Problem 1 - Algebra and Trigonometry

Question

If a system of linear equations has infinitely many solutions, then the system is called _____. If a system of linear equations has no solution, then the system is called _____.

Step-by-Step Solution

Answer

Dependent; Inconsistent.

Step 1: Understanding the Problem

We need to classify systems of linear equations based on the nature of their solutions. Specifically, systems that have infinitely many solutions and those with no solutions.

Step 2: Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, it means that there are infinitely many sets of values that satisfy all the equations simultaneously. Such a system is called a 'dependent system'.

Step 3: No Solution

If a system of linear equations has no solution, it indicates that no set of values exists that can satisfy all equations at once. This system is known as an 'inconsistent system'.

Key Concepts

Infinitely Many Solutions

When dealing with systems of linear equations, the term "infinitely many solutions" arises when the equations describe geometric lines that coincide on a graph, essentially lying on top of each other.
This means that the two equations represent the same line. As an outcome, each point on this line is a solution to the system.
This can happen in cases where one equation is a multiple or a linear combination of another.

• For example, consider the equations \( y = 2x + 3 \) and \( 2y = 4x + 6 \). After simplifying, both equations represent the same line.
• Their solution set is an infinite set of points lying on the line, compatible with each other.

When represented in matrix form, the rank of the coefficient matrix is less than the rank of the augmented matrix, indicating dependence amongst equations.

Dependent System

A dependent system is one where the equations in the system do not provide unique information.
They depend on each other, meaning one equation can be derived from another by multiplication or addition.

• This situation arises when two equations describe the same geometric feature—namely, the same line in a 2D graph.
• In practical terms, this means that not only do they have infinitely many solutions, but their solution set also forms a line rather than an intersection point of two distinct lines.

Knowing whether a system is dependent is crucial for solving multiple equations and understanding the relationship among them.
It often simplifies analysis as redundant equations can be excluded from consideration.

Inconsistent System

An inconsistent system occurs when there is no set of solutions that satisfy all the equations at the same time.
This often manifests on a graph as parallel lines that never meet, indicating no intersection point.

• For instance, the equations \( y = 2x + 3 \) and \( y = 2x + 5 \) are parallel; they have the same slope but different y-intercepts.
• This difference in intercepts prevents them from meeting, meaning there is no x-value for which both equations give the same y-value.

In mathematical terms, an inconsistent system typically has a scenario where the rows of the augmented matrix result in a contradiction, such as 0 = 1, during row reduction or other forms of matrix manipulation.