x + y − z = −5

x + y − z = −5

x + 2y + 3z = −12

2x − y − 13z = 13

$\text{Let Matrix } A = \begin{pmatrix} 1 & 1 & -1 \\ 1 & 2 & 3 \\ 2 & -1 & -13 \end{pmatrix}$

$\text{Let } \vec{r} = \begin{pmatrix} -5 \\ -12 \\ 13 \end{pmatrix}$

$\text{Let } \vec{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$

$\text{Let Matrix } A^{-1} = \begin{pmatrix} -23 & 14 & 5 \\ 19 & -11 & -4 \\ -5 & 3 & 1 \end{pmatrix}$

Then

$\begin{array}{|rcll|} \hline A\vec{x} &=& \vec{r} \\ \vec{x} &=& A^{-1}\vec{r} \\ \vec{x} &=& \begin{pmatrix} -23 & 14 & 5 \\ 19 & -11 & -4 \\ -5 & 3 & 1 \end{pmatrix}\cdot \begin{pmatrix} -5 \\ -12 \\ 13 \end{pmatrix} \\ \vec{x} &=& \begin{pmatrix} 12 \\ -15 \\ 2 \end{pmatrix} \\ \hline \end{array}$

x =  12

y = -15

z =  2