Solve ax^2-bx=ay^2+by

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

Solve for a (complex solution)

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

Steps for Solving Linear Equation

Subtract ay^{2} from both sides.

$$ax^{2}-bx-ay^{2}=by$$

Add bx to both sides.

$$ax^{2}-ay^{2}=by+bx$$

Combine all terms containing a.

$$\left(x^{2}-y^{2}\right)a=by+bx$$

The equation is in standard form.

$$\left(x^{2}-y^{2}\right)a=bx+by$$

Divide both sides by x^{2}-y^{2}.

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

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

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

Divide b\left(y+x\right) by x^{2}-y^{2}.

$$a=\frac{b}{x-y}$$

Solve for b (complex solution)

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax^{2}-bx-by=ay^{2}$$

Subtract ax^{2} from both sides.

$$-bx-by=ay^{2}-ax^{2}$$

Reorder the terms.

$$-bx-by=-ax^{2}+ay^{2}$$

Combine all terms containing b.

$$\left(-x-y\right)b=-ax^{2}+ay^{2}$$

The equation is in standard form.

$$\left(-x-y\right)b=ay^{2}-ax^{2}$$

Divide both sides by -x-y.

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

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

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

Divide a\left(x+y\right)\left(-x+y\right) by -x-y.

$$b=ax-ay$$

Solve for a

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

Steps for Solving Linear Equation

Subtract ay^{2} from both sides.

$$ax^{2}-bx-ay^{2}=by$$

Add bx to both sides.

$$ax^{2}-ay^{2}=by+bx$$

Combine all terms containing a.

$$\left(x^{2}-y^{2}\right)a=by+bx$$

The equation is in standard form.

$$\left(x^{2}-y^{2}\right)a=bx+by$$

Divide both sides by x^{2}-y^{2}.

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

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

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

Divide b\left(y+x\right) by x^{2}-y^{2}.

$$a=\frac{b}{x-y}$$

Solve for b

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

Steps for Solving Linear Equation

Subtract by from both sides.

$$ax^{2}-bx-by=ay^{2}$$

Subtract ax^{2} from both sides.

$$-bx-by=ay^{2}-ax^{2}$$

Reorder the terms.

$$-bx-by=-ax^{2}+ay^{2}$$

Combine all terms containing b.

$$\left(-x-y\right)b=-ax^{2}+ay^{2}$$

The equation is in standard form.

$$\left(-x-y\right)b=ay^{2}-ax^{2}$$

Divide both sides by -x-y.

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

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

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

Divide a\left(x+y\right)\left(-x+y\right) by -x-y.

$$b=ax-ay$$