Solve P(A)-A(B)=P(AB)

$$P ( A ) - A ( B ) = P ( A B )$$

Solve for A

$$\left\{\begin{matrix}\\A=0\text{, }&\text{unconditionally}\\A\in \mathrm{R}\text{, }&P=\frac{B}{1-B}\text{ and }B\neq 1\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract PAB from both sides.

$$PA-AB-PAB=0$$

Reorder the terms.

$$-ABP-AB+AP=0$$

Combine all terms containing A.

$$\left(-BP-B+P\right)A=0$$

The equation is in standard form.

$$\left(P-B-BP\right)A=0$$

Divide 0 by P-B-PB.

$$A=0$$

Solve for B

$$\left\{\begin{matrix}B=\frac{P}{P+1}\text{, }&P\neq -1\\B\in \mathrm{R}\text{, }&A=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract PAB from both sides.

$$PA-AB-PAB=0$$

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

$$-AB-PAB=-PA$$

Combine all terms containing B.

$$\left(-A-PA\right)B=-PA$$

The equation is in standard form.

$$\left(-AP-A\right)B=-AP$$

Divide both sides by -A-PA.

$$\frac{\left(-AP-A\right)B}{-AP-A}=-\frac{AP}{-AP-A}$$

Dividing by -A-PA undoes the multiplication by -A-PA.

$$B=-\frac{AP}{-AP-A}$$

Divide -PA by -A-PA.

$$B=\frac{P}{P+1}$$