Find all ordered triples (x,y,z)

[SilverSprite]

Find all ordered triples (x,y,z) to:

x+yz=6
y+xz=6
z+xy=6

Show proof!!

[kedaman]

z=6-xy
x=6-zy
y+x(6-xy)=6
y+(6-zy)(6-(6-zy)y)=6
y+(6-zy)(6-6y-zy^2)=6
y+36-36y-6zy^2-6zy+6zy^2+z^2y^3=6
-35y-30-6zy+z^2y^3=0
z=-6y/2y^3 +- sqrt( (-6y/2y^3)^2 - 35y-30)
z=-3/y^2 +- sqrt( 9/y^4 - 35y-30)
z= -3/y^2 + sqrt( 9/y^4 - 35y-30), x=6- 3/y - sqrt( 9/y^2 - 35y^3-30y^2) or
z= -3/y^2 - sqrt( 9/y^4 - 35y-30), x=6- 3/y + sqrt( 9/y^2 - 35y^3-30y^2)
for all pairs of X, Y and Z respectively, where the remaining variable is in R.
{ (x,y,z)|y in R, z= -3/y^2 + sqrt( 9/y^4 - 35y-30), x=6- 3/y - sqrt( 9/y^2 - 35y^3-30y^2) or
z= -3/y^2 - sqrt( 9/y^4 - 35y-30), x=6- 3/y + sqrt( 9/y^2 - 35y^3-30y^2) | (x,y,z) in permuations of(X,Y,Z) }