Answer
$u=\frac{26}{5}$
$w=-\frac{37}{15}$
$x=\frac{19}{5}$
$y=-\frac{26}{15}$
$z=\frac{16}{5}$
Work Step by Step
We have to solve the system of equations:
$\begin{cases}
u-3x+z=-3\\
w+y+z=-1\\
x+z=7\\
u+w-x+4y=-8\\
u+w+x+y+z=8
\end{cases}$
First identify the matrices $A,X,B$ and write the system in the form $AX=B$:
$A=\begin{bmatrix}1&0&-3&0&1\\0&1&0&1&0\\0&0&1&0&1\\1&1&-1&4&0\\1&1&1&1&1\end{bmatrix}$
$X=\begin{bmatrix}u\\w\\x\\y\\z\end{bmatrix}$
$B=\begin{bmatrix}-3\\-1\\7\\-8\\8\end{bmatrix}$
$\begin{bmatrix}1&0&-3&0&1\\0&1&0&1&0\\0&0&1&0&1\\1&1&-1&4&0\\1&1&1&1&1\end{bmatrix}\begin{bmatrix}u\\w\\x\\y\\z\end{bmatrix}=\begin{bmatrix}-3\\-1\\7\\-8\\8\end{bmatrix}$
We have to determine the solution of the system, $X$:
$X=A^{-1}B$
Determine $A^{-1}$:
$A^{-1}=\begin{bmatrix}\frac{1}{5}&-\frac{4}{5}&-\frac{1}{5}&0&\frac{4}{5}\\-\frac{2}{15}&\frac{13}{15}&-\frac{6}{5}&-\dfrac{1}{3}&\dfrac{7}{15}\\-\frac{1}{5}&-\frac{1}{5}&\frac{1}{5}&0&\frac{1}{5}\\-\frac{1}{15}&-\frac{1}{15}&\frac{2}{5}&\frac{1}{3}&-\frac{4}{15}\\\frac{1}{5}&\frac{1}{5}&\frac{4}{5}&0&-\frac{1}{5}\end{bmatrix}$
Determine the solution of the system:
$X=A^{-1}B$
$X=\begin{bmatrix}\frac{1}{5}&-\frac{4}{5}&-\frac{1}{5}&0&\frac{4}{5}\\-\frac{2}{15}&\frac{13}{15}&-\frac{6}{5}&-\dfrac{1}{3}&\dfrac{7}{15}\\-\frac{1}{5}&-\frac{1}{5}&\frac{1}{5}&0&\frac{1}{5}\\-\frac{1}{15}&-\frac{1}{15}&\frac{2}{5}&\frac{1}{3}&-\frac{4}{15}\\\frac{1}{5}&\frac{1}{5}&\frac{4}{5}&0&-\frac{1}{5}\end{bmatrix}\begin{bmatrix}-3\\-1\\7\\-8\\8\end{bmatrix}$
$=\begin{bmatrix}-\frac{3}{5}+\frac{4}{5}-\frac{7}{5}+0+\frac{32}{5}\\\frac{6}{15}-\frac{13}{15}-\frac{42}{5}+\frac{8}{3}+\frac{56}{15}\\\frac{3}{5}+\frac{1}{5}+\frac{7}{5}+0+\frac{8}{5}\\\frac{3}{15}+\frac{1}{15}+\frac{14}{5}-\frac{8}{3}-\frac{32}{15}\\-\frac{3}{5}-\frac{1}{5}+\frac{28}{5}+0-\frac{8}{5}\end{bmatrix}$
$=\begin{bmatrix}\frac{26}{5}\\-\frac{37}{15}\\\frac{19}{5}\\-\frac{26}{15}\\\frac{16}{5}\end{bmatrix}$
$\begin{bmatrix}u\\w\\x\\y\\z\end{bmatrix}=\begin{bmatrix}\frac{26}{5}\\-\frac{37}{15}\\\frac{19}{5}\\-\frac{26}{15}\\\frac{16}{5}\end{bmatrix}$
The solution is:
$u=\frac{26}{5}$
$w=-\frac{37}{15}$
$x=\frac{19}{5}$
$y=-\frac{26}{15}$
$z=\frac{16}{5}$
