2x +  y + 2z = 10
 x + 2y +  z =  8
3x +  y -  z =  2




Swap Rows 1 and 2 because a 1 is easier
to work with in the upper left corner
than a 2:


 
Get a 0 where the 2 is in Row 2
Multiply Row 1 by -2



Add Row 1 to row 2



Restore Row 1



Get a 0 where the 3 is in Row 3
Multiply Row 1 by -3



Add Row 1 to Row 3



Restore Row 1



Get a 1 where the -3 is in Row 2
Multiply Row 2 through by -1/3, or
you can divide Row 2 by -3 (same thing)



Get a 0 where the -5 is
Multiply Row 2 by 5



Add Row 2 to row 3



Restore Row 2



Get a 1 where the -4 is
Multiply Row 3 by -1/4, or
Divide Row 3 by -4 (same thing)



Interpret as a new system of equations
in echelon (triangular) form:

1x + 2y + 1z = 8
0x + 1y + 0z = 2
0x + 0y + 1z = 3

or

x + 2y +  z = 8
     y      = 2
          z = 3

Then we see that z = 3 and y = 2

Substitute in the top equation:

x + 2(2) +  3 = 8
    x + 4 + 3 = 8
        x + 7 = 8
            x = 1

Solution (x,y,z) = (1,2,3)

Edwin