For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, on the other hand, the ranks of these two matrices are equal, the system must have at least one solution.
22.
For if there are at least as many independent equations as unknowns that will eliminate any stretches of overlap of the equations'surfaces in the geometric space of the unknowns ( aside from possibly a single point ); and if the rank of the augmented matrix exceeds ( necessarily by one if at all ) the rank of the coefficient matrix then the equations jointly contradict each other.
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