Solve z^2-xy-yz-x=0

$${ z }^{ 2 } -xy-yz-x=0$$

Solve for x

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

Steps for Solving Linear Equation

Subtract z^{2} from both sides. Anything subtracted from zero gives its negation.

$$-xy-yz-x=-z^{2}$$

Add yz to both sides.

$$-xy-x=-z^{2}+yz$$

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by -y-1.

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

Dividing by -y-1 undoes the multiplication by -y-1.

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

Divide z\left(-z+y\right) by -y-1.

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

Solve for y

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

Steps for Solving Linear Equation

Subtract z^{2} from both sides. Anything subtracted from zero gives its negation.

$$-xy-yz-x=-z^{2}$$

Add x to both sides.

$$-xy-yz=-z^{2}+x$$

Combine all terms containing y.

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

The equation is in standard form.

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

Divide both sides by -x-z.

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

Dividing by -x-z undoes the multiplication by -x-z.

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

Divide x-z^{2} by -x-z.

$$y=-\frac{x-z^{2}}{x+z}$$