Illustration of gauss-jordan:

Here's an illustration of performing such substitutions by using MATLAB

>> a = [1 3 0; 2 1 3; 4 2 3]

a =

1 3 0

2 1 3

4 2 3

>> b = [1 6 3]'

b =

       1

       6

       3

>> ab = [a b]

ab =

1 3 0 1

2 1 3 6

4 2 3 3

>> ab(2,:) = ab(2,:) - 2*ab(1,:)

ab =

1  3 0 1

0 -5 3 4

4  2 3 3

>> ab(3,:) = ab(3,:) - 4 * ab(1,:)

ab =

1  3 0  1

0 -5 3  4

0 -10 3 -1

>> ab(3,:) = ab(3,:) - 2 * ab(2,:)

ab =

1  3  0  1

0 -5  3  4

0  0 -3 -9

>> ab(2,:) = ab(2,:) + ab(3,:)

ab =

1  3  0  1

0 -5  0 -5

0  0 -3 -9

>> ab(1,:) = ab(1,:) + 3/5*ab(2,:)

ab =

1  0  0 -2

0 -5  0 -5

0        0 -3 -9