Solve y=ax^2+by

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

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{y\left(b-1\right)}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{C}\text{, }&\left(b=1\text{ or }y=0\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$ax^{2}+by=y$$

Subtract by from both sides.

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

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

Divide y-yb by x^{2}.

$$a=\frac{y\left(1-b\right)}{x^{2}}$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$ax^{2}+by=y$$

Subtract ax^{2} from both sides.

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

Reorder the terms.

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

The equation is in standard form.

$$yb=y-ax^{2}$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{y-ax^{2}}{y}$$

Dividing by y undoes the multiplication by y.

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

Solve for a

$$\left\{\begin{matrix}a=-\frac{y\left(b-1\right)}{x^{2}}\text{, }&x\neq 0\\a\in \mathrm{R}\text{, }&\left(b=1\text{ or }y=0\right)\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$ax^{2}+by=y$$

Subtract by from both sides.

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

The equation is in standard form.

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

Divide both sides by x^{2}.

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

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

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

Divide y-yb by x^{2}.

$$a=\frac{y\left(1-b\right)}{x^{2}}$$

Solve for b

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

Steps for Solving Linear Equation

Swap sides so that all variable terms are on the left hand side.

$$ax^{2}+by=y$$

Subtract ax^{2} from both sides.

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

Reorder the terms.

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

The equation is in standard form.

$$yb=y-ax^{2}$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{y-ax^{2}}{y}$$

Dividing by y undoes the multiplication by y.

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