1)      x +  y +  z = 6
2)     2x -  y + 3z = 9
3)     -x + 2y + 2z = 9

Pick a letter to eliminate and a pair of equations that contain
it and eliminate it from them:

I choose to eliminate x from 1) and 3):

1)      x +  y +  z =  6
3)     -x + 2y + 2z =  9  Just add them term by term
------------------------
            3y + 3z = 15  Divide thru by 3
4)           y +  z =  5

Pick one of the equations that you just used along
with the equation you haven't used and eliminate the
SAME letter you just eliminated. 

I pick 1) and 2) to eliminate x from:

1)      x +  y +  z = 6
2)     2x -  y + 3z = 9 
----------------------- Multiply 10 thru by -2

      -2x - 2y - 2z = -12
2)     2x -  y + 3z =   9 
-------------------------
5)         -3y +  z =  -3

Solve the resulting 2x2 system of equations:

Now we take 4) and 5)

4)           y +  z =  5
5)         -3y +  z = -3

Eliminate z by multiplying 4) by -1

            -y -  z = -5
5)         -3y +  z = -3
------------------------
6)         -4y      = -8
                  y =  2

Substitute in 4)

4)            y + z = 5
              2 + z = 5
                  z = 3

 Substitute y = 2 and z = 3 in 1)

1)        x + y + z = 6
          x + 2 + 3 = 6
              x + 5 = 6
                  x = 1

Edwin