Solve y=x^2+px-q/x+r

$$y = \frac { x ^ { 2 } + p x - q } { x + r }$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=\frac{q+ry+xy-x^{2}}{x}\text{, }&x\neq 0\text{ and }x\neq -r\\p\in \mathrm{C}\text{, }&q=-ry\text{ and }x=0\text{ and }r\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+r.

$$y\left(x+r\right)=x^{2}+px-q$$

Use the distributive property to multiply y by x+r.

$$yx+yr=x^{2}+px-q$$

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

$$x^{2}+px-q=yx+yr$$

Subtract x^{2} from both sides.

$$px-q=yx+yr-x^{2}$$

Add q to both sides.

$$px=yx+yr-x^{2}+q$$

The equation is in standard form.

$$xp=q+ry+xy-x^{2}$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{q+ry+xy-x^{2}}{x}$$

Dividing by x undoes the multiplication by x.

$$p=\frac{q+ry+xy-x^{2}}{x}$$

Divide yx+yr-x^{2}+q by x.

$$p=\frac{ry+q}{x}+y-x$$

Solve for p

$$\left\{\begin{matrix}p=\frac{q+ry+xy-x^{2}}{x}\text{, }&x\neq 0\text{ and }x\neq -r\\p\in \mathrm{R}\text{, }&q=-ry\text{ and }x=0\text{ and }r\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+r.

$$y\left(x+r\right)=x^{2}+px-q$$

Use the distributive property to multiply y by x+r.

$$yx+yr=x^{2}+px-q$$

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

$$x^{2}+px-q=yx+yr$$

Subtract x^{2} from both sides.

$$px-q=yx+yr-x^{2}$$

Add q to both sides.

$$px=yx+yr-x^{2}+q$$

The equation is in standard form.

$$xp=q+ry+xy-x^{2}$$

Divide both sides by x.

$$\frac{xp}{x}=\frac{q+ry+xy-x^{2}}{x}$$

Dividing by x undoes the multiplication by x.

$$p=\frac{q+ry+xy-x^{2}}{x}$$

Divide yx+yr-x^{2}+q by x.

$$p=\frac{ry+q}{x}+y-x$$

Solve for q

$$q=-y\left(x+r\right)+px+x^{2}$$

$$x\neq -r$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x+r.

$$y\left(x+r\right)=x^{2}+px-q$$

Use the distributive property to multiply y by x+r.

$$yx+yr=x^{2}+px-q$$

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

$$x^{2}+px-q=yx+yr$$

Subtract x^{2} from both sides.

$$px-q=yx+yr-x^{2}$$

Subtract px from both sides.

$$-q=yx+yr-x^{2}-px$$

The equation is in standard form.

$$-q=ry-px+xy-x^{2}$$

Divide both sides by -1.

$$\frac{-q}{-1}=\frac{ry-px+xy-x^{2}}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$q=\frac{ry-px+xy-x^{2}}{-1}$$

Divide yx+yr-x^{2}-px by -1.

$$q=x^{2}+px-xy-ry$$