Solve ab=a(a+A)c

$$a b = a ( a + A ) c$$

Solve for A

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

Steps for Solving Linear Equation

Use the distributive property to multiply a by a+A.

$$ab=\left(a^{2}+aA\right)c$$

Use the distributive property to multiply a^{2}+aA by c.

$$ab=a^{2}c+aAc$$

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

$$a^{2}c+aAc=ab$$

Subtract a^{2}c from both sides.

$$aAc=ab-a^{2}c$$

Reorder the terms.

$$Aac=ab-ca^{2}$$

The equation is in standard form.

$$acA=ab-ca^{2}$$

Divide both sides by ac.

$$\frac{acA}{ac}=\frac{a\left(b-ac\right)}{ac}$$

Dividing by ac undoes the multiplication by ac.

$$A=\frac{a\left(b-ac\right)}{ac}$$

Divide a\left(b-ca\right) by ac.

$$A=-a+\frac{b}{c}$$

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a=-A+\frac{b}{c}\text{, }&c\neq 0\\a\in \mathrm{R}\text{, }&b=0\text{ and }c=0\end{matrix}\right.$$