Solve y=(m-2)x^n-1+3

$$y = ( m - 2 ) x ^ { n - 1 } + 3$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\left(3-y\right)x^{1-n}+2\text{, }&n=1\text{ or }x\neq 0\\m\in \mathrm{C}\text{, }&y=3\text{ and }x=0\text{ and }n\neq 1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m-2 by x^{n-1}.

$$y=mx^{n-1}-2x^{n-1}+3$$

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

$$mx^{n-1}-2x^{n-1}+3=y$$

Add 2x^{n-1} to both sides.

$$mx^{n-1}+3=y+2x^{n-1}$$

Subtract 3 from both sides.

$$mx^{n-1}=y+2x^{n-1}-3$$

The equation is in standard form.

$$x^{n-1}m=2x^{n-1}+y-3$$

Divide both sides by x^{n-1}.

$$\frac{x^{n-1}m}{x^{n-1}}=\frac{2x^{n-1}+y-3}{x^{n-1}}$$

Dividing by x^{n-1} undoes the multiplication by x^{n-1}.

$$m=\frac{2x^{n-1}+y-3}{x^{n-1}}$$

Divide y+2x^{n-1}-3 by x^{n-1}.

$$m=x^{1-n}\left(\frac{2}{x^{1-n}}+y-3\right)$$

Solve for m

$$\left\{\begin{matrix}m=-\left(3-y\right)x^{1-n}+2\text{, }&\left(x<0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }x>0\\m\in \mathrm{R}\text{, }&y=3\text{ and }x=0\text{ and }n>1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply m-2 by x^{n-1}.

$$y=mx^{n-1}-2x^{n-1}+3$$

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

$$mx^{n-1}-2x^{n-1}+3=y$$

Add 2x^{n-1} to both sides.

$$mx^{n-1}+3=y+2x^{n-1}$$

Subtract 3 from both sides.

$$mx^{n-1}=y+2x^{n-1}-3$$

The equation is in standard form.

$$x^{n-1}m=2x^{n-1}+y-3$$

Divide both sides by x^{n-1}.

$$\frac{x^{n-1}m}{x^{n-1}}=\frac{2x^{n-1}+y-3}{x^{n-1}}$$

Dividing by x^{n-1} undoes the multiplication by x^{n-1}.

$$m=\frac{2x^{n-1}+y-3}{x^{n-1}}$$

Divide y+2x^{n-1}-3 by x^{n-1}.

$$m=x^{1-n}\left(\frac{2}{x^{1-n}}+y-3\right)$$

Solve for n (complex solution)

$$\left\{\begin{matrix}n=\frac{\ln(x)+\ln(\frac{y-3}{m-2})}{\ln(x)}+\frac{2\pi n_{1}i}{\ln(x)}\text{, }n_{1}\in \mathrm{Z}\text{, }&y\neq 3\text{ and }x\neq 1\text{ and }x\neq 0\text{ and }m\neq 2\\n\in \mathrm{C}\text{, }&\left(x=0\text{ and }y=3\text{ and }m\neq 2\right)\text{ or }\left(x=1\text{ and }m=y-1\text{ and }y\neq 3\right)\text{ or }\left(y=3\text{ and }m=2\right)\end{matrix}\right.$$

Solve for n

$$\left\{\begin{matrix}n=\frac{\ln(x)+\ln(\frac{y-3}{m-2})}{\ln(x)}\text{, }&\left(y<3\text{ and }m<2\text{ and }x\neq 1\text{ and }x>0\right)\text{ or }\left(y>3\text{ and }m>2\text{ and }x\neq 1\text{ and }x>0\right)\\n\in \mathrm{R}\text{, }&\left(m=y-1\text{ and }y\neq 3\text{ and }x=1\right)\text{ or }\left(y=3\text{ and }m=2\text{ and }x>0\right)\text{ or }\left(m=5-y\text{ and }Denominator(n)\text{bmod}2=1\text{ and }Numerator(n-1)\text{bmod}2=1\text{ and }y\neq 3\text{ and }x=-1\right)\text{ or }\left(y=3\text{ and }m=2\text{ and }x<0\text{ and }Denominator(n)\text{bmod}2=1\right)\\n>1\text{, }&x=0\text{ and }y=3\end{matrix}\right.$$