Solve x/y-x=z/n

$$\frac { x } { y - x } = \frac { z } { n }$$

Solve for n

$$\left\{\begin{matrix}n=\frac{z\left(y-x\right)}{x}\text{, }&x\neq y\text{ and }z\neq 0\text{ and }x\neq 0\\n\neq 0\text{, }&x=0\text{ and }z=0\text{ and }y\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable n cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by n\left(-x+y\right), the least common multiple of y-x,n.

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

Use the distributive property to multiply -x+y by z.

$$nx=-xz+yz$$

The equation is in standard form.

$$xn=yz-xz$$

Divide both sides by x.

$$\frac{xn}{x}=\frac{z\left(y-x\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$n=\frac{z\left(y-x\right)}{x}$$

Variable n cannot be equal to 0.

$$n=\frac{z\left(y-x\right)}{x}\text{, }n\neq 0$$

Solve for x

$$\left\{\begin{matrix}x=\frac{yz}{z+n}\text{, }&y\neq 0\text{ and }n\neq 0\text{ and }z\neq -n\\x\neq 0\text{, }&y=0\text{ and }z=-n\text{ and }n\neq 0\end{matrix}\right.$$