Multiply the first equation by -1 and add it to the second one, getting
y-4z=-10
Multiply the first equation by 2 and add it to the third one, getting -2y+10z=26
As you can see, we eliminated x and turned it into two equations with 2 variables.
Multiply the first new equation by 2 and add it to the second one to get
2z=6 and z=3. Substitute it in to -2y+10z=26 to get y=2 and substitute it into -x+y+z=4 to get x=1