Consistent System
A consistent system of equations has at least one solution, meaning the equations are satisfied by some set of variable values.
Definition
A consistent system of equations has at least one solution, meaning some set of values for the variables makes all the equations true simultaneously. Graphed, the lines of a consistent system either cross at exactly one point (independent) or coincide entirely (dependent, with infinitely many solutions); an inconsistent system, by contrast, has no solution at all. Formally, a linear system $Ax = b$ is consistent if and only if $b$ lies in the column space of $A$, equivalently $\text{rank}(A) = \text{rank}([A|b])$ (the Rouche-Capelli theorem); if consistent and $\text{rank}(A) = n$ (the number of unknowns), the solution is unique, while if $\text{rank}(A) < n$ the solution space is an affine subspace of dimension $n - \text{rank}(A)$.
Example
$x + y = 5$ and $x - y = 1$ has the solution $(3, 2)$, so the system is consistent. $y = 3x + 1$ and $y = 3x + 5$ are parallel with no solution (inconsistent), while $y = 2x + 3$ and $y = -x + 6$ intersect at $(1, 5)$ (consistent and independent). For $A = [[1,2],[2,4]]$, $b = [[3],[6]]$, $\text{rank}(A) = 1 = \text{rank}([A|b])$ since $[3,6]$ lies in the column span of $A$, giving infinitely many solutions along $x + 2y = 3$.
Key Insight
Consistent means "agreeing": if a system's equations agree on at least one answer, it is consistent, while inconsistent systems describe impossible or contradictory conditions. Checking consistency via rank is a key feasibility step before attempting to solve any system, and it underlies methods used throughout linear programming and optimization.