Solve (x-2)(xy-1)=5

$$( x - 2 ) ( x y - 1 ) = 5$$

Solve for y

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

$$x\neq 2\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-2 by xy-1.

$$yx^{2}-x-2xy+2=5$$

Add x to both sides.

$$yx^{2}-2xy+2=5+x$$

Subtract 2 from both sides.

$$yx^{2}-2xy=5+x-2$$

Subtract 2 from 5 to get 3.

$$yx^{2}-2xy=3+x$$

Combine all terms containing y.

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

The equation is in standard form.

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

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

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

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

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

Divide x+3 by x^{2}-2x.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{4y^{2}+16y+1}+2y+1}{2y}\text{; }x=\frac{-\sqrt{4y^{2}+16y+1}+2y+1}{2y}\text{, }&y\neq 0\\x=-3\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{4y^{2}+16y+1}+2y+1}{2y}\text{; }x=\frac{-\sqrt{4y^{2}+16y+1}+2y+1}{2y}\text{, }&y\leq -\frac{\sqrt{15}}{2}-2\text{ or }\left(y\neq 0\text{ and }y\geq \frac{\sqrt{15}}{2}-2\right)\\x=-3\text{, }&y=0\end{matrix}\right.$$