Are you sure those equations are correct? All 3 terms in each line are the same, so the first line is actually equivalent to 3 * (x - y)^2 = 2.
Besides that, I think it might be easier to let
a = x - y
b = y - z
c = a + b
3a^2 = 2
3b^2 = 2
3(a + b)^2 = 2
So it doesn't have any solutions, since a, b = +/- sqrt(2 / 3), but these don't solve the 3rd equation.