Solve ax+by=xy

$$ax+by = xy$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax=xy-by$$

The equation is in standard form.

$$xa=xy-by$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{y\left(x-b\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{y\left(x-b\right)}{x}$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=xy-ax$$

The equation is in standard form.

$$yb=xy-ax$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{x\left(y-a\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{x\left(y-a\right)}{y}$$

Solve for a

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax=xy-by$$

The equation is in standard form.

$$xa=xy-by$$

Divide both sides by x.

$$\frac{xa}{x}=\frac{y\left(x-b\right)}{x}$$

Dividing by x undoes the multiplication by x.

$$a=\frac{y\left(x-b\right)}{x}$$

Solve for b

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=xy-ax$$

The equation is in standard form.

$$yb=xy-ax$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{x\left(y-a\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{x\left(y-a\right)}{y}$$