Inconsistent System

Definition

An inconsistent system is a system of linear equations that has no solution, meaning there are no values for the variables that satisfy all the equations in the system simultaneously. This concept is crucial in understanding the behavior of systems of linear equations and their graphical representations.

5 Must Know Facts For Your Next Test

1. In an inconsistent system, the equations are not compatible, and there is no single set of values for the variables that satisfies all the equations simultaneously.
2. Graphically, an inconsistent system of linear equations is represented by lines that do not intersect, or planes that do not intersect in the case of a system of three variables.
3. Inconsistent systems can arise when the equations in the system are contradictory or represent parallel lines (in the case of two variables) or parallel planes (in the case of three variables).
4. Identifying an inconsistent system is crucial in solving systems of linear equations, as it allows you to determine the feasibility of the system and the existence of a unique solution.
5. Gaussian elimination and Cramer's rule are two methods used to solve systems of linear equations, and both can be used to identify whether a system is inconsistent.

Review Questions

• Explain how an inconsistent system of linear equations is different from a consistent system.
  • An inconsistent system of linear equations has no solution, meaning there are no values for the variables that satisfy all the equations in the system simultaneously. In contrast, a consistent system of linear equations has at least one solution, where there are values for the variables that satisfy all the equations. Graphically, an inconsistent system is represented by lines or planes that do not intersect, while a consistent system has lines or planes that do intersect, representing the solution to the system.
• Describe how Gaussian elimination can be used to identify an inconsistent system of linear equations.
  • Gaussian elimination is a method used to solve systems of linear equations by transforming the system into an equivalent system with a simpler form. During the Gaussian elimination process, if a row of the augmented matrix becomes all zeros, except for the constant term, this indicates that the system is inconsistent. This is because the row of all zeros, except for the constant term, represents an equation that is always false, meaning the system has no solution.
• Explain how the determinant of the coefficient matrix can be used to determine if a system of linear equations is inconsistent when using Cramer's rule to solve the system.
  • Cramer's rule is a method for solving systems of linear equations that involves calculating the determinant of the coefficient matrix and the determinants of matrices formed by replacing the columns of the coefficient matrix with the constant terms. If the determinant of the coefficient matrix is zero, then the system is inconsistent, as the coefficient matrix is not invertible. This means that Cramer's rule cannot be used to solve the system, as it requires the coefficient matrix to be invertible in order to find the unique solution.

Related terms

System of Linear Equations: A system of linear equations is a set of two or more linear equations that share the same variables and must be solved together to find the values of the variables.

Consistent System: A consistent system of linear equations is a system that has at least one solution, meaning there are values for the variables that satisfy all the equations in the system.

Dependent System: A dependent system of linear equations is a consistent system where the equations are linearly dependent, meaning one equation can be expressed as a linear combination of the other equations.