Solve x^3+xy=0

$$x ^ { 3 } + x y = 0$$

Solve for y (complex solution)

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

Steps for Solving Linear Equation

Subtract x^{3} from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Divide -x^{3} by x.

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

Solve for y

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

Steps for Solving Linear Equation

Subtract x^{3} from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by x.

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

Dividing by x undoes the multiplication by x.

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

Divide -x^{3} by x.

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

Solve for x (complex solution)

$$x=i\sqrt{y}$$

$$x=0$$

$$x=-i\sqrt{y}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\sqrt{-y}\text{; }x=-\sqrt{-y}\text{, }&y\leq 0\end{matrix}\right.$$