Problem 2 - Intermediate Algebra

Question

Fill in the blanks. When a system of equations has at least one solution, it is called a ____ system. If a system has no solutions, it is called an ____ system.

Step-by-Step Solution

Answer

consistent; inconsistent

Step 1: Understanding the Definitions

In algebra, a system of equations is called 'consistent' if there is at least one set of values for the unknowns that satisfies all the equations in the system. Conversely, a system that has no possible solutions is referred to as 'inconsistent'.

Step 2: Applying Definitions to the Exercise

We need to fill in the blanks using the definitions we reviewed. The first blank deals with a scenario where a system has at least one solution, so this is a 'consistent' system. The second scenario, where the system has no solutions, is called 'inconsistent'.

Key Concepts

Consistent System

A consistent system of equations is one that has at least one solution. This means there exists a set of values for the variables that satisfy all the equations simultaneously. In other words, the graphs of these equations will intersect at one or more points in the coordinate plane. A consistent system can be:

• Independent: If this system has exactly one solution. This generally corresponds to lines that intersect at a single point on a graph.
• Dependent: If there are infinitely many solutions. This occurs when the equations represent the same line, thus overlapping entirely in the graph.

A simple example is the system of equations:\[ \begin{aligned} &x + y = 2 \ &x - y = 0 \end{aligned} \]Solving this, we find a consistent solution: \[ x = 1, \ y = 1 \] This set of values satisfies both equations.

Inconsistent System

An inconsistent system is one that has no solution. This happens when no set of values exists that can satisfy all equations in the system simultaneously. On a graph, this situation is visualized as lines or curves that do not intersect at any point.
A common scenario for an inconsistent system is when the equations represent parallel lines. Since parallel lines never meet, they have no common points, and thus, no solutions.For example, consider:\[ \begin{aligned} &x + 2y = 3 \ &x + 2y = 5 \end{aligned} \]Here, the two equations describe parallel lines. When we attempt to solve them simultaneously, we find no solution, confirming the system is inconsistent.

Algebraic Definitions

When working with systems of equations, understanding the underlying algebraic definitions is crucial.
These definitions help categorize the systems and predict the nature of their solutions. Here are some key definitions:

• System of equations: A set of two or more equations with the same variables.
• Solution of a system: A set of values for the variables that makes all equations true at once.
• Consistent system: Exists if there is at least one solution.
• Inconsistent system: Exists if there is no solution.

These algebraic definitions form the foundation of understanding larger concepts in linear algebra and real-world applications. By grasping these, you can better understand how to solve complex systems of equations encountered in mathematics and its applications.