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

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

Simplifying
x(x + -1y) + y(y + -1x) = 0
(x * x + -1y * x) + y(y + -1x) = 0

Reorder the terms:
(-1xy + x2) + y(y + -1x) = 0
(-1xy + x2) + y(y + -1x) = 0

Reorder the terms:
-1xy + x2 + y(-1x + y) = 0
-1xy + x2 + (-1x * y + y * y) = 0
-1xy + x2 + (-1xy + y2) = 0

Reorder the terms:
-1xy + -1xy + x2 + y2 = 0

Combine like terms: -1xy + -1xy = -2xy
-2xy + x2 + y2 = 0

Solving
-2xy + x2 + y2 = 0

Solving for variable 'x'.

Factor a trinomial.
(x + -1y)(x + -1y) = 0

Subproblem 1
Set the factor '(x + -1y)' equal to zero and attempt to solve:

Simplifying
x + -1y = 0

Solving
x + -1y = 0

Move all terms containing x to the left, all other terms to the right.

Add 'y' to each side of the equation.
x + -1y + y = 0 + y

Combine like terms: -1y + y = 0
x + 0 = 0 + y
x = 0 + y
Remove the zero:
x = y

Simplifying
x = y
Subproblem 2
Set the factor '(x + -1y)' equal to zero and attempt to solve:

Simplifying
x + -1y = 0

Solving
x + -1y = 0

Move all terms containing x to the left, all other terms to the right.

Add 'y' to each side of the equation.
x + -1y + y = 0 + y

Combine like terms: -1y + y = 0
x + 0 = 0 + y
x = 0 + y
Remove the zero:
x = y

Simplifying
x = y
Solution
x = {y, y}