# Classification of a system by number of solutions

Solving the system , we found a single solution: the ordered pair (3,5). So we say that the system is possible (has solution) and determined (single solution).

In the case of the system , we find that the ordered pairs (0.8), (1.7), (2.6), (3.5), (4.4), (5.3),… are some of their infinite solutions. Therefore, we say that the system is possible (has solution) and undetermined (infinite solutions).

For , we find that no ordered pair simultaneously satisfies the equations. Therefore, the system is impossible (there is no solution).

In short, a linear system can be:

a) possible and determined (single solution);
b) possible and undetermined (infinite solutions);
c) impossible (no solution).

## Normal system

A system is normal when it has the same number of equations (m) and unknowns (no) and the determinant of the incomplete matrix associated with the system is nonzero. If m = n and det A 0, so the system is normal.

## Cramer's Rule

Every normal system has a single solution given by: where i {1,2,3,…, n}, D = det A is the determinant of the incomplete matrix associated with the system, and Dxi is the determinant obtained by substituting, in the incomplete matrix, the column i by the column formed by the independent terms.

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