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

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

Simplifying
(a + -1b)(a + b + -1c) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Multiply (a + -1b) * (a + b + -1c)
(a(a + b + -1c) + -1b * (a + b + -1c)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y
((a * a + b * a + -1c * a) + -1b * (a + b + -1c)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
((ab + -1ac + a2) + -1b * (a + b + -1c)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y
((ab + -1ac + a2) + -1b * (a + b + -1c)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y
(ab + -1ac + a2 + (a * -1b + b * -1b + -1c * -1b)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
(ab + -1ac + a2 + (-1ab + 1bc + -1b2)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y
(ab + -1ac + a2 + (-1ab + 1bc + -1b2)) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
(ab + -1ab + -1ac + a2 + 1bc + -1b2) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Combine like terms: ab + -1ab = 0
(0 + -1ac + a2 + 1bc + -1b2) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y
(-1ac + a2 + 1bc + -1b2) + (b + -1c)(b + c + -1a) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + (b + -1c)(-1a + b + c) + (c + -1a)(c + a + -1b) = y

Multiply (b + -1c) * (-1a + b + c)
-1ac + a2 + 1bc + -1b2 + (b(-1a + b + c) + -1c * (-1a + b + c)) + (c + -1a)(c + a + -1b) = y
-1ac + a2 + 1bc + -1b2 + ((-1a * b + b * b + c * b) + -1c * (-1a + b + c)) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + ((-1ab + bc + b2) + -1c * (-1a + b + c)) + (c + -1a)(c + a + -1b) = y
-1ac + a2 + 1bc + -1b2 + ((-1ab + bc + b2) + -1c * (-1a + b + c)) + (c + -1a)(c + a + -1b) = y
-1ac + a2 + 1bc + -1b2 + (-1ab + bc + b2 + (-1a * -1c + b * -1c + c * -1c)) + (c + -1a)(c + a + -1b) = y
-1ac + a2 + 1bc + -1b2 + (-1ab + bc + b2 + (1ac + -1bc + -1c2)) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + (-1ab + 1ac + bc + -1bc + b2 + -1c2) + (c + -1a)(c + a + -1b) = y

Combine like terms: bc + -1bc = 0
-1ac + a2 + 1bc + -1b2 + (-1ab + 1ac + 0 + b2 + -1c2) + (c + -1a)(c + a + -1b) = y
-1ac + a2 + 1bc + -1b2 + (-1ab + 1ac + b2 + -1c2) + (c + -1a)(c + a + -1b) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (-1a + c)(c + a + -1b) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (-1a + c)(a + -1b + c) = y

Multiply (-1a + c) * (a + -1b + c)
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (-1a * (a + -1b + c) + c(a + -1b + c)) = y
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + ((a * -1a + -1b * -1a + c * -1a) + c(a + -1b + c)) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + ((1ab + -1ac + -1a2) + c(a + -1b + c)) = y
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + ((1ab + -1ac + -1a2) + c(a + -1b + c)) = y
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (1ab + -1ac + -1a2 + (a * c + -1b * c + c * c)) = y
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (1ab + -1ac + -1a2 + (ac + -1bc + c2)) = y

Reorder the terms:
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (1ab + -1ac + ac + -1a2 + -1bc + c2) = y

Combine like terms: -1ac + ac = 0
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (1ab + 0 + -1a2 + -1bc + c2) = y
-1ac + a2 + 1bc + -1b2 + -1ab + 1ac + b2 + -1c2 + (1ab + -1a2 + -1bc + c2) = y

Reorder the terms:
-1ab + 1ab + -1ac + 1ac + a2 + -1a2 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y

Combine like terms: -1ab + 1ab = 0
0 + -1ac + 1ac + a2 + -1a2 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y
-1ac + 1ac + a2 + -1a2 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y

Combine like terms: -1ac + 1ac = 0
0 + a2 + -1a2 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y
a2 + -1a2 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y

Combine like terms: a2 + -1a2 = 0
0 + 1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y
1bc + -1bc + -1b2 + b2 + -1c2 + c2 = y

Combine like terms: 1bc + -1bc = 0
0 + -1b2 + b2 + -1c2 + c2 = y
-1b2 + b2 + -1c2 + c2 = y

Combine like terms: -1b2 + b2 = 0
0 + -1c2 + c2 = y
-1c2 + c2 = y

Combine like terms: -1c2 + c2 = 0
0 = y

Solving
0 = y

Solving for variable 'y'.

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

Add '-1y' to each side of the equation.
0 + -1y = y + -1y
Remove the zero:
-1y = y + -1y

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

Divide each side by '-1'.
y = 0

Simplifying
y = 0