Solve x+z/x-z-x-z/x+z=z(2x^2+zy)/x^2-z^2

$$\frac{ x+z }{ x-z } - \frac{ x-z }{ x+z } = \frac{ z(2 { x }^{ 2 } +zy) }{ { x }^{ 2 } - { z }^{ 2 } }$$

Solve for y (complex solution)

$$\left\{\begin{matrix}y=-\frac{2x\left(x-2\right)}{z}\text{, }&z\neq 0\text{ and }x\neq z\text{ and }x\neq -z\\y\in \mathrm{C}\text{, }&z=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-z\right)\left(-x-z\right), the least common multiple of x-z,x+z,x^{2}-z^{2}.

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

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

$$-x^{2}-2xz-z^{2}-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

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

$$-x^{2}-2xz-z^{2}-\left(-x^{2}+2xz-z^{2}\right)=-z\left(2x^{2}+zy\right)$$

To find the opposite of -x^{2}+2xz-z^{2}, find the opposite of each term.

$$-x^{2}-2xz-z^{2}+x^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -x^{2} and x^{2} to get 0.

$$-2xz-z^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -2xz and -2xz to get -4xz.

$$-4xz-z^{2}+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -z^{2} and z^{2} to get 0.

$$-4xz=-z\left(2x^{2}+zy\right)$$

Use the distributive property to multiply -z by 2x^{2}+zy.

$$-4xz=-2zx^{2}-yz^{2}$$

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

$$-2zx^{2}-yz^{2}=-4xz$$

Add 2zx^{2} to both sides.

$$-yz^{2}=-4xz+2zx^{2}$$

The equation is in standard form.

$$\left(-z^{2}\right)y=2zx^{2}-4xz$$

Divide both sides by -z^{2}.

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

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

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

Divide 2xz\left(-2+x\right) by -z^{2}.

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

Solve for y

$$\left\{\begin{matrix}y=-\frac{2x\left(x-2\right)}{z}\text{, }&z\neq 0\text{ and }|x|\neq |z|\\y\in \mathrm{R}\text{, }&z=0\text{ and }x\neq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-z\right)\left(-x-z\right), the least common multiple of x-z,x+z,x^{2}-z^{2}.

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

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

$$-x^{2}-2xz-z^{2}-\left(-x+z\right)\left(x-z\right)=-z\left(2x^{2}+zy\right)$$

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

$$-x^{2}-2xz-z^{2}-\left(-x^{2}+2xz-z^{2}\right)=-z\left(2x^{2}+zy\right)$$

To find the opposite of -x^{2}+2xz-z^{2}, find the opposite of each term.

$$-x^{2}-2xz-z^{2}+x^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -x^{2} and x^{2} to get 0.

$$-2xz-z^{2}-2xz+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -2xz and -2xz to get -4xz.

$$-4xz-z^{2}+z^{2}=-z\left(2x^{2}+zy\right)$$

Combine -z^{2} and z^{2} to get 0.

$$-4xz=-z\left(2x^{2}+zy\right)$$

Use the distributive property to multiply -z by 2x^{2}+zy.

$$-4xz=-2zx^{2}-yz^{2}$$

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

$$-2zx^{2}-yz^{2}=-4xz$$

Add 2zx^{2} to both sides.

$$-yz^{2}=-4xz+2zx^{2}$$

The equation is in standard form.

$$\left(-z^{2}\right)y=2zx^{2}-4xz$$

Divide both sides by -z^{2}.

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

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

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

Divide 2xz\left(-2+x\right) by -z^{2}.

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