Solve y=px^n+q

$$y = p x ^ { n } + q$$

Solve for p (complex solution)

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

Steps for Solving Linear Equation

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

$$px^{n}+q=y$$

Subtract q from both sides.

$$px^{n}=y-q$$

The equation is in standard form.

$$x^{n}p=y-q$$

Divide both sides by x^{n}.

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

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

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

Solve for p

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

Steps for Solving Linear Equation

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

$$px^{n}+q=y$$

Subtract q from both sides.

$$px^{n}=y-q$$

The equation is in standard form.

$$x^{n}p=y-q$$

Divide both sides by x^{n}.

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

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

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

Solve for n (complex solution)

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

Solve for n

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