Solve {r}{1/x+1/y+1/z=1/a(x>0,y>0,z>0,a>0)}

$$\left. \begin{array} { r } { \frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = \frac { 1 } { a } ( x > 0 , y > 0 , z > 0 , a > 0 ) } \end{array} \right.$$

Solve for x

$$x=-\frac{ayz}{ay-yz+az}$$

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

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 axyz, the least common multiple of x,y,z,a.

$$ayz+axz+axy=xyz$$

Subtract xyz from both sides.

$$ayz+axz+axy-xyz=0$$

Subtract ayz from both sides. Anything subtracted from zero gives its negation.

$$axz+axy-xyz=-ayz$$

Combine all terms containing x.

$$\left(az+ay-yz\right)x=-ayz$$

The equation is in standard form.

$$\left(ay-yz+az\right)x=-ayz$$

Divide both sides by az+ya-yz.

$$\frac{\left(ay-yz+az\right)x}{ay-yz+az}=-\frac{ayz}{ay-yz+az}$$

Dividing by az+ya-yz undoes the multiplication by az+ya-yz.

$$x=-\frac{ayz}{ay-yz+az}$$

Variable x cannot be equal to 0.

$$x=-\frac{ayz}{ay-yz+az}\text{, }x\neq 0$$