Answer
$w=-\frac{22}{5}$
$x=\frac{17}{5}$
$y=\frac{11}{5}$
$z=\frac{14}{5}$
Work Step by Step
We have to solve the system of equations:
$\begin{cases}
w+x+y+z=4\\
w+3x-2y+2z=7\\
2w+2x+y+z=3\\
w-x+2y+3z=-5
\end{cases}$
First identify the matrices $A,X,B$ and write the system in the form $AX=B$:
$A=\begin{bmatrix}1&1&1&1\\1&3&-2&2\\2&2&1&1\\1&-1&2&3\end{bmatrix}$
$X=\begin{bmatrix}w\\x\\y\\z\end{bmatrix}$
$B=\begin{bmatrix}4\\7\\3\\5\end{bmatrix}$
$\begin{bmatrix}1&1&1&1\\1&3&-2&2\\2&2&1&1\\1&-1&2&3\end{bmatrix}\begin{bmatrix}w\\x\\y\\z\end{bmatrix}=\begin{bmatrix}4\\7\\3\\5\end{bmatrix}$
We have to determine the solution of the system, $X$:
$X=A^{-1}B$
Determine $A^{-1}$:
$A^{-1}=\begin{bmatrix}-\frac{27}{10}&-\frac{1}{10}&\frac{17}{10}&\frac{2}{5}\\\frac{17}{10}&\frac{1}{10}&-\frac{7}{10}&-\dfrac{2}{5}\\\frac{8}{5}&-\frac{1}{5}&-\frac{3}{5}&-\frac{1}{5}\\\frac{2}{5}&\frac{1}{5}&-\frac{2}{5}&\frac{1}{5}\end{bmatrix}$
Determine the solution of the system:
$X=A^{-1}B$
$X=\begin{bmatrix}-\frac{27}{10}&-\frac{1}{10}&\frac{17}{10}&\frac{2}{5}\\\frac{17}{10}&\frac{1}{10}&-\frac{7}{10}&-\dfrac{2}{5}\\\frac{8}{5}&-\frac{1}{5}&-\frac{3}{5}&-\frac{1}{5}\\\frac{2}{5}&\frac{1}{5}&-\frac{2}{5}&\frac{1}{5}\end{bmatrix}\begin{bmatrix}4\\7\\3\\5\end{bmatrix}$
$=\begin{bmatrix}-\frac{108}{10}-\frac{7}{10}+\frac{51}{10}+2\\\frac{68}{10}+\frac{7}{10}-\frac{21}{10}-2\\\frac{32}{5}-\frac{7}{5}-\frac{9}{5}-1\\\frac{8}{5}+\frac{7}{5}-\frac{6}{5}+1\end{bmatrix}$
$=\begin{bmatrix}-\frac{22}{5}\\\frac{17}{5}\\\frac{11}{5}\\\frac{14}{5}\end{bmatrix}$
$\begin{bmatrix}w\\x\\y\\z\end{bmatrix}=\begin{bmatrix}-\frac{22}{5}\\\frac{17}{5}\\\frac{11}{5}\\\frac{14}{5}\end{bmatrix}$
The solution is:
$w=-\frac{22}{5}$
$x=\frac{17}{5}$
$y=\frac{11}{5}$
$z=\frac{14}{5}$
