Solve ax+by=cquad(a)

$$a x + b y = c \quad ( a )$$

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Subtract ca from both sides.

$$ax+by-ca=0$$

Subtract by from both sides. Anything subtracted from zero gives its negation.

$$ax-ca=-by$$

Reorder the terms.

$$ax-ac=-by$$

Combine all terms containing a.

$$\left(x-c\right)a=-by$$

Divide both sides by x-c.

$$\frac{\left(x-c\right)a}{x-c}=-\frac{by}{x-c}$$

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

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

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=ca-ax$$

The equation is in standard form.

$$yb=ac-ax$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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

Solve for a

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

Steps for Solving Linear Equation

Subtract ca from both sides.

$$ax+by-ca=0$$

Subtract by from both sides. Anything subtracted from zero gives its negation.

$$ax-ca=-by$$

Reorder the terms.

$$ax-ac=-by$$

Combine all terms containing a.

$$\left(x-c\right)a=-by$$

Divide both sides by x-c.

$$\frac{\left(x-c\right)a}{x-c}=-\frac{by}{x-c}$$

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

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

Solve for b

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

Steps for Solving Linear Equation

Subtract ax from both sides.

$$by=ca-ax$$

The equation is in standard form.

$$yb=ac-ax$$

Divide both sides by y.

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

Dividing by y undoes the multiplication by y.

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