Example 2, find the positive integer solutions $(x, y, z)$ of the equation $\frac{1}{x}+\frac{1}{4}+\frac{1}{z}=1(x \geqslant y \geqslant z)$.
Solution: $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$.
Without loss of generality, assume $x \geqslant y \geqslant z \geqslant 1$, then $1=\frac{1}{x}+\frac{1}{y}+\frac{1}{z} \leqslant \frac{3}{z}, 1 \leqslant z \leqslant 3$.
Since $z$ is an integer, $z=1, 2, 3$.
Below, we discuss the cases $z=1, 2, 3$, and finally obtain the positive integer solutions of the equation $(x, y, z)=(4, 2), (6, 3, 2), (3, 3, 3)$.