Solve ABZ-AB=AA1

$$A B Z - A B = A A 1$$

Solve for A

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&A_{1}=B\left(Z-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract AA_{1} from both sides.

$$ABZ-AB-AA_{1}=0$$

Combine all terms containing A.

$$\left(BZ-B-A_{1}\right)A=0$$

The equation is in standard form.

$$\left(BZ-A_{1}-B\right)A=0$$

Divide 0 by BZ-B-A_{1}.

$$A=0$$

Solve for A_1

$$\left\{\begin{matrix}\\A_{1}=B\left(Z-1\right)\text{, }&\text{unconditionally}\\A_{1}\in \mathrm{R}\text{, }&A=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$AA_{1}=ABZ-AB$$

Divide both sides by A.

$$\frac{AA_{1}}{A}=\frac{AB\left(Z-1\right)}{A}$$

Dividing by A undoes the multiplication by A.

$$A_{1}=\frac{AB\left(Z-1\right)}{A}$$

Divide AB\left(-1+Z\right) by A.

$$A_{1}=B\left(Z-1\right)$$