Solve (x-3)(xy-1)=7

$$( x - 3 ) ( x y - 1 ) = 7$$

Solve for y

$$y=\frac{x+4}{x\left(x-3\right)}$$

$$x\neq 3\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-3 by xy-1.

$$yx^{2}-x-3xy+3=7$$

Add x to both sides.

$$yx^{2}-3xy+3=7+x$$

Subtract 3 from both sides.

$$yx^{2}-3xy=7+x-3$$

Subtract 3 from 7 to get 4.

$$yx^{2}-3xy=4+x$$

Combine all terms containing y.

$$\left(x^{2}-3x\right)y=4+x$$

The equation is in standard form.

$$\left(x^{2}-3x\right)y=x+4$$

Divide both sides by x^{2}-3x.

$$\frac{\left(x^{2}-3x\right)y}{x^{2}-3x}=\frac{x+4}{x^{2}-3x}$$

Dividing by x^{2}-3x undoes the multiplication by x^{2}-3x.

$$y=\frac{x+4}{x^{2}-3x}$$

Divide x+4 by x^{2}-3x.

$$y=\frac{x+4}{x\left(x-3\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{9y^{2}+22y+1}+3y+1}{2y}\text{; }x=\frac{-\sqrt{9y^{2}+22y+1}+3y+1}{2y}\text{, }&y\neq 0\\x=-4\text{, }&y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{9y^{2}+22y+1}+3y+1}{2y}\text{; }x=\frac{-\sqrt{9y^{2}+22y+1}+3y+1}{2y}\text{, }&y\leq \frac{-4\sqrt{7}-11}{9}\text{ or }\left(y\neq 0\text{ and }y\geq \frac{4\sqrt{7}-11}{9}\right)\\x=-4\text{, }&y=0\end{matrix}\right.$$