0 0 10 | -3 1 1
Replace the second row with 10 times it minus 13 times the third row:
1 0 -21 | 7 -5 0
0 10 0 | -1 17 -13
0 0 10 | -3 1 1
Finally, you can either start using fractions or deal with somewhat large numbers. I'll replace the first row with 10 times it plus 21 times the third row:
10 0 0 | 7 -29 21
0 10 0 | -1 17 -13
0 0 10 | -3 1 1
Dividing each row by 10, we get
1 0 0 | 7/10 -29/10 21/10
0 1 0 | -1/10 17/10 -13/10
0 0 1 | -3/10 1/10 1/10
Sometimes it's better to start using decimals, but for problems they expect you to do by hand, the fractions involved (so hence the integers used) stay small. In general, you make the first column contain all 0's except for the first entry. You then make the second column contain all 0's below the diagonal, and do the same thing to the remaining columns. This gives 0's below the diagonal. Then you make everything above the diagonal 0 in the same fashion. Finally you divide each row so that its element on the diagonal is 1, and you're done.