Solve ax^2+by^2=c^2

$$a x ^ { 2 } + b y ^ { 2 } = c ^ { 2 }$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{by^{2}-c^{2}}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&\left(b=\frac{c^{2}}{y^{2}}\text{ and }y\neq 0\text{ and }x=0\right)\text{ or }\left(c=0\text{ and }y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract by^{2} from both sides.

$$ax^{2}=c^{2}-by^{2}$$

Reorder the terms.

$$ax^{2}=-by^{2}+c^{2}$$

The equation is in standard form.

$$x^{2}a=c^{2}-by^{2}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{c^{2}-by^{2}}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{c^{2}-by^{2}}{x^{2}}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}b=-\frac{ax^{2}-c^{2}}{y^{2}}\text{, }&y\neq 0\\b\in \mathrm{C}\text{, }&\left(a=\frac{c^{2}}{x^{2}}\text{ and }x\neq 0\text{ and }y=0\right)\text{ or }\left(c=0\text{ and }x=0\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides.

$$by^{2}=c^{2}-ax^{2}$$

Reorder the terms.

$$by^{2}=-ax^{2}+c^{2}$$

The equation is in standard form.

$$y^{2}b=c^{2}-ax^{2}$$

Divide both sides by y^{2}.

$$\frac{y^{2}b}{y^{2}}=\frac{c^{2}-ax^{2}}{y^{2}}$$

Dividing by y^{2} undoes the multiplication by y^{2}.

$$b=\frac{c^{2}-ax^{2}}{y^{2}}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{by^{2}-c^{2}}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&\left(b=\frac{c^{2}}{y^{2}}\text{ and }y\neq 0\text{ and }x=0\right)\text{ or }\left(c=0\text{ and }y=0\text{ and }x=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract by^{2} from both sides.

$$ax^{2}=c^{2}-by^{2}$$

Reorder the terms.

$$ax^{2}=-by^{2}+c^{2}$$

The equation is in standard form.

$$x^{2}a=c^{2}-by^{2}$$

Divide both sides by x^{2}.

$$\frac{x^{2}a}{x^{2}}=\frac{c^{2}-by^{2}}{x^{2}}$$

Dividing by x^{2} undoes the multiplication by x^{2}.

$$a=\frac{c^{2}-by^{2}}{x^{2}}$$

Solve for b

$$\left\{\begin{matrix}b=-\frac{ax^{2}-c^{2}}{y^{2}}\text{, }&y\neq 0\\b\in \mathrm{R}\text{, }&\left(a=\frac{c^{2}}{x^{2}}\text{ and }x\neq 0\text{ and }y=0\right)\text{ or }\left(c=0\text{ and }x=0\text{ and }y=0\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract ax^{2} from both sides.

$$by^{2}=c^{2}-ax^{2}$$

Reorder the terms.

$$by^{2}=-ax^{2}+c^{2}$$

The equation is in standard form.

$$y^{2}b=c^{2}-ax^{2}$$

Divide both sides by y^{2}.

$$\frac{y^{2}b}{y^{2}}=\frac{c^{2}-ax^{2}}{y^{2}}$$

Dividing by y^{2} undoes the multiplication by y^{2}.

$$b=\frac{c^{2}-ax^{2}}{y^{2}}$$