x+y+z=9

3x-y+z=-1

-2x+2y-3z=-2

translates into the augmented matrix:

    1    1    1    9

    3   -1    1   -1

   -2    2   -3   -2

Added -3 times row 1 to row 2.

     1     1     1     9

     0    -4    -2   -28

    -2     2    -3    -2

Added 2 times row 1 to row 3.

     1     1     1     9

     0    -4    -2   -28

     0     4    -1    16

Replaced row 2 with -1/4 times row 2.

     1     1     1     9

     0     1   1/2     7

     0     4    -1    16

Added -1 times row 2 to row 1.

     1     0   1/2     2

     0     1   1/2     7

     0     4    -1    16

Added -4 times row 2 to row 3.

     1     0   1/2     2

     0     1   1/2     7

     0     0    -3   -12

Replaced row 3 with -1/3 times row 3.

     1     0   1/2     2

     0     1   1/2     7

     0     0     1     4

Added -1/2 times row 3 to row 1.

     1     0     0     0

     0     1   1/2     7

     0     0     1     4

Added -1/2 times row 3 to row 2.

   1   0   0   0

   0   1   0   5

   0   0   1   4

which translates back into:

x = 0

y = 5

z = 4 

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