Theorem BNS says that if we find the vector form of the solutions to the homogeneous system $\homosystem{A}$, then the fixed vectors (one per free variable) will have the desired properties. Row-reduce $A$, viewing it as the augmented matrix of a homogeneous system with an invisible columns of zeros as the last column,
\begin{equation*}
\begin{bmatrix}
\leading{1} & 0 & 4 & -5\\
0 & \leading{1} & 2 & -3\\
0 & 0 & 0 & 0
\end{bmatrix}
\end{equation*}
Moving to the vector form of the solutions (Theorem VFSLS), with free variables $x_3$ and $x_4$, solutions to the consistent system (it is homogeneous, Theorem HSC) can be expressed as
\begin{equation*}
\colvector{x_1\\x_2\\x_3\\x_4}
=
x_3\colvector{-4\\-2\\1\\0}+
x_4\colvector{5\\3\\0\\1}
\end{equation*}
Then with $S$ given by
\begin{equation*}
S=\set{\colvector{-4\\-2\\1\\0},\,\colvector{5\\3\\0\\1}}
\end{equation*}
Theorem BNS guarantees the set has the desired properties.