Answer
Geometrically, the solution set of the homogenous equation is translated by vector $\begin{bmatrix}
-5\\
3\\
0
\end{bmatrix}$
Work Step by Step
$\begin{bmatrix}
1 & 3 & -5 & 4\\
1 & 4 & -8 & 7\\
-3 & -7 & 9 & -6
\end{bmatrix}
$ Subtract row 1 from row 2
$\begin{bmatrix}
1 & 3 & -5 & 4\\
0 & 1 & -3 & 3\\
-3 & -7 & 9 & -6
\end{bmatrix}
$ Add 3 times row 1 to row 3
$\begin{bmatrix}
1 & 3 & -5 & 4\\
0 & 1 & -3 & 3\\
0 & 2 & -6 & 6
\end{bmatrix}
$ Subtract 2 times row 2 from row 3
$\begin{bmatrix}
1 & 3 & -5 & 4\\
0 & 1 & -3 & 3\\
0 & 0 & 0 & 0
\end{bmatrix}
$ Subtract 3 times row 2 from row 1
$\begin{bmatrix}
1 & 0 & 4 & -5\\
0 & 1 & -3 & 3\\
0 & 0 & 0 & 0
\end{bmatrix}
$ Subtract 3 times row 2 from row 1
$x_1=-5-4x_3$
$x_2=3+3x_3$
$x_3$ is free
$x=\begin{bmatrix}
x_1\\
x_2\\
x_3
\end{bmatrix}=\begin{bmatrix}
-5-4x_3\\
3+3x_3\\
x_3
\end{bmatrix}=\begin{bmatrix}
-5\\
3\\
0
\end{bmatrix}+x_3\begin{bmatrix}
-4\\
3\\
1
\end{bmatrix}
$
This means geometrically, the solution set of the homogenous equation is translated by vector $\begin{bmatrix}
-5\\
3\\
0
\end{bmatrix}$
