Inconsistent system

Definition

An inconsistent system is a set of equations that has no solution. This typically occurs when the equations represent parallel lines that never intersect.

5 Must Know Facts For Your Next Test

1. Inconsistent systems have no points of intersection, meaning they have no common solution.
2. Graphically, inconsistent systems are represented by parallel lines in two-variable cases.
3. For three-variable systems, inconsistency can occur if the planes do not intersect at a common point or line.
4. Algebraically, an inconsistent system results in a contradiction such as $0 = 1$ when solving the equations.
5. The determinant of the coefficient matrix in an inconsistent system is zero, but this alone does not confirm inconsistency without further evaluation.

Review Questions

• What does it mean for a system of linear equations to be inconsistent?
• How can you identify an inconsistent system when solving algebraically?
• What is the graphical representation of an inconsistent system with two variables?

Related terms

Consistent System:

A set of equations that has at least one solution.

Dependent System:

A consistent system where all equations represent the same line or plane, resulting in infinitely many solutions.

Independent System: A consistent system with exactly one unique solution.