a(x-y)+b(y-x)=a(x-y)+b(-x+y)

Solution for a(x-y)+b(y-x)=a(x-y)+b(-x+y) equation:

Simplifying
a(x + -1y) + b(y + -1x) = a(x + -1y) + b(-1x + y)
(x * a + -1y * a) + b(y + -1x) = a(x + -1y) + b(-1x + y)
(ax + -1ay) + b(y + -1x) = a(x + -1y) + b(-1x + y)

Reorder the terms:
ax + -1ay + b(-1x + y) = a(x + -1y) + b(-1x + y)
ax + -1ay + (-1x * b + y * b) = a(x + -1y) + b(-1x + y)
ax + -1ay + (-1bx + by) = a(x + -1y) + b(-1x + y)
ax + -1ay + -1bx + by = (x * a + -1y * a) + b(-1x + y)
ax + -1ay + -1bx + by = (ax + -1ay) + b(-1x + y)
ax + -1ay + -1bx + by = ax + -1ay + (-1x * b + y * b)
ax + -1ay + -1bx + by = ax + -1ay + (-1bx + by)

Add '-1ax' to each side of the equation.
ax + -1ay + -1bx + -1ax + by = ax + -1ay + -1bx + -1ax + by

Reorder the terms:
ax + -1ax + -1ay + -1bx + by = ax + -1ay + -1bx + -1ax + by

Combine like terms: ax + -1ax = 0
0 + -1ay + -1bx + by = ax + -1ay + -1bx + -1ax + by
-1ay + -1bx + by = ax + -1ay + -1bx + -1ax + by

Reorder the terms:
-1ay + -1bx + by = ax + -1ax + -1ay + -1bx + by

Combine like terms: ax + -1ax = 0
-1ay + -1bx + by = 0 + -1ay + -1bx + by
-1ay + -1bx + by = -1ay + -1bx + by

Add 'ay' to each side of the equation.
-1ay + -1bx + ay + by = -1ay + -1bx + ay + by

Reorder the terms:
-1ay + ay + -1bx + by = -1ay + -1bx + ay + by

Combine like terms: -1ay + ay = 0
0 + -1bx + by = -1ay + -1bx + ay + by
-1bx + by = -1ay + -1bx + ay + by

Reorder the terms:
-1bx + by = -1ay + ay + -1bx + by

Combine like terms: -1ay + ay = 0
-1bx + by = 0 + -1bx + by
-1bx + by = -1bx + by

Add 'bx' to each side of the equation.
-1bx + bx + by = -1bx + bx + by

Combine like terms: -1bx + bx = 0
0 + by = -1bx + bx + by
by = -1bx + bx + by

Combine like terms: -1bx + bx = 0
by = 0 + by
by = by

Add '-1by' to each side of the equation.
by + -1by = by + -1by

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

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

Solving
0 = 0

Couldn't find a variable to solve for.

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