Solve r/1-r+x-2/r=1

$$\frac { r } { 1 - r } + \frac { x - 2 } { r } = 1$$

Solve for x

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

$$r\neq 1\text{ and }r\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by r\left(r-1\right), the least common multiple of 1-r,r.

$$-rr+\left(r-1\right)\left(x-2\right)=r\left(r-1\right)$$

Multiply r and r to get r^{2}.

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

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

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

Use the distributive property to multiply r by r-1.

$$-r^{2}+rx-2r-x+2=r^{2}-r$$

Add r^{2} to both sides.

$$rx-2r-x+2=r^{2}-r+r^{2}$$

Combine r^{2} and r^{2} to get 2r^{2}.

$$rx-2r-x+2=2r^{2}-r$$

Add 2r to both sides.

$$rx-x+2=2r^{2}-r+2r$$

Combine -r and 2r to get r.

$$rx-x+2=2r^{2}+r$$

Subtract 2 from both sides.

$$rx-x=2r^{2}+r-2$$

Combine all terms containing x.

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

Divide both sides by r-1.

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

Dividing by r-1 undoes the multiplication by r-1.

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

Solve for r (complex solution)

$$\left\{\begin{matrix}\\r=\frac{\sqrt{x^{2}-10x+17}+x-1}{4}\text{, }&\text{unconditionally}\\r=\frac{-\sqrt{x^{2}-10x+17}+x-1}{4}\text{, }&x\neq 2\end{matrix}\right.$$

Solve for r

$$\left\{\begin{matrix}r=\frac{-\sqrt{x^{2}-10x+17}+x-1}{4}\text{, }&\left(x\leq 5-2\sqrt{2}\text{ and }x\neq 2\right)\text{ or }x\geq 2\sqrt{2}+5\\r=\frac{\sqrt{x^{2}-10x+17}+x-1}{4}\text{, }&x\geq 2\sqrt{2}+5\text{ or }x\leq 5-2\sqrt{2}\end{matrix}\right.$$