Solve xyz+xy+xz+y+x+z=1000

$$xyz+xy+xz+y+x+z=1000$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{y+z-1000}{yz+y+z+1}\text{, }&z\neq -1\text{ and }y\neq -1\\x\in \mathrm{R}\text{, }&\left(y=-1\text{ and }z=1001\right)\text{ or }\left(y=1001\text{ and }z=-1\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract y from both sides.

$$xyz+xy+xz+x+z=1000-y$$

Subtract z from both sides.

$$xyz+xy+xz+x=1000-y-z$$

Combine all terms containing x.

$$\left(yz+y+z+1\right)x=1000-y-z$$

The equation is in standard form.

$$\left(yz+y+z+1\right)x=1000-z-y$$

Divide both sides by yz+y+z+1.

$$\frac{\left(yz+y+z+1\right)x}{yz+y+z+1}=\frac{1000-z-y}{yz+y+z+1}$$

Dividing by yz+y+z+1 undoes the multiplication by yz+y+z+1.

$$x=\frac{1000-z-y}{yz+y+z+1}$$

Solve for y

$$\left\{\begin{matrix}y=-\frac{xz+x+z-1000}{xz+x+1}\text{, }&z=-1\text{ or }x\neq -\frac{1}{z+1}\\y\in \mathrm{R}\text{, }&z=1001\text{ and }x=-\frac{1}{1002}\end{matrix}\right.$$

Steps for Solving Linear Equation

Subtract xz from both sides.

$$xyz+xy+y+x+z=1000-xz$$

Subtract x from both sides.

$$xyz+xy+y+z=1000-xz-x$$

Subtract z from both sides.

$$xyz+xy+y=1000-xz-x-z$$

Combine all terms containing y.

$$\left(xz+x+1\right)y=1000-xz-x-z$$

The equation is in standard form.

$$\left(xz+x+1\right)y=1000-z-x-xz$$

Divide both sides by xz+x+1.

$$\frac{\left(xz+x+1\right)y}{xz+x+1}=\frac{1000-z-x-xz}{xz+x+1}$$

Dividing by xz+x+1 undoes the multiplication by xz+x+1.

$$y=\frac{1000-z-x-xz}{xz+x+1}$$