Solve 1/x+1/y+1/z=1/2

$$\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 1 } { 2 }$$

Solve for x

$$x=-\frac{2yz}{2z+2y-yz}$$

$$y\neq 0\text{ and }z\neq 0\text{ and }\left(z=2\text{ or }y\neq -\frac{2z}{2-z}\right)$$

Steps for Solving Linear Equation

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2xyz, the least common multiple of x,y,z,2.

$$2yz+2xz+2xy=xyz$$

Subtract xyz from both sides.

$$2yz+2xz+2xy-xyz=0$$

Subtract 2yz from both sides. Anything subtracted from zero gives its negation.

$$2xz+2xy-xyz=-2yz$$

Combine all terms containing x.

$$\left(2z+2y-yz\right)x=-2yz$$

Divide both sides by 2y+2z-yz.

$$\frac{\left(2z+2y-yz\right)x}{2z+2y-yz}=-\frac{2yz}{2z+2y-yz}$$

Dividing by 2y+2z-yz undoes the multiplication by 2y+2z-yz.

$$x=-\frac{2yz}{2z+2y-yz}$$

Variable x cannot be equal to 0.

$$x=-\frac{2yz}{2z+2y-yz}\text{, }x\neq 0$$

Solve for y

$$y=-\frac{2xz}{2z+2x-xz}$$

$$x\neq 0\text{ and }z\neq 0\text{ and }\left(z=2\text{ or }x\neq -\frac{2z}{2-z}\right)$$

Steps for Solving Linear Equation

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2xyz, the least common multiple of x,y,z,2.

$$2yz+2xz+2xy=xyz$$

Subtract xyz from both sides.

$$2yz+2xz+2xy-xyz=0$$

Subtract 2xz from both sides. Anything subtracted from zero gives its negation.

$$2yz+2xy-xyz=-2xz$$

Combine all terms containing y.

$$\left(2z+2x-xz\right)y=-2xz$$

Divide both sides by 2x+2z-xz.

$$\frac{\left(2z+2x-xz\right)y}{2z+2x-xz}=-\frac{2xz}{2z+2x-xz}$$

Dividing by 2x+2z-xz undoes the multiplication by 2x+2z-xz.

$$y=-\frac{2xz}{2z+2x-xz}$$

Variable y cannot be equal to 0.

$$y=-\frac{2xz}{2z+2x-xz}\text{, }y\neq 0$$