Problem 64 - Introductory Algebra for College Students

Question

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. In dependent systems, the two equations represent the same line.

Step-by-Step Solution

Answer

The statement makes sense because it accurately describes a property of dependent systems of equations in that they represent the same line when graphed.

Step 1: Understand the Properties of a Dependent System

In linear algebra, a system of equations is said to be dependent when there are infinite solutions. This happens when the equations given in the system are not distinct but multiples of each other. In visual terms, they represent the same line on a graph.

Step 2: Compare Definitions

With the provided understanding of a dependent system, compare that to the given statement. The statement says that in dependent systems, the two equations represent the same line.

Step 3: Final Evaluation

The statement is consistent with the definition of a dependent system. Therefore, it makes sense because, in a dependent system, the equations given are multiples of each other, and when graphed, they would represent the same line.

Key Concepts

Linear Algebra

Linear algebra is an essential branch of mathematics that deals with vectors, vector spaces, and linear equations. It provides methods for solving systems of linear equations, which are collections of two or more linear equations involving the same set of variables. For instance, the equations given by \( ax + by = c \) and \( dx + ey = f \) form a system if you're looking to find values for \( x \) and \( y \) that satisfy both equations simultaneously.

In relation to our original exercise, a critical part of linear algebra is understanding the nature of these systems: whether they're independent, dependent, or inconsistent. Knowing how to distinguish these systems allows us to predict the number of solutions we might expect and identify the relationships between the variables involved.

Systems of Equations

A system of equations consists of two or more equations with a common set of variables. These systems can be classified based on the number of solutions they possess:

• Independent: Exactly one solution, meaning the lines intersect at a single point.
• Dependent: Infinite solutions, signifying the equations represent the same line.
• Inconsistent: No solution, indicating the lines are parallel and never intersect.

When solving these systems, various methods such as substitution, elimination, and graphing are applied to find the solutions. Our focus within this context is on dependent systems, where all the equations describe the same geometric line, hence having an infinite number of common solutions.

Infinite Solutions

When dealing with a system of equations, 'infinite solutions' refers to the scenario where the set of equations describes the same line or plane. This means any point on the line (or plane) is a solution to all the equations in the system, leading to an endless number of solutions.

For example, if one equation in a system is \( y = 2x + 3 \) and another is \( 2y = 4x + 6 \), these are essentially the same line since the second is merely a multiple of the first. Hence, rather than looking for a single solution, it's understood that there are infinitely many solutions–an attribute of a dependent system. This concept is a step away from more straightforward mathematical problems that yield a single, unique solution, and it requires a deeper understanding of the underlying relationships between equations.

Graphical Representation of Equations

Graphical representation of equations provides a visual way to analyze systems of equations and understand their solutions. In a two-variable system of linear equations, each equation can be plotted as a straight line on the Cartesian coordinate system.

Here's how different systems would appear when graphed:

• An independent system displays two lines crossing at a single point, symbolizing the unique solution.
• A dependent system shows lines that coincide completely, indicating infinite solutions as the lines are essentially the same.
• An inconsistent system illustrates parallel lines that never meet, suggesting that there are no solutions.

The graphical approach not only aids in finding solutions but also in understanding the relationship between equations. For students grappling with dependent systems, recognizing that coinciding lines correspond to the infinite solutions can be enlightening and affirm the concept of multiple solutions that has been discussed theoretically.