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-8`x` 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 ]