Solve y^n+xy=0

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-y^{n-1}\text{, }&y\neq 0\\x\in \mathrm{C}\text{, }&y=0\text{ and }n\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract y^{n} from both sides. Anything subtracted from zero gives its negation.

$$xy=-y^{n}$$

The equation is in standard form.

$$yx=-y^{n}$$

Divide both sides by y.

$$\frac{yx}{y}=-\frac{y^{n}}{y}$$

Dividing by y undoes the multiplication by y.

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

Divide -y^{n} by y.

$$x=-y^{n-1}$$

Solve for x

$$\left\{\begin{matrix}x=-y^{n-1}\text{, }&\left(Denominator(n)\text{bmod}2=1\text{ and }y<0\right)\text{ or }y>0\\x\in \mathrm{R}\text{, }&y=0\text{ and }n>0\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract y^{n} from both sides. Anything subtracted from zero gives its negation.

$$xy=-y^{n}$$

The equation is in standard form.

$$yx=-y^{n}$$

Divide both sides by y.

$$\frac{yx}{y}=-\frac{y^{n}}{y}$$

Dividing by y undoes the multiplication by y.

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

Divide -y^{n} by y.

$$x=-y^{n-1}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\log_{y}\left(-xy\right)+\frac{2\pi n_{1}i}{\ln(y)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 0\text{ and }x\neq 0\text{ and }y\neq 1\\n\in \mathrm{C}\text{, }&y=0\text{ or }\left(x=-1\text{ and }y=1\right)\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\log_{y}\left(-xy\right)\text{, }&x<0\text{ and }y>0\text{ and }y\neq 1\\n\in \mathrm{R}\text{, }&\left(x=-1\text{ and }y=1\right)\text{ or }\left(x=-1\text{ and }y=-1\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n)\text{bmod}2=1\right)\\n>0\text{, }&y=0\end{matrix}\right.$$