Solve a^y=x+1/x-1

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

Solve for x

$$\left\{\begin{matrix}x=\frac{a^{y}+1}{a^{y}-1}\text{, }&\left(a=0\text{ and }y>0\right)\text{ or }\left(y\neq 0\text{ and }a\neq -1\text{ and }Denominator(y)\text{bmod}2=1\text{ and }a<0\right)\text{ or }\left(a<0\text{ and }Numerator(y)\text{bmod}2=1\text{ and }Denominator(y)\text{bmod}2=1\right)\text{ or }\left(y\neq 0\text{ and }a\neq 1\text{ and }a>0\right)\\x\neq 1\text{, }&\left(Numerator(y)\text{bmod}2=1\text{ and }Numerator(y)\text{bmod}2=0\text{ and }a=1\text{ and }Denominator(y)\text{bmod}2=1\right)\text{ or }\left(a=\left(-1\right)^{\frac{1}{y}}\text{ and }Denominator(y)\text{bmod}2=1\text{ and }Denominator(y)\text{bmod}2=0\text{ and }Numerator(y)\text{bmod}2=1\right)\text{ or }\left(Numerator(y)\text{bmod}2=1\text{ and }Numerator(y)\text{bmod}2=0\text{ and }a=-1\text{ and }Denominator(y)\text{bmod}2=1\right)\end{matrix}\right.$$

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.

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

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

$$xa^{y}-a^{y}=x+1$$

Subtract x from both sides.

$$xa^{y}-a^{y}-x=1$$

Add a^{y} to both sides.

$$xa^{y}-x=1+a^{y}$$

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by a^{y}-1.

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

Dividing by a^{y}-1 undoes the multiplication by a^{y}-1.

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

Variable x cannot be equal to 1.

$$x=\frac{a^{y}+1}{a^{y}-1}\text{, }x\neq 1$$