Solve ay-x=z+by

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

Solve for a

$$\left\{\begin{matrix}a=\frac{x+by+z}{y}\text{, }&y\neq 0\\a\in \mathrm{R}\text{, }&x=-z\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Add x to both sides.

$$ay=z+by+x$$

The equation is in standard form.

$$ya=x+by+z$$

Divide both sides by y.

$$\frac{ya}{y}=\frac{x+by+z}{y}$$

Dividing by y undoes the multiplication by y.

$$a=\frac{x+by+z}{y}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{x-ay+z}{y}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&x=-z\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$z+by=ay-x$$

Subtract z from both sides.

$$by=ay-x-z$$

The equation is in standard form.

$$yb=-x+ay-z$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{-x+ay-z}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{-x+ay-z}{y}$$