1) The only way for the sum of two squares to equal zero is if both squares are equal to zero. Therefore, we have:
(x - y)^2 = 0
y - 3 = 0

From the first equation, we get:
x - y = 0
x = y

Substitute x = y into the second equation:
y - 3 = 0
y = 3

Thus, the solution to the equation is x = 3, y = 3.

2) The given equation can be simplified as follows:
(3x - y + 1)^2 = - x^2 + 4xy - 4y^2
9x^2 - 6xy + y^2 + 6x - 2y + 1 = - x^2 + 4xy - 4y^2
10x^2 - 10xy + 5y^2 + 6x - 2y + 1 = 0

This equation can be further simplified, but it does not seem to have a straightforward solution. It might be easier to solve by graphing.

3) This equation involves the absolute value function, and can be simplified into two cases:
1) 2x - 4y - 10 + (3x + y - 1)^2 = 0
2) -2x + 4y + 10 + (3x + y - 1)^2 = 0

To find the solutions, you need to solve both equations separately and check if the solutions satisfy the original equation.