Consistent System
AlgebraA 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 is one that has at least one solution. This means there exists at least one pair of values $(x, y)$ that makes all equations true.
Example
$x + y = 5$ and $x - y = 1$. This has the solution $(3, 2)$. Since a solution exists, the system is consistent.
Key Insight
Consistent means "agreeing." If the equations are consistent with each other, they agree on at least one answer.
Definition
A system is consistent if it has one or more solutions. When graphed, the lines of a consistent system either intersect at exactly one point (independent) or are the same line (dependent, infinitely many solutions). An inconsistent system, by contrast, has no solution.
Example
$y = 3x + 1$ and $y = 3x + 5$: parallel lines, no solution, inconsistent. But $y = 2x + 3$ and $y = -x + 6$: intersect at $(1, 5)$, consistent and independent.
Key Insight
A consistent system means the conditions described by the equations can all be true at the same time. Inconsistent systems describe impossible or contradictory conditions.
Definition
A linear system $Ax = b$ is consistent if and only if $b$ lies in the column space of $A$, equivalently $\text{rank}(A) = \text{rank}([A|b])$ (Rouche-Capelli theorem). For a consistent system, if $\text{rank}(A) = n$ (number of unknowns), the solution is unique; if $\text{rank}(A) < n$, the solution space is an affine subspace of dimension $n - \text{rank}(A)$.
Example
$A = [[1,2],[2,4]]$, $b = [[3],[6]]$. $\text{rank}(A) = 1 = \text{rank}([A|b])$ since $[3,6]$ is in the column span of $A$. Consistent, with infinitely many solutions: $x + 2y = 3$.
Key Insight
Consistency is the fundamental feasibility condition in linear programming and optimization. Checking consistency via rank is a key step before attempting any solution method.