Solve y=p(x-1)+4/x

$$y = p ( x - 1 ) + \frac { 4 } { x }$$

Solve for p (complex solution)

$$\left\{\begin{matrix}p=-\frac{4-xy}{x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\p\in \mathrm{C}\text{, }&x=1\text{ and }y=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$yx=p\left(x-1\right)x+4$$

Use the distributive property to multiply p by x-1.

$$yx=\left(px-p\right)x+4$$

Use the distributive property to multiply px-p by x.

$$yx=px^{2}-px+4$$

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

$$px^{2}-px+4=yx$$

Subtract 4 from both sides.

$$px^{2}-px=yx-4$$

Combine all terms containing p.

$$\left(x^{2}-x\right)p=yx-4$$

The equation is in standard form.

$$\left(x^{2}-x\right)p=xy-4$$

Divide both sides by x^{2}-x.

$$\frac{\left(x^{2}-x\right)p}{x^{2}-x}=\frac{xy-4}{x^{2}-x}$$

Dividing by x^{2}-x undoes the multiplication by x^{2}-x.

$$p=\frac{xy-4}{x^{2}-x}$$

Divide yx-4 by x^{2}-x.

$$p=\frac{xy-4}{x\left(x-1\right)}$$

Solve for p

$$\left\{\begin{matrix}p=-\frac{4-xy}{x\left(x-1\right)}\text{, }&x\neq 1\text{ and }x\neq 0\\p\in \mathrm{R}\text{, }&x=1\text{ and }y=4\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$yx=p\left(x-1\right)x+4$$

Use the distributive property to multiply p by x-1.

$$yx=\left(px-p\right)x+4$$

Use the distributive property to multiply px-p by x.

$$yx=px^{2}-px+4$$

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

$$px^{2}-px+4=yx$$

Subtract 4 from both sides.

$$px^{2}-px=yx-4$$

Combine all terms containing p.

$$\left(x^{2}-x\right)p=yx-4$$

The equation is in standard form.

$$\left(x^{2}-x\right)p=xy-4$$

Divide both sides by x^{2}-x.

$$\frac{\left(x^{2}-x\right)p}{x^{2}-x}=\frac{xy-4}{x^{2}-x}$$

Dividing by x^{2}-x undoes the multiplication by x^{2}-x.

$$p=\frac{xy-4}{x^{2}-x}$$

Divide yx-4 by x^{2}-x.

$$p=\frac{xy-4}{x\left(x-1\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{y^{2}+2py+p^{2}-16p}-p-y}{2p}\text{; }x=\frac{\sqrt{y^{2}+2py+p^{2}-16p}+p+y}{2p}\text{, }&p\neq 0\\x=\frac{4}{y}\text{, }&p=0\text{ and }y\neq 0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{y^{2}+2py+p^{2}-16p}-p-y}{2p}\text{; }x=\frac{\sqrt{y^{2}+2py+p^{2}-16p}+p+y}{2p}\text{, }&p\neq 0\text{ and }\left(y\geq -p+4\sqrt{p}\text{ or }y\leq -p-4\sqrt{p}\text{ or }p<0\right)\\x=\frac{4}{y}\text{, }&p=0\text{ and }y\neq 0\end{matrix}\right.$$