# How do you solve -5x+6y-z= -26 and x-4y-4z= -27 and 6x+y+z = 38 using matrices?

Jun 28, 2016

$\left[\begin{matrix}1 + 0 + 0 | + 5 \\ 0 + 1 + 0 | + 1 \\ 0 + 0 + 1 | + 7\end{matrix}\right]$

x=5; y=1; z=7

#### Explanation:

AS a check

"Target values"->x=5; y=1; z=7
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{red}{\text{These can go wrong very easily. Particularly if fractions are involved.}}$

$\text{ "x" "y" "z" } =$
$\left[\begin{matrix}- 5 + 6 - 1 | - 26 \\ 1 - 4 - 4 | - 27 \\ 6 + 1 + 1 | 38\end{matrix}\right]$
$R o w 1 \div \left(- 5\right) \text{ and } R o w 3 \div 6$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 - \frac{6}{5} + \frac{1}{5} | + \frac{26}{5} \\ 1 - 4 - 4 | - 27 \\ 1 + \frac{1}{6} + \frac{1}{6} | \frac{19}{3}\end{matrix}\right]$
$R o w 2 - R o w 1$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 - \frac{6}{5} + \frac{1}{5} | + \frac{26}{5} \\ 0 - \frac{14}{5} - \frac{21}{5} | - \frac{161}{5} \\ 1 + \frac{1}{6} + \frac{1}{6} | \frac{19}{3}\end{matrix}\right]$
$R o w 2 \times \left(- \frac{5}{14}\right)$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 - \frac{6}{5} + \frac{1}{5} | + \frac{26}{5} \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 1 + \frac{1}{6} + \frac{1}{6} | \frac{19}{3}\end{matrix}\right]$
$R o w 1 + \frac{6}{5} R o w 2$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 2 | + 19 \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 1 + \frac{1}{6} + \frac{1}{6} | \frac{19}{3}\end{matrix}\right]$
$R o w 3 - R o w 1$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 2 | + 19 \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 0 + \frac{1}{6} - \frac{11}{6} | - \frac{38}{3}\end{matrix}\right]$
$R o w 3 \times 6$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 2 | + 19 \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 0 + 1 - 11 | - 76\end{matrix}\right]$
$R o w 3 - R o w 2$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 2 | + 19 \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 0 + 0 - \frac{25}{2} | - \frac{175}{2}\end{matrix}\right]$
$R o w 3 \times \left(- \frac{2}{25}\right)$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 2 | + 19 \\ 0 + 1 + \frac{3}{2} | + \frac{23}{2} \\ 0 + 0 + 1 | + 7\end{matrix}\right]$
$R o w 1 - 2 R o w 3 \text{ and } R o w 2 - \frac{3}{2} R o w 3$
$\textcolor{red}{\text{ } \downarrow}$

$\left[\begin{matrix}1 + 0 + 0 | + 5 \\ 0 + 1 + 0 | + 1 \\ 0 + 0 + 1 | + 7\end{matrix}\right]$