$y=\frac{1}{xz}$
$\left(x\neq 0\text{ and }z=-1\right)\text{ or }\left(z\neq 1\text{ and }x=-\frac{1}{z}\text{ and }z\neq 0\text{ and }z\neq -1\right)\text{ or }\left(x\neq -\frac{1}{z+1}\text{ and }z\neq 0\text{ and }x\neq -1\text{ and }x\neq 0\text{ and }z\neq -1\text{ and }x\neq -\frac{1}{z}\right)\text{ or }\left(x=-1\text{ and }z\neq 0\right)$

$x=\frac{1}{yz}$
$\left(y=-\frac{1}{z}\text{ or }y\neq 0\right)\text{ and }y\neq -1-\frac{1}{z}\text{ and }z\neq 0$

$y=\frac{1}{xz}$
$\left(x=-1\text{ and }z\neq 0\right)\text{ or }\left(x\neq -\frac{1}{z+1}\text{ and }z\neq 0\text{ and }x\neq -1\text{ and }x\neq 0\text{ and }z\neq -1\text{ and }x\neq -\frac{1}{z}\right)\text{ or }\left(x=-\frac{1}{z}\text{ and }z\neq 0\text{ and }|z|\neq 1\right)\text{ or }\left(x\neq 0\text{ and }z=-1\right)$