Definition: when one equation in a system can be expressed as a combination of the other equations.

Example: in the system of equations shown below, eq. (3) is not independent, since (3) = 2 × (2) + (1).

(1)  x + y + z = 10

(2) x + 2y + z = 12

(3) 3x + 5y + 3z = 34

Definition: system with m equations and n unknowns where m > n.

In general there will be no consistent solution for all m equations, though some systems like this may have a solution.

Definition: system with m equations and n unknowns where m < n.

There are not enough constraints for the n unknowns, so there are infinitely many possible solutions.

Definition: system with m equations and n unknowns where m = n and the equations are linearly independent.