Solve z=12x^2+2xy+2y

$$z = 12 x ^ { 2 } + 2 x y + 2 y$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{12x^{2}-z}{2\left(x+1\right)}\text{, }&x\neq -1\\y\in \mathrm{R}\text{, }&z=12\text{ and }x=-1\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$12x^{2}+2xy+2y=z$$

Subtract 12x^{2} from both sides.

$$2xy+2y=z-12x^{2}$$

Combine all terms containing y.

$$\left(2x+2\right)y=z-12x^{2}$$

Divide both sides by 2x+2.

$$\frac{\left(2x+2\right)y}{2x+2}=\frac{z-12x^{2}}{2x+2}$$

Dividing by 2x+2 undoes the multiplication by 2x+2.

$$y=\frac{z-12x^{2}}{2x+2}$$

Divide z-12x^{2} by 2x+2.

$$y=\frac{z-12x^{2}}{2\left(x+1\right)}$$

Solve for x

$$x=\frac{\sqrt{y^{2}-24y+12z}-y}{12}$$

$$x=\frac{-\sqrt{y^{2}-24y+12z}-y}{12}\text{, }z\geq -\frac{y^{2}}{12}+2y$$