Consistency of Linear Systems: Explained with Examples

A system of linear equations is called consistent if it has at least one solution, and inconsistent if it has none. For example, x + y = 5 and x − y = 1 are consistent because (3, 2) satisfies both, while x + y = 5 and x + y = 7 are inconsistent because no pair can satisfy both at once.

Consistency matters because it tells you whether a problem even has an answer before you spend effort solving it. In engineering and economics, detecting an inconsistent model early saves you from chasing a solution that does not exist, while spotting infinitely many solutions tells you extra conditions are needed.

Key Concepts

• Consistent system: has at least one solution, either exactly one or infinitely many.
• Inconsistent system: has no solution at all, usually because the equations contradict each other.
• Unique solution: exactly one solution; geometrically, lines or planes meeting at a single point.
• Infinitely many solutions: the equations are dependent, describing the same line or plane.
• Homogeneous system: AX = 0; always consistent because x = 0, y = 0, … is always a solution (the trivial solution).

How It Works

• Test for n equations in n unknowns: form the coefficient matrix A. If det(A) ≠ 0, the system is consistent with a unique solution X = A⁻¹B.
• If det(A) = 0, compare the equations. If every equation is a multiple of the others (coefficients and constants in the same proportion), the system is consistent with infinitely many solutions. Otherwise it is inconsistent.
• Geometric picture for two equations: intersecting lines give one solution, parallel distinct lines give none, and coincident lines give infinitely many.

Worked Example

Check the consistency of: x + 2y = 6 and 2x + 4y = 12.

1. det(A) = 1 × 4 − 2 × 2 = 4 − 4 = 0, so there is no unique solution; it is either none or infinitely many.
2. Compare proportions: 1/2 = 2/4 = 6/12. The second equation is exactly twice the first, so they describe the same line.
3. The system is consistent with infinitely many solutions. Writing y = t gives x = 6 − 2t for any number t.
4. Check with t = 1: x = 4, y = 1. Then 4 + 2(1) = 6 and 2(4) + 4(1) = 12. Both hold.
5. Contrast: if the second equation were 2x + 4y = 14, the proportions 1/2 = 2/4 ≠ 6/14 would break, and the system would be inconsistent with no solution.

Common Misconceptions

• det(A) = 0 does not mean "no solution". It means "no unique solution" — the answer could be infinitely many solutions.
• Homogeneous systems are never inconsistent. The all-zero solution always works, so AX = 0 always has at least the trivial solution.
• Two equations, two unknowns does not guarantee one answer. Dependent or contradictory equations break that expectation.

Real-World Applications

• Optimization: checking whether a set of constraints can all be satisfied before searching for the best one.
• Circuit analysis: detecting contradictory measurements or redundant equations in a network.
• Structural design: finding whether support reactions are uniquely determined or the structure is under-specified.
• Data fitting: recognizing when data points are redundant versus genuinely conflicting.

A linear system is consistent when its equations agree with each other, giving one solution if det(A) ≠ 0 or infinitely many when the equations are dependent. When the proportions break, the system is inconsistent and has no solution at all.