Algebra.com does not display matrices well. So I am just going to show you the rectangular array of numbers without the usual brackets.
Initial matrix:
2 2 3 -6
3 5 4 3
2 3 4 -10
Replace row 1 with 1/2 times row 1:
1 1 3/2 -3
3 5 4 3
2 3 4 -10
Add -3 times row 1 to row 2:
1 1 3/2 -3
0 2 -1/2 12
2 3 4 -10
Add -2 times row 1 to row 3:
1 1 3/2 -3
0 2 -1/2 12
0 1 1 -4
Replace row 2 with 1/2 times row 2:
1 1 3/2 -3
0 1 -1/4 6
0 1 1 -4
Add -1 times row 2 to row 1:
1 0 7/4 -9
0 1 -1/4 6
0 1 1 -4
Add -1 times row 2 to row 3:
1 0 7/4 -9
0 1 -1/4 6
0 0 5/4 -10
Replace row 3 with 4/5 times row 3:
1 0 7/4 -9
0 1 -1/4 6
0 0 1 -8
Add -7/4 times row 3 to row 1:
1 0 0 5
0 1 -1/4 6
0 0 1 -8
Add 1/4 times row 3 to row 2:
1 0 0 5
0 1 0 4
0 0 1 -8
which translates back into:
x = 5
y = 4
z = -8