Solve (y-1)x^2-2xy+y+1=0

$$(y-1) { x }^{ 2 } -2xy+y+1=0$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=\frac{x+1}{x-1}\text{, }&x\neq 1\\y\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y-1 by x^{2}.

$$yx^{2}-x^{2}-2xy+y+1=0$$

Add x^{2} to both sides. Anything plus zero gives itself.

$$yx^{2}-2xy+y+1=x^{2}$$

Subtract 1 from both sides.

$$yx^{2}-2xy+y=x^{2}-1$$

Combine all terms containing y.

$$\left(x^{2}-2x+1\right)y=x^{2}-1$$

Divide both sides by x^{2}-2x+1.

$$\frac{\left(x^{2}-2x+1\right)y}{x^{2}-2x+1}=\frac{x^{2}-1}{x^{2}-2x+1}$$

Dividing by x^{2}-2x+1 undoes the multiplication by x^{2}-2x+1.

$$y=\frac{x^{2}-1}{x^{2}-2x+1}$$

Divide x^{2}-1 by x^{2}-2x+1.

$$y=\frac{x+1}{x-1}$$

Solve for y

$$\left\{\begin{matrix}y=\frac{x+1}{x-1}\text{, }&x\neq 1\\y\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply y-1 by x^{2}.

$$yx^{2}-x^{2}-2xy+y+1=0$$

Add x^{2} to both sides. Anything plus zero gives itself.

$$yx^{2}-2xy+y+1=x^{2}$$

Subtract 1 from both sides.

$$yx^{2}-2xy+y=x^{2}-1$$

Combine all terms containing y.

$$\left(x^{2}-2x+1\right)y=x^{2}-1$$

Divide both sides by x^{2}-2x+1.

$$\frac{\left(x^{2}-2x+1\right)y}{x^{2}-2x+1}=\frac{x^{2}-1}{x^{2}-2x+1}$$

Dividing by x^{2}-2x+1 undoes the multiplication by x^{2}-2x+1.

$$y=\frac{x^{2}-1}{x^{2}-2x+1}$$

Divide x^{2}-1 by x^{2}-2x+1.

$$y=\frac{x+1}{x-1}$$

Solve for x

$$\left\{\begin{matrix}\\x=1\text{, }&\text{unconditionally}\\x=\frac{y+1}{y-1}\text{, }&y\neq 1\end{matrix}\right.$$