Solve x+n=x^n*r*(1-n)

$$x + n = x ^ { n } \cdot r \cdot ( 1 - n )$$

Solve for r (complex solution)

$$\left\{\begin{matrix}r=-\frac{x+n}{\left(n-1\right)x^{n}}\text{, }&\left(x\neq 0\text{ and }n\neq 1\right)\text{ or }n=0\\r\in \mathrm{C}\text{, }&x=-1\text{ and }n=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{n}r by 1-n.

$$x+n=x^{n}r-x^{n}rn$$

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

$$x^{n}r-x^{n}rn=x+n$$

Combine all terms containing r.

$$\left(x^{n}-x^{n}n\right)r=x+n$$

The equation is in standard form.

$$\left(x^{n}-nx^{n}\right)r=x+n$$

Divide both sides by x^{n}-x^{n}n.

$$\frac{\left(x^{n}-nx^{n}\right)r}{x^{n}-nx^{n}}=\frac{x+n}{x^{n}-nx^{n}}$$

Dividing by x^{n}-x^{n}n undoes the multiplication by x^{n}-x^{n}n.

$$r=\frac{x+n}{x^{n}-nx^{n}}$$

Divide x+n by x^{n}-x^{n}n.

$$r=\frac{x+n}{\left(1-n\right)x^{n}}$$

Solve for r

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

Steps for Solving Linear Equation

Use the distributive property to multiply x^{n}r by 1-n.

$$x+n=x^{n}r-x^{n}rn$$

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

$$x^{n}r-x^{n}rn=x+n$$

Combine all terms containing r.

$$\left(x^{n}-x^{n}n\right)r=x+n$$

The equation is in standard form.

$$\left(x^{n}-nx^{n}\right)r=x+n$$

Divide both sides by x^{n}-x^{n}n.

$$\frac{\left(x^{n}-nx^{n}\right)r}{x^{n}-nx^{n}}=\frac{x+n}{x^{n}-nx^{n}}$$

Dividing by x^{n}-x^{n}n undoes the multiplication by x^{n}-x^{n}n.

$$r=\frac{x+n}{x^{n}-nx^{n}}$$

Divide x+n by x^{n}-x^{n}n.

$$r=\frac{x+n}{\left(1-n\right)x^{n}}$$