Solve y-1=1/x^2-1

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

Solve for y (complex solution)

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

$$x\neq -1\text{ and }x\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$\left(x-1\right)\left(x+1\right)y+\left(x-1\right)\left(x+1\right)\left(-1\right)=1$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

Use the distributive property to multiply x-1 by x+1 and combine like terms.

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

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

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

Add x^{2} to both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

$$x^{2}y-y=x^{2}$$

Combine all terms containing y.

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

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

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

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

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

Solve for y

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

$$|x|\neq 1$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-1\right)\left(x+1\right).

$$\left(x-1\right)\left(x+1\right)y+\left(x-1\right)\left(x+1\right)\left(-1\right)=1$$

Consider \left(x-1\right)\left(x+1\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 1.

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

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

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

Use the distributive property to multiply x-1 by x+1 and combine like terms.

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

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

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

Add x^{2} to both sides.

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

Subtract 1 from both sides.

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

Subtract 1 from 1 to get 0.

$$x^{2}y-y=x^{2}$$

Combine all terms containing y.

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

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

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

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

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

Solve for x (complex solution)

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

$$x=\left(y-1\right)^{-\frac{1}{2}}\sqrt{y}\text{, }y\neq 1$$

Solve for x

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

$$x=-\sqrt{\frac{y}{y-1}}\text{, }y>1\text{ or }y\leq 0$$