Solve xyz+4xy+3xz+2yz+12x+8y+6z=96

$$xyz+4xy+3xz+2yz+12x+8y+6z=96$$

Solve for x

$$x=-\frac{2\left(yz+4y+3z-48\right)}{\left(y+3\right)\left(z+4\right)}$$

$$z\neq -4\text{ and }y\neq -3$$

Steps for Solving Linear Equation

Subtract 2yz from both sides.

$$xyz+4xy+3xz+12x+8y+6z=96-2yz$$

Subtract 8y from both sides.

$$xyz+4xy+3xz+12x+6z=96-2yz-8y$$

Subtract 6z from both sides.

$$xyz+4xy+3xz+12x=96-2yz-8y-6z$$

Combine all terms containing x.

$$\left(yz+4y+3z+12\right)x=96-2yz-8y-6z$$

The equation is in standard form.

$$\left(yz+4y+3z+12\right)x=96-6z-8y-2yz$$

Divide both sides by yz+4y+3z+12.

$$\frac{\left(yz+4y+3z+12\right)x}{yz+4y+3z+12}=\frac{96-6z-8y-2yz}{yz+4y+3z+12}$$

Dividing by yz+4y+3z+12 undoes the multiplication by yz+4y+3z+12.

$$x=\frac{96-6z-8y-2yz}{yz+4y+3z+12}$$

Divide 96-2yz-8y-6z by yz+4y+3z+12.

$$x=\frac{2\left(48-3z-4y-yz\right)}{\left(y+3\right)\left(z+4\right)}$$

Solve for y

$$y=-\frac{3\left(xz+4x+2z-32\right)}{\left(x+2\right)\left(z+4\right)}$$

$$z\neq -4\text{ and }x\neq -2$$

Steps for Solving Linear Equation

Subtract 3xz from both sides.

$$xyz+4xy+2yz+12x+8y+6z=96-3xz$$

Subtract 12x from both sides.

$$xyz+4xy+2yz+8y+6z=96-3xz-12x$$

Subtract 6z from both sides.

$$xyz+4xy+2yz+8y=96-3xz-12x-6z$$

Combine all terms containing y.

$$\left(xz+4x+2z+8\right)y=96-3xz-12x-6z$$

The equation is in standard form.

$$\left(xz+4x+2z+8\right)y=96-6z-12x-3xz$$

Divide both sides by xz+4x+2z+8.

$$\frac{\left(xz+4x+2z+8\right)y}{xz+4x+2z+8}=\frac{96-6z-12x-3xz}{xz+4x+2z+8}$$

Dividing by xz+4x+2z+8 undoes the multiplication by xz+4x+2z+8.

$$y=\frac{96-6z-12x-3xz}{xz+4x+2z+8}$$

Divide 96-3xz-12x-6z by xz+4x+2z+8.

$$y=\frac{3\left(32-2z-4x-xz\right)}{\left(x+2\right)\left(z+4\right)}$$