Inconsistent System
AlgebraAn 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 equations true at the same time.
Example
$x + y = 3$ and $x + y = 7$. These say the same two numbers add to both $3$ and $7$, which is impossible. No solution.
Key Insight
Inconsistent means "contradicting." If someone says "it is raining" and "it is not raining" at the same moment, that is inconsistent - and impossible.
Definition
A system of linear equations is inconsistent when the lines represented by the equations are parallel but distinct - they have the same slope but different y-intercepts. When you attempt to solve algebraically, you get a false statement (like $0 = 5$), which signals no solution.
Example
$2x + y = 4$ and $2x + y = 9$. Subtract: $0 = -5$. This false statement means no solution. Graphically, both lines have slope $-2$ but different intercepts.
Key Insight
Any time algebra leads to a contradiction like $3 = 8$ or $0 = 1$, the system is inconsistent. The algebra is telling you honestly that no solution exists.
Definition
A system $Ax = b$ is inconsistent if and only if $\text{rank}(A) < \text{rank}([A|b])$, meaning $b$ does not lie in the column space of $A$. In least-squares problems, the inconsistent system $Ax = b$ is replaced by the normal equations $A^T A x = A^T b$ to find the closest approximate solution (minimizing $||Ax - b||^2$).
Example
$A = [[1,1],[1,1]]$, $b = [[3],[7]]$. $\text{rank}(A) = 1$, $\text{rank}([A|b]) = 2$. Inconsistent. Least-squares solution minimizes $(x+y-3)^2 + (x+y-7)^2$, giving $x + y = 5$.
Key Insight
Inconsistent systems are common in overdetermined problems (more equations than unknowns), as in regression. Least-squares replaces exact consistency with minimization of the squared residual, providing a useful approximate solution.