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\left(1+y+x+xy\right)\left(1+z\right)=\left(1-x\right)\left(1-y\right)\left(1-z\right)
Use the distributive property to multiply 1+x by 1+y.
1+z+y+yz+x+xz+xy+xyz=\left(1-x\right)\left(1-y\right)\left(1-z\right)
Use the distributive property to multiply 1+y+x+xy by 1+z.
1+z+y+yz+x+xz+xy+xyz=\left(1-y-x+xy\right)\left(1-z\right)
Use the distributive property to multiply 1-x by 1-y.
1+z+y+yz+x+xz+xy+xyz=1-z-y+yz-x+xz+xy-xyz
Use the distributive property to multiply 1-y-x+xy by 1-z.
1+z+y+yz+x+xz+xy+xyz+x=1-z-y+yz+xz+xy-xyz
Add x to both sides.
1+z+y+yz+2x+xz+xy+xyz=1-z-y+yz+xz+xy-xyz
Combine x and x to get 2x.
1+z+y+yz+2x+xz+xy+xyz-xz=1-z-y+yz+xy-xyz
Subtract xz from both sides.
1+z+y+yz+2x+xy+xyz=1-z-y+yz+xy-xyz
Combine xz and -xz to get 0.
1+z+y+yz+2x+xy+xyz-xy=1-z-y+yz-xyz
Subtract xy from both sides.
1+z+y+yz+2x+xyz=1-z-y+yz-xyz
Combine xy and -xy to get 0.
1+z+y+yz+2x+xyz+xyz=1-z-y+yz
Add xyz to both sides.
1+z+y+yz+2x+2xyz=1-z-y+yz
Combine xyz and xyz to get 2xyz.
z+y+yz+2x+2xyz=1-z-y+yz-1
Subtract 1 from both sides.
z+y+yz+2x+2xyz=-z-y+yz
Subtract 1 from 1 to get 0.
y+yz+2x+2xyz=-z-y+yz-z
Subtract z from both sides.
y+yz+2x+2xyz=-2z-y+yz
Combine -z and -z to get -2z.
yz+2x+2xyz=-2z-y+yz-y
Subtract y from both sides.
yz+2x+2xyz=-2z-2y+yz
Combine -y and -y to get -2y.
2x+2xyz=-2z-2y+yz-yz
Subtract yz from both sides.
2x+2xyz=-2z-2y
Combine yz and -yz to get 0.
\left(2+2yz\right)x=-2z-2y
Combine all terms containing x.
\left(2yz+2\right)x=-2y-2z
The equation is in standard form.
\frac{\left(2yz+2\right)x}{2yz+2}=\frac{-2y-2z}{2yz+2}
Divide both sides by 2yz+2.
x=\frac{-2y-2z}{2yz+2}
Dividing by 2yz+2 undoes the multiplication by 2yz+2.
x=-\frac{y+z}{yz+1}
Divide -2z-2y by 2yz+2.
\left(1+y+x+xy\right)\left(1+z\right)=\left(1-x\right)\left(1-y\right)\left(1-z\right)
Use the distributive property to multiply 1+x by 1+y.
1+z+y+yz+x+xz+xy+xyz=\left(1-x\right)\left(1-y\right)\left(1-z\right)
Use the distributive property to multiply 1+y+x+xy by 1+z.
1+z+y+yz+x+xz+xy+xyz=\left(1-y-x+xy\right)\left(1-z\right)
Use the distributive property to multiply 1-x by 1-y.
1+z+y+yz+x+xz+xy+xyz=1-z-y+yz-x+xz+xy-xyz
Use the distributive property to multiply 1-y-x+xy by 1-z.
1+z+y+yz+x+xz+xy+xyz+y=1-z+yz-x+xz+xy-xyz
Add y to both sides.
1+z+2y+yz+x+xz+xy+xyz=1-z+yz-x+xz+xy-xyz
Combine y and y to get 2y.
1+z+2y+yz+x+xz+xy+xyz-yz=1-z-x+xz+xy-xyz
Subtract yz from both sides.
1+z+2y+x+xz+xy+xyz=1-z-x+xz+xy-xyz
Combine yz and -yz to get 0.
1+z+2y+x+xz+xy+xyz-xy=1-z-x+xz-xyz
Subtract xy from both sides.
1+z+2y+x+xz+xyz=1-z-x+xz-xyz
Combine xy and -xy to get 0.
1+z+2y+x+xz+xyz+xyz=1-z-x+xz
Add xyz to both sides.
1+z+2y+x+xz+2xyz=1-z-x+xz
Combine xyz and xyz to get 2xyz.
z+2y+x+xz+2xyz=1-z-x+xz-1
Subtract 1 from both sides.
z+2y+x+xz+2xyz=-z-x+xz
Subtract 1 from 1 to get 0.
2y+x+xz+2xyz=-z-x+xz-z
Subtract z from both sides.
2y+x+xz+2xyz=-2z-x+xz
Combine -z and -z to get -2z.
2y+xz+2xyz=-2z-x+xz-x
Subtract x from both sides.
2y+xz+2xyz=-2z-2x+xz
Combine -x and -x to get -2x.
2y+2xyz=-2z-2x+xz-xz
Subtract xz from both sides.
2y+2xyz=-2z-2x
Combine xz and -xz to get 0.
\left(2+2xz\right)y=-2z-2x
Combine all terms containing y.
\left(2xz+2\right)y=-2x-2z
The equation is in standard form.
\frac{\left(2xz+2\right)y}{2xz+2}=\frac{-2x-2z}{2xz+2}
Divide both sides by 2xz+2.
y=\frac{-2x-2z}{2xz+2}
Dividing by 2xz+2 undoes the multiplication by 2xz+2.
y=-\frac{x+z}{xz+1}
Divide -2z-2x by 2xz+2.