Apparently you have be registered with "virtualhighschool" to be able to view that site- and I'm not. But if it is just a symmetric equation, you can, but don't have to, derive parametric equations but if your equations are of the form f(x,y,z)= g(x,y,z)= h(x,y,z), you can separate that into two equations, f(x,y,z)= g(x,y,z) and g(x,y,z)= h(x,y,z). Choose whatever value you like for one of the variables, say x, put that into the two equations, and solve the two equations for y and z.

For example if you equations is 2x+ y- z= x+ y+ 2z= x- y+ z, you can separate them as 2x+ y- z= x+ 2y+ 2z and x+ y+ 2z= x- y+ z. Now:

If x= 0, these give y-z= 2y+ 2z and 2y+ 2z= y+ z. Those are equivalent to y+ 3z= 0 and y+ z= 0. The only solution to that is (0, 0, 0).

If x= 1, 2+ y- z= 1+ 2y+ 2z and 1+y+ 2z= 1- y+ z. Those are equivalent to y+ 3z= 1 and 2y+z= 0. z= -2y so y+ 3(-2y)= -5y= 1. y= -1/5 and z= 2/5. The point is (1, -1/5, 2/5).