Pick a pair of equations and a letter to eliminate.
[Note: There are 6 different ways to start, all correct!]

I will pick the first two equations and y to eliminate,
because it happens to be the easiest because all you have to
do is add them and y will be eliminated:


 

Now pick a different pair of equations and eliminate the
SAME letter eliminated first. 
[Note: we have to pick one of the same equations again, but
we must pick the third equation this time.]

So I will pick the first and third equations and y to eliminate:



To eliminate y we multiply the first equation through by 3


 

Now we have 2 equations in 2 letters to solve:



We can eliminate x by multiplying the first equation through by -2

 
        

Substitute that in 







Substitute z=-2 and x=5 in any one of the original equations:







Answer: (x,y,z) = (5,1,-2)

Edwin