Solve ax+b(x+c)=d(x+e)

$$a x + b ( x + c ) = d ( x + e )$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{dx-bx-bc+ed}{x}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&\left(x=0\text{ and }c\neq e\text{ and }b\neq 0\text{ and }d=\frac{bc}{e}\right)\text{ or }\left(b=d\text{ and }c=e\text{ and }x=0\right)\text{ or }\left(d=0\text{ and }b=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply b by x+c.

$$ax+bx+bc=d\left(x+e\right)$$

Use the distributive property to multiply d by x+e.

$$ax+bx+bc=dx+de$$

Subtract bx from both sides.

$$ax+bc=dx+de-bx$$

Subtract bc from both sides.

$$ax=dx+de-bx-bc$$

The equation is in standard form.

$$xa=dx-bx-bc+ed$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{dx-bx-bc+ed}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{dx-bx-bc+ed}{x}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax-dx-ed}{x+c}\text{, }&x\neq -c\\b\in \mathrm{C}\text{, }&\left(a=-\frac{d\left(e-c\right)}{c}\text{ and }x=-c\text{ and }c\neq 0\right)\text{ or }\left(d=0\text{ and }x=0\text{ and }c=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply b by x+c.

$$ax+bx+bc=d\left(x+e\right)$$

Use the distributive property to multiply d by x+e.

$$ax+bx+bc=dx+de$$

Subtract ax from both sides.

$$bx+bc=dx+de-ax$$

Combine all terms containing b.

$$\left(x+c\right)b=dx+de-ax$$

The equation is in standard form.

$$\left(x+c\right)b=dx-ax+ed$$

Divide both sides by x+c.

$$\frac{\left(x+c\right)b}{x+c}=\frac{dx-ax+ed}{x+c}$$

Dividing by x+c undoes the multiplication by x+c.

$$b=\frac{dx-ax+ed}{x+c}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{dx-bx-bc+ed}{x}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&\left(x=0\text{ and }c\neq e\text{ and }b\neq 0\text{ and }d=\frac{bc}{e}\right)\text{ or }\left(b=d\text{ and }c=e\text{ and }x=0\right)\text{ or }\left(d=0\text{ and }b=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply b by x+c.

$$ax+bx+bc=d\left(x+e\right)$$

Use the distributive property to multiply d by x+e.

$$ax+bx+bc=dx+de$$

Subtract bx from both sides.

$$ax+bc=dx+de-bx$$

Subtract bc from both sides.

$$ax=dx+de-bx-bc$$

The equation is in standard form.

$$xa=dx-bx-bc+ed$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{dx-bx-bc+ed}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{dx-bx-bc+ed}{x}$$

Solve for b

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

Steps for Solving Linear Equation

Use the distributive property to multiply b by x+c.

$$ax+bx+bc=d\left(x+e\right)$$

Use the distributive property to multiply d by x+e.

$$ax+bx+bc=dx+de$$

Subtract ax from both sides.

$$bx+bc=dx+de-ax$$

Combine all terms containing b.

$$\left(x+c\right)b=dx+de-ax$$

The equation is in standard form.

$$\left(x+c\right)b=dx-ax+ed$$

Divide both sides by x+c.

$$\frac{\left(x+c\right)b}{x+c}=\frac{dx-ax+ed}{x+c}$$

Dividing by x+c undoes the multiplication by x+c.

$$b=\frac{dx-ax+ed}{x+c}$$