 11 ?

(I've tried a number of obvious things with no success.)

What's the most straightforward way to ask Mathematica to find all solutions of an equation like

$$3x + 2y + 4z = 0 \pmod {11}$$

(for instance), where either $x$, $y$, $z$ can be considered to be integers in the range $-5\dots 5$, or equivalently they belong to the ring $Z/(11Z)$ of integers modulo $11$?

(I've tried a number of obvious things with no success.)