Solve x-y+z/x^2-2x=b

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

Solve for b

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

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

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(x-2\right).

$$x-y+z=bx\left(x-2\right)$$

Use the distributive property to multiply bx by x-2.

$$x-y+z=bx^{2}-2bx$$

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

$$bx^{2}-2bx=x-y+z$$

Combine all terms containing b.

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

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

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

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

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

Divide x-y+z by x^{2}-2x.

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{\sqrt{1+4b+4b^{2}+4bz-4by}+2b+1}{2b}\text{, }&\left(b\neq \frac{1}{2}\text{ and }arg(1-2b)<\pi \text{ and }b\neq 0\text{ and }b\neq -\frac{1}{2}\text{ and }arg(2b+1)<\pi \right)\text{ or }\left(b\neq \frac{1}{2}\text{ and }arg(1-2b)<\pi \text{ and }b\neq 0\text{ and }y\neq z\right)\text{ or }\left(y\neq z+2\text{ and }b\neq 0\text{ and }b\neq -\frac{1}{2}\text{ and }arg(2b+1)<\pi \right)\text{ or }\left(y\neq z+2\text{ and }b\neq 0\text{ and }y\neq z\right)\\x=-\frac{\sqrt{1+4b+4b^{2}+4bz-4by}-2b-1}{2b}\text{, }&\left(arg(2b+1)\geq \pi \text{ and }arg(1-2b)\geq \pi \right)\text{ or }\left(arg(1-2b)\geq \pi \text{ and }b\neq \frac{1}{2}\text{ and }y\neq z\right)\text{ or }\left(arg(2b+1)\geq \pi \text{ and }y\neq z+2\text{ and }b\neq -\frac{1}{2}\right)\text{ or }\left(b\neq 0\text{ and }y\neq z+2\text{ and }y\neq z\right)\\x=y-z\text{, }&y\neq z+2\text{ and }y\neq z\text{ and }b=0\end{matrix}\right.$$