First, we'll recall the rectangular form of a complex number z:
z = x + i*y, where x represents the real part and y represents the imaginary part.
Since z' is the conjugate of z, we'll write z':
z' = x - i*y
Now, we'll re-write the expression provided using the rectangular forms:
x + i*y + 2x - 2i*y = 3 + i
We'll combine real parts and imaginary parts from the left side:
3x - i*y = 3 + i
Comparing both sides, we'll get:
3x = 3 => x = 1
-y = 1 => y = -1
Since we know now the real and imaginary parts, we'll determine the modulus of z:
|z| = sqrt(x^2 + y^2)
|z| = sqrt(1+1)
|z| = sqrt2
The requested modulus of the complex number z is; |z| = sqrt2.