Solve ax+by=cz

$$a x + b y = c z$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{by-cz}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&\left(b=\frac{cz}{y}\text{ and }y\neq 0\text{ and }x=0\right)\text{ or }\left(z=0\text{ and }y=0\text{ and }x=0\right)\text{ or }\left(c=0\text{ and }y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax=cz-by$$

The equation is in standard form.

$$xa=cz-by$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{cz-by}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{cz-by}{x}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax-cz}{y}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&\left(a=\frac{cz}{x}\text{ and }x\neq 0\text{ and }y=0\right)\text{ or }\left(z=0\text{ and }y=0\text{ and }x=0\right)\text{ or }\left(c=0\text{ and }y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=cz-ax$$

The equation is in standard form.

$$yb=cz-ax$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{cz-ax}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{cz-ax}{y}$$