Wolfram Alpha:

```
Augmented Matrices
------------------
Use to solve linear equatons of form:

a1x + b1y + c1z = d1
a2x + b2y + c2z = d2
a3X + b3y + c3z = d3

=>
-            -
| a1 b1 c1 | d1 |
| a2 b2 c2 | d2 |
| a3 b3 c3 | d3 |
-            -

Can do:

- interchange rows
- multiply a row by a constant
- add a multiple of a row to another row

Ex.

3x - 2y = 14
x + 3y =  1
-         -
| 3 -2 | 14 |
| 1  3 |  1 |
-         -
row interchange =>
-         -
| 1  3 |  1 |
| 3 -2 | 14 |
-         -

-3 * row 1 + row 2 =>

-          -
| 1   3 |  1 |
| 0 -11 | 11 |
-          -

divide row 2 by -11 =>

-         -
| 1  3 |  1 |
| 0  1 | -1 |
-         -

row 1 - 3 * row 2 =>

-         -
| 1  0 |  4 |
| 0  1 | -1 |
-         -

x = 4, y = -1

Cramer's Rule
-------------

Another way of solving linear equations of the above form.

-            -
| a1 b1 c1 | d1 |
| a2 b2 c2 | d2 |
| a3 b3 c3 | d3 |
-            -

| a1 b1 c1 |
D = | a2 b2 c2 |
| a3 b3 c3 |

To solve D write as:

| a1 b1 c1 | a1 b1
| a2 b2 c2 | a2 b2
| a3 b3 c3 | a3 b3

(a1b2c3 + b1c2a3 + c1a2b3) - (a2b2c1 + b3c2a1 + c3a2b1)

To get x sub first column:

| d1 b1 c1 |
Dx = | d2 b2 c2 |
| d3 b3 c3 |

x = Dx/D

To get y:

| a1 d1 c1 |
Dy = | a2 d2 c2 |
| a3 d3 c3 |

y = Dy/D

To get z:

| a1 b1 d1 |
Dz = | a2 b2 d2 |
| a3 b3 d3 |

z = Dz/D

Note: if D = 0 the equations are inconsistent and there is no solution.
Ex.

3x - 2y = 14
x + 3y =  1

D = |3  -2|  = 9 + 2 = 11
|1   3|

Dx = |14 -2| = 42 + 2 = 44
| 1  3|

x = 44/11 = 4

Dy = | 3 14| = 3 - 14 = -11
| 1  1|

y = -11/11 = -1```