Inconsistent System
An inconsistent system of equations has no solution because the equations contradict each other, corresponding to parallel lines on a graph.
Definition
An inconsistent system has no solution: no matter what values you try for the variables, you cannot make all the equations true at once. Graphically, this corresponds to lines that are parallel but distinct, sharing the same slope but different y-intercepts, and algebraically, attempting to solve the system produces a false statement (like $0 = 5$ or $3 = 8$) that signals no solution exists. Formally, a system $Ax = b$ is inconsistent exactly when $\text{rank}(A) < \text{rank}([A|b])$, meaning $b$ does not lie in the column space of $A$; in least-squares problems, such an inconsistent system is replaced by the normal equations $A^TAx = A^Tb$ to find the closest approximate solution, minimizing $||Ax - b||^2$.
Example
$x + y = 3$ and $x + y = 7$ say the same two numbers add to both $3$ and $7$, which is impossible. Subtracting $2x + y = 4$ from $2x + y = 9$ gives $0 = -5$, a false statement confirming no solution (both lines have slope $-2$ but different intercepts). For $A = [[1,1],[1,1]]$, $b = [[3],[7]]$, $\text{rank}(A) = 1$ but $\text{rank}([A|b]) = 2$: the system is inconsistent, and its least-squares solution minimizes $(x+y-3)^2 + (x+y-7)^2$, giving $x + y = 5$.
Key Insight
Inconsistent means "contradicting," like someone saying "it is raining" and "it is not raining" at the same moment: impossible, and any time algebra leads to a contradiction, the algebra is honestly telling you no solution exists. Inconsistent systems are common in overdetermined problems, more equations than unknowns, as in regression, where least-squares replaces exact consistency with minimizing the squared residual to find a useful approximate answer.