I am completely lost on how to solve augmented matrices. 
If you could show me step by step how to solve the 
following equation it would be great.  

3x + 4y = 11
 x + 3y = 2
 
I know you have to break it down like this 

[ 3 4 11]
[ 1 3  2]

After this however I am  lost.

You have to end up with a matrix that
looks like this:

[ 1  0  #]
[ 0  1  #]

Then the solutions for x and y will appear
where the #'s are.

We get the 0's first, then we get the 1's:

Your augmented matrix is:

[ 3 4 11]
[ 1 3  2]

Get a 0 where the 1 is: 

Multiply the bottom row through by -3

[ 3  4 11]
[-3 -9 -6]

Add the top row to the bottom row but
leave the top row as it is:

[ 3  4 11]
[ 0 -5  5]

Get a 0 where the 4 is:

Multiply the top row by 5 and the bottom
row by 4:

[15  20  55]
[ 0 -20  20]

Add the bottom row to the top row but keep
the bottom row as it is.

[15   0  75]
[ 0 -20  20]

Get a 1 where the 15 is by dividing the
top row through by 15

Get a 1 where the -20 is by dividing the
bottom row through by -20

[ 1   0   5]
[ 0   1  -1]

The two numbers in the right-most column  
are the answers:

x = 5, and y = -1

Edwin