Consistent and Inconsistent Systems
Whether a system of linear equations has a solution is decided by whether the system is consistent or not.
If a system is inconsistent, iff there exists a row such that:
$$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & | & c \end{bmatrix} $$
Otherwise, the matrix is consistent.
A good rule of thumb to keep in mind is that if a matrice can be written in echeleon form, it is consistent.