augmented matrix method?

Question

I need to know how to do augmented matrix method. For example show how to solve this (please show steps so I can understand) :

x.1 - 2x.2 = 1
2x.1 - x.2 = 5

Answer 1

Write down the coefficients in the matrix
1 -2 | 1
2 -1 | 5
Your first step is to get a 1 in the first row of the first column. As you can see there is a 1 there already.
Your next step is to get a zero below that 1.
Multiply the first row by -2, add that to row 2, and then replace row 2 with the new row
1 -2 | 1
0 3 | 3
Now you need to change the second entry of the second row into a 1. You can do that by dividing every element in the second row by 3
1 -2 | 1
0 1 | 1
Now you need a 0 above that 1. Multiply the second row by 2, add that to the first row and replace the first row with the new row.
1 0 | 3
0 1 | 1
Thus, x.1 = 3 and x.2 = 1
You can check your answer by plugging that into the original problem.
3 - 2(1) = 1
3 - 2 = 1
1 = 1
2(3) - 1 = 5
6 - 1 = 5
5 = 5
Thus, your answer is x.1 = 3 and x.2 = 1

Answer 2

The augmented matrix is

1 -2 1
2 -1 5

Now perform Gaussian elimination on this matrix. Subtract twice the first row (2 -4 2) from the second, and you get

1 -2 1
0 3 3

The second equation is now 3x.2 = 3, so x.2 = 1. Substitute this into the first equation, and you have x.1 - 2 = 1, so x.1 = 3.

Answer 3

|x1| |1 -2| |1|
| |x | | = | |
|x2| | 2 1| |5|

then

|x1| |1 -2| -1 |1|
| | = | | x | |
|x2| | 2 1| |5|