Given the following system of three equations,
write as an augmented matrix.

3x - 2y + z = 4
2x + y = 3
x + 2y - 3z = 5

Line up the like variables on the left,
the equal signs and the constant terms 
on the right like this:

3x - 2y +  z = 4
2x +  y      = 3
 x + 2y - 3z = 5

Put 1 coefficients where they are understood:

3x - 2y + 1z = 4
2x + 1y      = 3
1x + 2y - 3z = 5

There is no z term in the 2nd equation. So
fill that space in with " + 0z "

3x - 2y + 1z = 4
2x + 1y + 0z = 3
1x + 2y - 3z = 5

Now erase all the variables

3  - 2  + 1  = 4
2  + 1  + 0  = 3
1  + 2  - 3  = 5

Place the signs close to the
numbers.

3   -2   +1  = 4
2   +1   +0  = 3
1   +2   -3  = 5

Erase the + signs:

3   -2    1  = 4
2    1    0  = 3
1    2   -3  = 5

Replace the equal signs with
a dotted line:

3   -2    1  | 4
2    1    0  | 3
1    2   -3  | 5

Now place brackets around the whole 
thing:

æ3   -2    1  | 4ö
ç2    1    0  | 3÷
è1    2   -3  | 5ø

That's the augmented matrix.

Edwin