Solve the following matrix
           [ 2  8  22 ]
           [ 4  1  14 ]
I think you need to multiply the top by negative 2 and then subtract from the bottom. but I am confused from there.  
  

2x + 8y = 22
4x +  y = 14
 
[ 2 8 22]
[ 4 1 14]

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:

[ 2 8 22]
[ 4 1 14]

Get a 0 where the 4 is: 

Multiply the top row through by -2

[-4 -16 -44]
[ 4   1  14]

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

[-4 -16 -44]
[ 0 -15 -30]

Let's stop and simplify by dividing the
top row through by -4, and the bottom row
through by -15

[ 1  4 11]
[ 0  1  2]

Get a 0 where the 4 is:

Multiply the bottom row by -4:

[ 1  4 11]
[ 0 -4 -8]


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

[ 1  0   3]
[ 0 -4  -8]


Get a 1 where the -4 is by dividing the
bottom row through by -4

[ 1   0   3]
[ 0   1   2]

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

x = 3, and y = 2

Edwin