Consistent System

A system of linear equations is called consistent if it has at least one set of values for the variables that satisfies all the equations in the system. In other words, the consistent system has a solution or solutions.

Types of Consistent Systems

1. Independent Consistent System: In this type of system, there is exactly one solution that satisfies all the equations. The lines or planes represented by the equations intersect at a single point.
2. Dependent Consistent System: This type of system has infinitely many solutions. The equations represent the same line or plane, so every point on the line or plane is a solution.

Methods to Determine Consistency

There are various methods to determine the consistency of a system of linear equations:

1. Graphical Method: Plot the graphs of the equations and see if they intersect at a single point, indicating an independent consistent system, or if they overlap, indicating a dependent consistent system.
2. Algebraic Method: Use methods such as substitution, elimination, or matrix operations to solve the system and determine if it has a unique solution, no solution, or infinitely many solutions.

Study Guide for Consistent System

Here are some key points to remember when studying consistent systems:

• Understand the concept of a consistent system and how it relates to the solutions of the system of equations.
• Practice identifying the type of consistent system (independent or dependent) using graphical and algebraic methods.
• Review the methods for solving consistent systems, such as substitution, elimination, and using matrices.
• Work through examples and practice problems to reinforce your understanding of consistent systems.

Understanding consistent systems is crucial in the study of linear algebra and solving real-world problems using systems of equations. Mastery of this topic will set a strong foundation for further studies in mathematics and related fields.