Archetype B is a system of linear equations with coefficient matrix, \begin{equation*} B= \begin{bmatrix} -7&-6&-12\\ 5&5&7\\ 1&0&4 \end{bmatrix} \end{equation*}
Do the columns of this matrix form a linearly independent or dependent set? By Example NM we know that $B$ is nonsingular. According to the definition of nonsingular matrices, Definition NM, the homogeneous system $\homosystem{A}$ has a unique solution. So by Theorem LIVHS, the columns of $B$ form a linearly independent set.