Solve z=x^2+3xy

$$z= { x }^{ 2 } +3xy$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{x}{3}+\frac{z}{3x}\text{, }&x\neq 0\\y\in \mathrm{R}\text{, }&z=0\text{ and }x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$x^{2}+3xy=z$$

Subtract x^{2} from both sides.

$$3xy=z-x^{2}$$

Divide both sides by 3x.

$$\frac{3xy}{3x}=\frac{z-x^{2}}{3x}$$

Dividing by 3x undoes the multiplication by 3x.

$$y=\frac{z-x^{2}}{3x}$$

Divide z-x^{2} by 3x.

$$y=-\frac{x}{3}+\frac{z}{3x}$$

Solve for x

$$x=\frac{-\sqrt{9y^{2}+4z}-3y}{2}$$

$$x=\frac{\sqrt{9y^{2}+4z}-3y}{2}\text{, }z\geq -\frac{9y^{2}}{4}$$