Answer
$\left\{ \left( \frac{-2z}{11}+\frac{81}{11},\ \frac{z}{22}+\frac{10}{11},\ \frac{4z}{11}-\frac{8}{11},\ z \right) \right\}$
Work Step by Step
Convert the provided system of equations into matrix form.
$\left[ \left. \begin{align}
& \begin{matrix}
1 & \ 2 & \ \ 3 & -1 \\
\end{matrix} \\
& \begin{matrix}
0 & \ 2 & -3 & \ 1 \\
\end{matrix} \\
& \begin{matrix}
1 & -4 & \ 1 & \ 0 \\
\end{matrix} \\
\end{align} \right|\begin{matrix}
7 \\
4 \\
3 \\
\end{matrix} \right]$
Solve the above matrix as below:
$\text{By},\ {{R}_{1}}-{{R}_{3}}\to {{R}_{3}}$
$\left[ \left. \begin{align}
& \begin{matrix}
1 & \ 2 & \ \ 3 & \ -1 \\
\end{matrix} \\
& \begin{matrix}
0 & \ 2 & -3 & \ \ \ 1 \\
\end{matrix} \\
& \begin{matrix}
0 & \ 6 & \ \ 2 & \ -1 \\
\end{matrix} \\
\end{align} \right|\begin{matrix}
7 \\
4 \\
4 \\
\end{matrix} \right]$
$\text{By},\ {{R}_{3}}-3{{R}_{2}}\to {{R}_{3}}$
$\left[ \left. \begin{align}
& \begin{matrix}
1 & \ 2 & \ \ 3 & \ \ -1 \\
\end{matrix} \\
& \begin{matrix}
0 & \ 2 & -3 & \ \ \ \ 1 \\
\end{matrix} \\
& \begin{matrix}
0 & \ 0 & \ \ 11 & \ -4 \\
\end{matrix} \\
\end{align} \right|\begin{matrix}
7 \\
4 \\
-8 \\
\end{matrix} \right]$
Convert the above matrix system into the linear equations.
$\begin{align}
w+2x+3y-z=7 & \\
2x-3y+z=4 & \\
11y-4z=-8 & \\
\end{align}$
Calculate the value of y in terms of z as below:
$\begin{align}
& 11y-4z=-8 \\
& 11y=-8+4z \\
& y=\frac{-8}{11}+\frac{4}{11}z
\end{align}$
Calculate the value of x in terms of z, by substituting the value of y, z as below:
$\begin{align}
& 2x-3\left( \frac{-8}{11}+\frac{4}{11}z \right)+z=4 \\
& 2x+\frac{24}{11}-\frac{12}{11}z+z=4 \\
& 2x=4-\frac{24}{11}+\frac{1}{11}z \\
& x=\frac{10}{11}+\frac{1}{22}z
\end{align}$
Calculate the value of w in terms of z, by substituting the value of x, y, z as below:
$\begin{align}
& w+2\left( \frac{10}{11}+\frac{1}{22}z \right)+3\left( \frac{-8}{11}+\frac{4}{11}z \right)-z=7 \\
& w+\frac{20}{11}+\frac{2z}{22}-\frac{24}{11}+\frac{12z}{11}-z=7 \\
& w=7+\frac{4}{11}-\frac{2}{11}z \\
& w=\frac{81}{11}-\frac{2}{11}z
\end{align}$
The value of w, x and y in terms of z are, $\begin{align}
& w=\frac{-2z}{11}+\frac{81}{11} \\
& x=\frac{10}{11}+\frac{z}{22} \\
& y=\frac{4z}{11}-\frac{8}{11} \\
& z=z \\
\end{align}$
Hence the numbers satisfying the set of solutions are:
$\left\{ \left( \frac{-2z}{11}+\frac{81}{11},\ \frac{z}{22}+\frac{10}{11},\ \frac{4z}{11}-\frac{8}{11},\ z \right) \right\}$