# How do you solve 3y-z=-1, x+5y-z=-4,  -3x+6y+2z=11 using matrices?

Feb 21, 2016

Use Cramer's Rule (with Determinants) to get
$\textcolor{w h i t e}{\text{XXX}} \left(x , y , z\right) = \left(- 3 , 0 , 1\right)$

#### Explanation:

Using the coefficient of $x , y , \mathmr{and} z$ plus the equated constants as columns we can write these equations in matrix form.

If $D$ is the determinant of the variable coefficient matrix
and ${D}_{a} , a \in \left\{x , y , z\right\}$ is the determinant of the variable coefficient matrix with the column for variable $a$ replaced by the equated constants column,

Cramer's Rule tells us that:
$\textcolor{w h i t e}{\text{XXX}} a = {D}_{a} / D$ for $a \in \left\{x , y , z\right\}$

An interesting point to observe, because of the way computers perform (what is called "floating point") arithmetic the value of ${D}_{y}$ shows up as being
$\textcolor{w h i t e}{\text{XXX}} 3E-16 = 0.0000000000000003$ instead of $0$
which, in turn, cause $y$ to display as
$\textcolor{w h i t e}{\text{XXX}} - 1.9E-17 = - 0.000000000000000019$ instead of $0$

You must be prepared to apply reasonable interpretations when working with computer generated outputs.