Solve c+i/c-i=a+bi

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

Solve for a

$$a=-\frac{i+c-b-ibc}{i-c}$$

$$c\neq i$$

Steps for Solving Linear Equation

Multiply both sides of the equation by c-i.

$$c+i=\left(c-i\right)a+bi\left(c-i\right)$$

Use the distributive property to multiply c-i by a.

$$c+i=ca-ia+bi\left(c-i\right)$$

Use the distributive property to multiply bi by c-i.

$$c+i=ca-ia+ibc-ibi$$

Multiply -i and i to get 1.

$$c+i=ca-ia+ibc+b$$

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

$$ca-ia+ibc+b=c+i$$

Subtract ibc from both sides.

$$ca-ia+b=c+i-ibc$$

Subtract b from both sides.

$$ca-ia=c+i-ibc-b$$

Reorder the terms.

$$ac-ia=-ibc-b+c+i$$

Combine all terms containing a.

$$\left(c-i\right)a=-ibc-b+c+i$$

The equation is in standard form.

$$\left(c-i\right)a=i+c-b-ibc$$

Divide both sides by c-i.

$$\frac{\left(c-i\right)a}{c-i}=\frac{i+c-b-ibc}{c-i}$$

Dividing by c-i undoes the multiplication by c-i.

$$a=\frac{i+c-b-ibc}{c-i}$$

Solve for b

$$b=\frac{i+c+ia-ac}{ic+1}$$

$$c\neq i$$

Steps for Solving Linear Equation

Multiply both sides of the equation by c-i.

$$c+i=\left(c-i\right)a+bi\left(c-i\right)$$

Use the distributive property to multiply c-i by a.

$$c+i=ca-ia+bi\left(c-i\right)$$

Use the distributive property to multiply bi by c-i.

$$c+i=ca-ia+ibc-ibi$$

Multiply -i and i to get 1.

$$c+i=ca-ia+ibc+b$$

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

$$ca-ia+ibc+b=c+i$$

Subtract ca from both sides.

$$-ia+ibc+b=c+i-ca$$

Subtract -ia from both sides.

$$ibc+b=c+i-ca-\left(-ia\right)$$

Multiply -1 and -i to get i.

$$ibc+b=c+i-ca+ia$$

Combine all terms containing b.

$$\left(ic+1\right)b=c+i-ca+ia$$

The equation is in standard form.

$$\left(ic+1\right)b=i+c+ia-ac$$

Divide both sides by ic+1.

$$\frac{\left(ic+1\right)b}{ic+1}=\frac{i+c+ia-ac}{ic+1}$$

Dividing by ic+1 undoes the multiplication by ic+1.

$$b=\frac{i+c+ia-ac}{ic+1}$$