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

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

Solve for x

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

$$|r|\neq 1$$

Steps for Solving Linear Equation

Variable x cannot be equal to 1 since division by zero is not defined. Multiply both sides of the equation by x-1.

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

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

$$xr^{2}-r^{2}=x+1$$

Subtract x from both sides.

$$xr^{2}-r^{2}-x=1$$

Add r^{2} to both sides.

$$xr^{2}-x=1+r^{2}$$

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by r^{2}-1.

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

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

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

Variable x cannot be equal to 1.

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

Solve for r

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

$$r=-\sqrt{\frac{x+1}{x-1}}\text{, }x>1\text{ or }x\leq -1$$