Solve Ax+by=cx

$$A x + b y = c x$$

Solve for A (complex solution)

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$Ax=cx-by$$

The equation is in standard form.

$$xA=cx-by$$

Divide both sides by x.

$$\frac{xA}{x}=\frac{cx-by}{x}$$

Dividing by x undoes the multiplication by x.

$$A=\frac{cx-by}{x}$$

Divide cx-by by x.

$$A=-\frac{by}{x}+c$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract Ax from both sides.

$$by=cx-Ax$$

The equation is in standard form.

$$yb=cx-Ax$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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

Solve for A

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$Ax=cx-by$$

The equation is in standard form.

$$xA=cx-by$$

Divide both sides by x.

$$\frac{xA}{x}=\frac{cx-by}{x}$$

Dividing by x undoes the multiplication by x.

$$A=\frac{cx-by}{x}$$

Divide cx-by by x.

$$A=-\frac{by}{x}+c$$

Solve for b

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

Steps for Solving Linear Equation

Subtract Ax from both sides.

$$by=cx-Ax$$

The equation is in standard form.

$$yb=cx-Ax$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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