Solve 2ac-2bd+4ad=b

$$2 a c - 2 b d + 4 a d = b$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=\frac{b\left(2d+1\right)}{2\left(c+2d\right)}\text{, }&c\neq -2d\\a\in \mathrm{C}\text{, }&\left(b=0\text{ and }c=-2d\right)\text{ or }\left(d=-\frac{1}{2}\text{ and }c=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 2bd to both sides.

$$2ac+4ad=b+2bd$$

Combine all terms containing a.

$$\left(2c+4d\right)a=b+2bd$$

The equation is in standard form.

$$\left(2c+4d\right)a=2bd+b$$

Divide both sides by 2c+4d.

$$\frac{\left(2c+4d\right)a}{2c+4d}=\frac{2bd+b}{2c+4d}$$

Dividing by 2c+4d undoes the multiplication by 2c+4d.

$$a=\frac{2bd+b}{2c+4d}$$

Divide b+2bd by 2c+4d.

$$a=\frac{b\left(2d+1\right)}{2\left(c+2d\right)}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=\frac{2a\left(c+2d\right)}{2d+1}\text{, }&d\neq -\frac{1}{2}\\b\in \mathrm{C}\text{, }&\left(a=0\text{ or }c=1\right)\text{ and }d=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract b from both sides.

$$2ac-2bd+4ad-b=0$$

Subtract 2ac from both sides. Anything subtracted from zero gives its negation.

$$-2bd+4ad-b=-2ac$$

Subtract 4ad from both sides.

$$-2bd-b=-2ac-4ad$$

Combine all terms containing b.

$$\left(-2d-1\right)b=-2ac-4ad$$

Divide both sides by -2d-1.

$$\frac{\left(-2d-1\right)b}{-2d-1}=\frac{-2ac-4ad}{-2d-1}$$

Dividing by -2d-1 undoes the multiplication by -2d-1.

$$b=\frac{-2ac-4ad}{-2d-1}$$

Divide -2ac-4ad by -2d-1.

$$b=\frac{2a\left(c+2d\right)}{2d+1}$$

Solve for a

$$\left\{\begin{matrix}a=\frac{b\left(2d+1\right)}{2\left(c+2d\right)}\text{, }&c\neq -2d\\a\in \mathrm{R}\text{, }&\left(b=0\text{ and }c=-2d\right)\text{ or }\left(d=-\frac{1}{2}\text{ and }c=1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Add 2bd to both sides.

$$2ac+4ad=b+2bd$$

Combine all terms containing a.

$$\left(2c+4d\right)a=b+2bd$$

The equation is in standard form.

$$\left(2c+4d\right)a=2bd+b$$

Divide both sides by 2c+4d.

$$\frac{\left(2c+4d\right)a}{2c+4d}=\frac{2bd+b}{2c+4d}$$

Dividing by 2c+4d undoes the multiplication by 2c+4d.

$$a=\frac{2bd+b}{2c+4d}$$

Divide b+2bd by 2c+4d.

$$a=\frac{b\left(2d+1\right)}{2\left(c+2d\right)}$$

Solve for b

$$\left\{\begin{matrix}b=\frac{2a\left(c+2d\right)}{2d+1}\text{, }&d\neq -\frac{1}{2}\\b\in \mathrm{R}\text{, }&\left(a=0\text{ or }c=1\right)\text{ and }d=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract b from both sides.

$$2ac-2bd+4ad-b=0$$

Subtract 2ac from both sides. Anything subtracted from zero gives its negation.

$$-2bd+4ad-b=-2ac$$

Subtract 4ad from both sides.

$$-2bd-b=-2ac-4ad$$

Combine all terms containing b.

$$\left(-2d-1\right)b=-2ac-4ad$$

Divide both sides by -2d-1.

$$\frac{\left(-2d-1\right)b}{-2d-1}=\frac{-2ac-4ad}{-2d-1}$$

Dividing by -2d-1 undoes the multiplication by -2d-1.

$$b=\frac{-2ac-4ad}{-2d-1}$$

Divide -2ac-4ad by -2d-1.

$$b=\frac{2a\left(c+2d\right)}{2d+1}$$