What makes a system of three equations with three variables inconsistent?

Name: Problem 64, Chapter 12: Elementary and Intermediate Algebra
Uploaded: 2020-04-20T00:35:58-08:00
Duration: 2 min 38 s
Channel: Dena Bischoff
Description: What makes a system of three equations with three variables inconsistent?

step 1 All right. This is question number 64. What makes a system with three equations and three variab

What makes a system of three equations with three variables...

Key Concepts

Rank and the Augmented Matrix

The concept of rank, particularly of the coefficient matrix versus the augmented matrix, plays a crucial role in characterizing consistency. According to the Rouché–Capelli theorem, a system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. This disparity signifies that, while the equations may align in some respects, the additional constants in the system force a conflict, leading to a lack of a common intersection point among the equations.

System of Linear Equations

A system of linear equations involves multiple linear equations that are solved simultaneously, typically with several variables. In the context of three equations with three variables, each equation represents a plane in three-dimensional space, and the solution set is the intersection of these planes. Understanding how these equations interact is fundamental to determining whether they have a unique solution, infinitely many solutions, or no solution at all.

Consistency and Inconsistency

A system is consistent if there exists at least one solution that satisfies all the equations concurrently; it is said to be inconsistent if no such solution exists. Inconsistency occurs when the equations, taken together, lead to a contradiction—such as deriving an equation that simplifies to an impossible statement (e.g., 0 = 5). Such contradictions indicate that the equations describe conditions that cannot be true simultaneously.

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