What distinguishes homogeneous systems from non-homogeneous systems of equations?

A homogeneous system of equations is defined by the property that all of its constant terms are zero. In mathematical terms, if we express a system of linear equations in matrix form as (Ax = b), a homogeneous system would have (b = 0). This leads to the equation (Ax = 0).

One of the key characteristics of homogeneous systems is that they always have at least one solution: the trivial solution, where all variables are equal to zero. This is true regardless of the number of equations or the specific coefficients in the matrix (A). Therefore, it can be concluded that a homogeneous system has a solution that is the zero vector, which directly supports the correctness of this answer.

In contrast, non-homogeneous systems have at least one of their constant terms as non-zero, meaning (b \neq 0). Such systems may have no solution, one solution, or infinitely many solutions, depending on the relationships between the equations involved. Thus, a key distinction lies in the presence of the zero vector as a guaranteed solution for homogeneous systems.