(This is too long for a comment so I post it as a solution).
In order to solve any equation with multiple variables, a good way to approach is to think how much "information" each equation gives.
We see the first equation, so it gives us one information. The second gives us a second information, because it cannot be derived from previous information.
But looking at the third equation (or number 6), we see that you can get this result by adding the two previous equations together. Therefore, it does not give us any new information. Now we have three unknowns and two equations that give us information about them. These two equations are said to be independent because they give us information about the variables. The third equation is a linear combination of the other two, so it's NOT independent of the two others.
Because we have two equations and three unknowns, we have an infinite amount of solutions. Apparently, in the example, we are interested in obtaining at least one of them. Due to the linearity of the equations, we we are free to set any value to any of the variables, and we will obtain the other two. In this case, the author has decided to set $z=0$ in order to get a solution.