(1)          3x - 2y -  z = 14
(2)           x - 3y - 3z =  8
(3)          2x - 4y - 2z =  8

Multiply equation (2) by -3

          -3(x - 3y - 3z) = -3(8)
            -3x + 9y + 9z = -24

Add it to the equation (1)

(1)          3x - 2y -  z =  14
            -3x + 9y + 9z = -24
            -------------------
(4)               7y + 8z = -10 

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Multiply equation (2) by -2

          -2(x - 3y - 3z) = -2(8)
            -2x + 6y + 6z = -16

Add it to the equation (3)

(3)          2x - 4y - 2z =  8
            -2x + 6y + 6z = -16
            -------------------
                  2y + 4z =  -8

Divide that through by 2

(5)               y + 2z =  -4

Now we have a system of 2 equations with only 2 unknowns:

(4)               7y + 8z = -10 
(5)                y + 2z =  -4

Multiply eq. (5) by -4

               -4(y + 2z) =  -4(-4)

                 -4y - 8z =  16

add that to eq (4):

(4)               7y + 8z = -10 
                 -4y - 8z =  16
                 --------------
                  3y      =   6
                        y = 2

 Substitute in eq (4)

                7(2) + 8z = -10
                  14 + 8z = -10
                       8z = -10-14
                       8z = -24
                        z = -3   

Substute y = 2 and z = -3 in eq. (1)

(1)          3x - 2y -  z = 14
         3x - 2(2) - (-3) = 14
               3x - 4 + 3 = 14
                   3x - 1 = 14
                       3x = 15
                        x = 5

So the solution (x,y,z) = (5,2,-3)

Edwin