Solve -2y(y^2-6y-3)=ay^3+12y^2+by

$$- 2 y ( y ^ { 2 } - 6 y - 3 ) = a y ^ { 3 } + 12 y ^ { 2 } + b y$$

Solve for a (complex solution)

$$\left\{\begin{matrix}a=-\frac{2y^{2}+b-6}{y^{2}}\text{, }&y\neq 0\\a\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2y by y^{2}-6y-3.

$$-2y^{3}+12y^{2}+6y=ay^{3}+12y^{2}+by$$

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

$$ay^{3}+12y^{2}+by=-2y^{3}+12y^{2}+6y$$

Subtract 12y^{2} from both sides.

$$ay^{3}+by=-2y^{3}+12y^{2}+6y-12y^{2}$$

Combine 12y^{2} and -12y^{2} to get 0.

$$ay^{3}+by=-2y^{3}+6y$$

Subtract by from both sides.

$$ay^{3}=-2y^{3}+6y-by$$

The equation is in standard form.

$$y^{3}a=6y-by-2y^{3}$$

Divide both sides by y^{3}.

$$\frac{y^{3}a}{y^{3}}=\frac{y\left(6-b-2y^{2}\right)}{y^{3}}$$

Dividing by y^{3} undoes the multiplication by y^{3}.

$$a=\frac{y\left(6-b-2y^{2}\right)}{y^{3}}$$

Divide y\left(-2y^{2}+6-b\right) by y^{3}.

$$a=\frac{6-b-2y^{2}}{y^{2}}$$

Solve for b (complex solution)

$$\left\{\begin{matrix}\\b=6-2y^{2}-ay^{2}\text{, }&\text{unconditionally}\\b\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2y by y^{2}-6y-3.

$$-2y^{3}+12y^{2}+6y=ay^{3}+12y^{2}+by$$

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

$$ay^{3}+12y^{2}+by=-2y^{3}+12y^{2}+6y$$

Subtract ay^{3} from both sides.

$$12y^{2}+by=-2y^{3}+12y^{2}+6y-ay^{3}$$

Subtract 12y^{2} from both sides.

$$by=-2y^{3}+12y^{2}+6y-ay^{3}-12y^{2}$$

Combine 12y^{2} and -12y^{2} to get 0.

$$by=-2y^{3}+6y-ay^{3}$$

The equation is in standard form.

$$yb=6y-2y^{3}-ay^{3}$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{y\left(6-2y^{2}-ay^{2}\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{y\left(6-2y^{2}-ay^{2}\right)}{y}$$

Divide y\left(-2y^{2}+6-ay^{2}\right) by y.

$$b=6-2y^{2}-ay^{2}$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{2y^{2}+b-6}{y^{2}}\text{, }&y\neq 0\\a\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2y by y^{2}-6y-3.

$$-2y^{3}+12y^{2}+6y=ay^{3}+12y^{2}+by$$

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

$$ay^{3}+12y^{2}+by=-2y^{3}+12y^{2}+6y$$

Subtract 12y^{2} from both sides.

$$ay^{3}+by=-2y^{3}+12y^{2}+6y-12y^{2}$$

Combine 12y^{2} and -12y^{2} to get 0.

$$ay^{3}+by=-2y^{3}+6y$$

Subtract by from both sides.

$$ay^{3}=-2y^{3}+6y-by$$

The equation is in standard form.

$$y^{3}a=6y-by-2y^{3}$$

Divide both sides by y^{3}.

$$\frac{y^{3}a}{y^{3}}=\frac{y\left(6-b-2y^{2}\right)}{y^{3}}$$

Dividing by y^{3} undoes the multiplication by y^{3}.

$$a=\frac{y\left(6-b-2y^{2}\right)}{y^{3}}$$

Divide y\left(-2y^{2}+6-b\right) by y^{3}.

$$a=\frac{6-b-2y^{2}}{y^{2}}$$

Solve for b

$$\left\{\begin{matrix}\\b=6-2y^{2}-ay^{2}\text{, }&\text{unconditionally}\\b\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply -2y by y^{2}-6y-3.

$$-2y^{3}+12y^{2}+6y=ay^{3}+12y^{2}+by$$

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

$$ay^{3}+12y^{2}+by=-2y^{3}+12y^{2}+6y$$

Subtract ay^{3} from both sides.

$$12y^{2}+by=-2y^{3}+12y^{2}+6y-ay^{3}$$

Subtract 12y^{2} from both sides.

$$by=-2y^{3}+12y^{2}+6y-ay^{3}-12y^{2}$$

Combine 12y^{2} and -12y^{2} to get 0.

$$by=-2y^{3}+6y-ay^{3}$$

The equation is in standard form.

$$yb=6y-2y^{3}-ay^{3}$$

Divide both sides by y.

$$\frac{yb}{y}=\frac{y\left(6-2y^{2}-ay^{2}\right)}{y}$$

Dividing by y undoes the multiplication by y.

$$b=\frac{y\left(6-2y^{2}-ay^{2}\right)}{y}$$

Divide y\left(-2y^{2}+6-ay^{2}\right) by y.

$$b=6-2y^{2}-ay^{2}$$