If we're not doing this using Matrix algebra, then we usually want to try and reduce our system to one of fewer variables. This is most easily done by expressing one variable in terms of others and then substituting.
You'll notice that your third line has only two variables in it, making it an excellent candidate for substitution since there is a one-to-one linear dependency between the two variables. We can isolate a single variable (say z) giving
z=8-2x
then substitute this into the first and second equations. After doing this, we will have reduced our system to one of two variable, and two equations. From here we can either repeat the process above to get a solution in terms of a single variable, or other similar method of solving 2x2 systems.