x(y-z)+y(z-x)+z(x-y)=0

Solution for x(y-z)+y(z-x)+z(x-y)=0 equation:

Simplifying
x(y + -1z) + y(z + -1x) + z(x + -1y) = 0
(y * x + -1z * x) + y(z + -1x) + z(x + -1y) = 0
(xy + -1xz) + y(z + -1x) + z(x + -1y) = 0

Reorder the terms:
xy + -1xz + y(-1x + z) + z(x + -1y) = 0
xy + -1xz + (-1x * y + z * y) + z(x + -1y) = 0
xy + -1xz + (-1xy + yz) + z(x + -1y) = 0
xy + -1xz + -1xy + yz + (x * z + -1y * z) = 0
xy + -1xz + -1xy + yz + (xz + -1yz) = 0

Reorder the terms:
xy + -1xy + -1xz + xz + yz + -1yz = 0

Combine like terms: xy + -1xy = 0
0 + -1xz + xz + yz + -1yz = 0
-1xz + xz + yz + -1yz = 0

Combine like terms: -1xz + xz = 0
0 + yz + -1yz = 0
yz + -1yz = 0

Combine like terms: yz + -1yz = 0
0 = 0

Solving
0 = 0

Couldn't find a variable to solve for.

This equation is an identity, all real numbers are solutions.