Answer:
[tex]\large\boxed{z=\dfrac{xy}{xy-x-y}}[/tex]
Step-by-step explanation:
[tex]\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1\qquad\text{multiply both sides by}\ xyz\neq0\\\\xyz\cdot\dfrac{1}{x}+xyz\cdot\dfrac{1}{y}+xyz\cdot\dfrac{1}{z}=xyz\cdot1\qquad\text{simplify}\\\\yz+xz+xy=xyz\qquad\text{subtract}\ xy\ \text{from both sides}\\\\xz+yz=xyz-xy\qquad\text{subtract}\ xyz\ \text{from both sides}\\\\xz+yz-xyz=-xy\qquad\text{distribute}\\\\(x+y-xy)z=-xy\qquad\text{divide both sides by}\ (x+y-xy)\\\\z=\dfrac{-xy}{x+y-xy}\\\\z=\dfrac{-xy}{-(xy-x-y)}\\\\z=\dfrac{xy}{xy-x-y}[/tex]
