I have done some eliminations also and they all led to similar equations, hence, infinitely many solutions.
The book's (0,4,-2) is one of those many solutions. It is a "trick". It assigned a value to one of the 3 variables to get the other two variables. Maybe it let x = 0, to get the y and z
.... or let y = 4, to get the x and z
.... or let z = -2, to get the x and y