Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the statement is true or false. Justify your answer. Writing Briefly explain whether or not it is possible for a consistent system of linear equations to have exactly two solutions.

Step-by-Step Solution

Answer

The statement is false. A consistent system of linear equations can't possess exactly two solutions; it can only have one unique solution, or infinitely many solutions. The possible outputs are determined by the relative equations and unknowns in the system.

Step 1: Understanding the Types of Solution Sets

In linear algebra, it is acknowledged that a consistent system of linear equations can either have one unique solution or infinitely many solutions. The unique solution condition ensues when there are the same number of equations as there are unknowns, and none of the equations are proportional to each other. The infinitely many solutions condition arises when there are fewer equations than unknowns, and no equation can be considered as a single line parallel to any of the other equations.

Step 2: Analyzing the Statement

The given statement suggests that a consistent system of linear equations may have exactly two solutions. This proposition contradicts the understood principles of linear algebra outlined in step 1.

Step 3: Verifying the Statement

Given our understanding of the types of solution sets for consistent systems, it is not possible for a consistent system of linear equations to have exactly two solutions. For a linear system to have exactly two solutions, at least one of the equations would have to be nonlinear, which is not the case here.

Key Concepts

Unique Solution

In the realm of linear algebra, when we talk about a *unique solution*, we are referring to a situation where a system of linear equations intersects at exactly one point. This signifies there's only one set of values for the variables that satisfies all the equations simultaneously.

• To achieve a unique solution, there should be as many equations as there are unknowns.
• No two equations in the system should be proportional; this means no equation can be a multiple of another.
• When plotted on a graph, these equations meet at a single point if all these conditions are fulfilled.

Understanding this concept helps in identifying that a consistent system with a unique solution is often termed as "having full rank." This indicates that each equation provides entirely new information about the variables in the system.

Infinitely Many Solutions

Infinitely many solutions occur when a system of linear equations doesn't just intersect at one point, but rather along a line or a plane. This happens when:

• There are more unknowns than there are equations, offering each variable the flexibility to take on numerous values.
• The equations are linearly dependent, meaning that some equations are effectively duplicates or combinations of the others.

In graphical terms, this means that instead of converging at a single spot, a consistent system which has infinitely many solutions may represent equations that overlap each other completely. For instance, if you visualize these equations as lines on a 2D plane, they might be the same line, suggesting every point on the line is a solution.

Linear Algebra

Linear algebra is a branch of mathematics concerned with vectors, spaces, and the mappings between these spaces. It is the foundation for understanding systems of linear equations, which is crucial for solving problems in various scientific and engineering fields.

• A consistent system refers to one that has at least one solution; it is never devoid of solutions.
• In linear algebra, systems are often represented in matrices, making it easier to visually and computationally manage the data structure.
• Many techniques in linear algebra, such as Gaussian elimination and determinants, help in determining the number and nature of solutions.

Grasping these foundational elements of linear algebra allows one to solve systems efficiently and understand the underlying geometric interpretations of solutions, whether they're unique, infinite, or otherwise."