Question

How do you solve 1/x + 1/y + 1/z = 1 for z?

Answer 1

`\large\boxed{z=(xy)/(xy-x-y)}`

Explanation:

`(1)/(x)+(1)/(y)+(1)/(z)=1\qquad\text{multiply both sides by}\ xyz\\eq0\\\\xyz\cdot(1)/(x)+xyz\cdot(1)/(y)+xyz\cdot(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=(-xy)/(x+y-xy)\\\\z=(-xy)/(-(xy-x-y))\\\\z=(xy)/(xy-x-y)`