Solve x/1+z*x/y+y/z=1/y-z

$$\frac { x } { 1 } + \frac { z \cdot x } { y } + \frac { y } { z } = \frac { 1 } { y - z }$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y\left(y^{2}-yz-z\right)}{z\left(y^{2}-z^{2}\right)}\text{, }&z\neq 0\text{ and }y\neq 0\text{ and }|z|\neq |y|\\x\in \mathrm{R}\text{, }&z=\frac{1}{2}\text{ and }y=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by yz\left(y-z\right), the least common multiple of y,z,y-z.

$$yz\left(y-z\right)x+z\left(y-z\right)zx+y\left(y-z\right)y=yz$$

Use the distributive property to multiply yz by y-z.

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

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

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

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

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

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

$$zy^{2}x-yz^{2}x+\left(z^{2}y-z^{3}\right)x+y\left(y-z\right)y=yz$$

Use the distributive property to multiply z^{2}y-z^{3} by x.

$$zy^{2}x-yz^{2}x+z^{2}yx-z^{3}x+y\left(y-z\right)y=yz$$

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

$$zy^{2}x-z^{3}x+y\left(y-z\right)y=yz$$

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

$$zy^{2}x-z^{3}x+y^{2}\left(y-z\right)=yz$$

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

$$zy^{2}x-z^{3}x+y^{3}-y^{2}z=yz$$

Subtract y^{3} from both sides.

$$zy^{2}x-z^{3}x-y^{2}z=yz-y^{3}$$

Add y^{2}z to both sides.

$$zy^{2}x-z^{3}x=yz-y^{3}+y^{2}z$$

Combine all terms containing x.

$$\left(zy^{2}-z^{3}\right)x=yz-y^{3}+y^{2}z$$

The equation is in standard form.

$$\left(zy^{2}-z^{3}\right)x=zy^{2}+yz-y^{3}$$

Divide both sides by zy^{2}-z^{3}.

$$\frac{\left(zy^{2}-z^{3}\right)x}{zy^{2}-z^{3}}=\frac{y\left(z+yz-y^{2}\right)}{zy^{2}-z^{3}}$$

Dividing by zy^{2}-z^{3} undoes the multiplication by zy^{2}-z^{3}.

$$x=\frac{y\left(z+yz-y^{2}\right)}{zy^{2}-z^{3}}$$

Divide y\left(z-y^{2}+yz\right) by zy^{2}-z^{3}.

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