Other (sometimes easier) ways to solve multivariate equations with matrices:
The equation used will be as follows:

``` x + 3y + 2z =  7
2x -  y      = -3
-x + 2y + 5z = 12
```

Its coefficient matrix would be:

```[ 1  3 2]
[ 2 -1 0]
[-1  2 5]
```

On TI-8x calculators, you can edit matrices in the matrix editor, then paste the variable to the screen, or you can enter them inline, like so:

```[3 2]
[4 7]
```

would be entered in as:

```[[3,2][4,7]]
```

### Cramers Rule:

Set up the coefficient matrix. Take the determinant of that matrix. Each variable will be equal to a numerator divided by that determinant. To find the numerator, replace the column for that variable with the constants in the original equations.

Example:

```    | 7  3 2|
|-3 -1 0|
|12  2 5|    22
x = --------- = --- = -22/29
| 1  3 2|   -29
| 2 -1 0|
|-1  2 5|

| 1  7 2|
| 2 -3 0|
|-1 12 5|   -43
y = --------- = --- = 43/29
| 1  3 2|   -29
| 2 -1 0|
|-1  2 5|

| 1  3  7|
| 2 -1 -3|
|-1  2 12|   -48
z = ---------- = --- = 48/29
| 1  3  2|   -29
| 2 -1  0|
|-1  2  5|
```

### Reduced Row-Echelon Form:

This is even easier: you just use the rref() function on your calculator, and input the coefficient matrix with the constants appended as another column.

```[ 1  3 2  7]
[ 2 -1 0 -3]
[-1  2 5 12]
```

is your matrix. You just use rref(matrix) so if you wanted to do this inline, you would enter: rref([[1,3,2,7][2,-1,0,-3][-1,2,5,12]]) and it would spit out something like:

```[1 0 0 -.7586206897]
[0 1 0 1.482758621 ]
[0 0 1 1.655172414 ]
```

which you could use the handy >FRAC command to get:

```[1 0 0 -22/29]
[0 1 0 43/29 ]
[0 0 1 48/29 ]```