Solve ax(a-3)+x=x(a+1)+a

$$a x ( a - 3 ) + x = x ( a + 1 ) + a$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{1}{a-4}\text{, }&a\neq 4\\x\in \mathrm{C}\text{, }&a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax by a-3.

$$xa^{2}-3ax+x=x\left(a+1\right)+a$$

Use the distributive property to multiply x by a+1.

$$xa^{2}-3ax+x=xa+x+a$$

Subtract xa from both sides.

$$xa^{2}-3ax+x-xa=x+a$$

Combine -3ax and -xa to get -4ax.

$$xa^{2}-4ax+x=x+a$$

Subtract x from both sides.

$$xa^{2}-4ax+x-x=a$$

Combine x and -x to get 0.

$$xa^{2}-4ax=a$$

Combine all terms containing x.

$$\left(a^{2}-4a\right)x=a$$

Divide both sides by a^{2}-4a.

$$\frac{\left(a^{2}-4a\right)x}{a^{2}-4a}=\frac{a}{a^{2}-4a}$$

Dividing by a^{2}-4a undoes the multiplication by a^{2}-4a.

$$x=\frac{a}{a^{2}-4a}$$

Divide a by a^{2}-4a.

$$x=\frac{1}{a-4}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{1}{a-4}\text{, }&a\neq 4\\x\in \mathrm{R}\text{, }&a=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply ax by a-3.

$$xa^{2}-3ax+x=x\left(a+1\right)+a$$

Use the distributive property to multiply x by a+1.

$$xa^{2}-3ax+x=xa+x+a$$

Subtract xa from both sides.

$$xa^{2}-3ax+x-xa=x+a$$

Combine -3ax and -xa to get -4ax.

$$xa^{2}-4ax+x=x+a$$

Subtract x from both sides.

$$xa^{2}-4ax+x-x=a$$

Combine x and -x to get 0.

$$xa^{2}-4ax=a$$

Combine all terms containing x.

$$\left(a^{2}-4a\right)x=a$$

Divide both sides by a^{2}-4a.

$$\frac{\left(a^{2}-4a\right)x}{a^{2}-4a}=\frac{a}{a^{2}-4a}$$

Dividing by a^{2}-4a undoes the multiplication by a^{2}-4a.

$$x=\frac{a}{a^{2}-4a}$$

Divide a by a^{2}-4a.

$$x=\frac{1}{a-4}$$

Solve for a

$$\left\{\begin{matrix}\\a=0\text{, }&\text{unconditionally}\\a=4+\frac{1}{x}\text{, }&x\neq 0\end{matrix}\right.$$