Solve ax+by=1

$$a x + b y = 1$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax=1-by$$

The equation is in standard form.

$$xa=1-by$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{1-by}{x}$$

Dividing by x undoes the multiplication by x.

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=1-ax$$

The equation is in standard form.

$$yb=1-ax$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{1-ax}{y}$$

Dividing by y undoes the multiplication by y.

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

Solve for a

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax=1-by$$

The equation is in standard form.

$$xa=1-by$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{1-by}{x}$$

Dividing by x undoes the multiplication by x.

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

Solve for b

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=1-ax$$

The equation is in standard form.

$$yb=1-ax$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{1-ax}{y}$$

Dividing by y undoes the multiplication by y.

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