here's an example of gauss solution.
http://www.mathwords.com/g/gaussian_elimination.htm
here's an example of aguss-jordan solution.
http://www.mathwords.com/g/gauss-jordan_elimination.htm
gauss takes it to the point where you can solve for each of the variables by back tracking.
gauss-jordan takes it one step further and does the back tracking for you and sets each row so that the solutuion is already there with no additional effort required on your part.
your final solution with gauss is:
x y z constant
1 1 1 -3
0 1 5 -6
0 0 1 -2
you know that z = -2 from the last row of the solution.
replace z with -2 in the second row and solve for y get y = 4.
replace y with 4 and z with -2 in the first row and solve for x to get x = 1
your final solution is:
x = 1
y = 4
z = -2
your final solution with gaugs-jordan is:
x y z constant
1 0 0 1
0 1 0 4
0 0 1 -2
the solution is already there.
your final solution is:
x = 1
y = 4
z = -2
mo additional processing is required.