Let's solve the first equation:

(2x-1)^2 + |x-1/2|=0

Since the square of any real number is non-negative, the first term is always non-negative. Therefore, for the equation to be satisfied, the absolute value term must be equal to zero.

|x-1/2| = 0

This means that x - 1/2 = 0, so x = 1/2.

Now let's solve the second equation:

|3x-1| = |x+3|

To solve this equation, we need to consider two cases:

Case 1: 3x - 1 = x + 3

Simplifying this equation, we get:

2x = 4

x = 2

Case 2: 3x - 1 = -(x + 3)

Simplifying this equation, we get:

3x - 1 = -x - 3

4x = -2

x = -1/2

Therefore, the solutions to the second equation are x = 2 and x = -1/2.

In summary, the solutions to the given system of equations are x = 1/2, x = 2, and x = -1/2.