Solve (a+bi)(x+yi)=1

$$( a + b i ) ( x + y i ) = 1$$

Solve for a

$$a=-\frac{ibx-by-1}{x+iy}$$

$$x\neq -iy$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+bi by x+yi.

$$ax+iay+ibx-yb=1$$

Subtract ibx from both sides.

$$ax+iay-yb=1-ibx$$

Add yb to both sides.

$$ax+iay=1-ibx+yb$$

Reorder the terms.

$$ax+iay=-ibx+by+1$$

Combine all terms containing a.

$$\left(x+iy\right)a=-ibx+by+1$$

The equation is in standard form.

$$\left(x+iy\right)a=1+by-ibx$$

Divide both sides by x+iy.

$$\frac{\left(x+iy\right)a}{x+iy}=\frac{1+by-ibx}{x+iy}$$

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

$$a=\frac{1+by-ibx}{x+iy}$$

Solve for b

$$b=-\frac{ax+iay-1}{ix-y}$$

$$x\neq -iy$$

Steps for Solving Linear Equation

Use the distributive property to multiply a+bi by x+yi.

$$ax+iay+ibx-yb=1$$

Subtract ax from both sides.

$$iay+ibx-yb=1-ax$$

Subtract iay from both sides.

$$ibx-yb=1-ax-iay$$

Reorder the terms.

$$ibx-by=-ax-iay+1$$

Combine all terms containing b.

$$\left(ix-y\right)b=-ax-iay+1$$

The equation is in standard form.

$$\left(ix-y\right)b=1-iay-ax$$

Divide both sides by ix-y.

$$\frac{\left(ix-y\right)b}{ix-y}=\frac{1-iay-ax}{ix-y}$$

Dividing by ix-y undoes the multiplication by ix-y.

$$b=\frac{1-iay-ax}{ix-y}$$