Archetype D and Archetype E are linear systems of equations, with an identical $3\times 4$ coefficient matrix, which we call $A$ here. However, Archetype D is consistent, while Archetype E is not. We can explain this difference by employing the column space of the matrix $A$.
The column vector of constants, $\vect{b}$, in Archetype D is given below, and one solution listed for $\linearsystem{A}{\vect{b}}$ is $\vect{x}$, \begin{align*} \vect{b}&=\colvector{8\\-12\\4} & \vect{x}&=\colvector{7\\8\\1\\3} \end{align*}
By Theorem SLSLC, we can summarize this solution as a linear combination of the columns of $A$ that equals $\vect{b}$, \begin{equation*} 7\colvector{2\\-3\\1}+ 8\colvector{1\\4\\1}+ 1\colvector{7\\-5\\4}+ 3\colvector{-7\\-6\\-5}= \colvector{8\\-12\\4}=\vect{b}. \end{equation*}
This equation says that $\vect{b}$ is a linear combination of the columns of $A$, and then by Definition CSM, we can say that $\vect{b}\in\csp{A}$.
On the other hand, Archetype E is the linear system $\linearsystem{A}{\vect{c}}$, where the vector of constants is \begin{equation*} \vect{c}=\colvector{2\\3\\2} \end{equation*} and this system of equations is inconsistent. This means $\vect{c}\not\in\csp{A}$, for if it were, then it would equal a linear combination of the columns of $A$ and Theorem SLSLC would lead us to a solution of the system $\linearsystem{A}{\vect{c}}$.