Teorema de Rouché–Capelli

The Teorema de Rouché–Capelli states that a system of linear equations has solutions (it's consistent) only if the rank of the coefficient matrix (the matrix of variables) is equal to the rank of the augmented matrix (which includes the constants). If these ranks are equal, solutions exist; if not, the system is inconsistent (no solutions). When solutions exist, the system may have a unique solution or infinitely many, depending on the rank relative to the number of variables. Essentially, the theorem provides a criterion to determine whether a set of linear equations can be solved simultaneously.