A symmetric 3-variable system

Polynomial. By Vieta's formulas, $x,y,z$ are roots of
$$t^3-St^2+Qt-P=0$$
with $S=6$, $Q=11$, $P=6$:
$$t^3-6t^2+11t-6=0.$$

Factor. $t=1$ is a root: $1-6+11-6=0$. Divide: $t^3-6t^2+11t-6=(t-1)(t^2-5t+6)=(t-1)(t-2)(t-3)$.

Roots. $\{x,y,z\}=\{1,2,3\}$ (in any order).

Conclusion. The solutions are the six permutations of $(1,2,3)$.