Solve xyz+1=x+y+z+xy+yz+zx

$$xyz+1=x+y+z+xy+yz+zx$$

Solve for x

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

$$z=1\text{ or }y\neq \frac{z+1}{z-1}$$

Steps for Solving Linear Equation

Subtract x from both sides.

$$xyz+1-x=y+z+xy+yz+zx$$

Subtract xy from both sides.

$$xyz+1-x-xy=y+z+yz+zx$$

Subtract zx from both sides.

$$xyz+1-x-xy-zx=y+z+yz$$

Subtract 1 from both sides.

$$xyz-x-xy-zx=y+z+yz-1$$

Combine all terms containing x.

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

The equation is in standard form.

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

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

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

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

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

Solve for y

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

$$z=1\text{ or }x\neq \frac{z+1}{z-1}$$

Steps for Solving Linear Equation

Subtract y from both sides.

$$xyz+1-y=x+z+xy+yz+zx$$

Subtract xy from both sides.

$$xyz+1-y-xy=x+z+yz+zx$$

Subtract yz from both sides.

$$xyz+1-y-xy-yz=x+z+zx$$

Subtract 1 from both sides.

$$xyz-y-xy-yz=x+z+zx-1$$

Combine all terms containing y.

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

The equation is in standard form.

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

Divide both sides by xz-1-x-z.

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

Dividing by xz-1-x-z undoes the multiplication by xz-1-x-z.

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