Problem 66 - Introductory Algebra for College Students

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If a linear system has graphs with equal slopes, the system must be inconsistent.

Step-by-Step Solution

Answer

The statement is false. The correct statement would be: If a linear system has graphs with equal slopes, the system could be either inconsistent or dependent.

Step 1: Assessing the Statement

A linear system with graphs having equal slopes can either be inconsistent or dependent. Inconsistency means that there are no solutions to the system, because the lines do not intersect. Dependency means that there are infinitely many solutions, because the lines coincide.

Step 2: Determining the Truth Value

Given the above explanation, the statement that if a linear system has graphs with equal slopes, the system must be inconsistent is false. The system can also be dependent if the lines coincide exactly.

Step 3: Correcting the Statement

If a linear system has graphs with equal slopes, the system could be either inconsistent (if the lines are parallel and distinct), or dependent (if the lines coincide exactly).

Key Concepts

Inconsistent Systems

In a linear system, two lines can either intersect at a single point, be parallel, or coincide. When we talk about inconsistent systems, we mean that the lines are parallel but do not coincide. This implies the two lines have the same slope but different y-intercepts.
Inconsistent systems have a key characteristic: they do not have any solutions. This is because parallel lines never meet, meaning there's no single point that satisfies both equations. For example, solving a system of equations such as \( y = 2x + 3 \) and \( y = 2x - 1 \) will reveal that there's no value of \( x \) that reconciles both equations.
Being able to identify inconsistent systems is important when solving linear equations, because it helps us quickly realize there are no solutions. This saves time and guides us on how to approach further problem-solving steps.

Dependent Systems

A dependent system arises when two or more linear equations describe the same line. In other words, they are essentially different expressions of the same relationship. As such, these lines directly overlap each other—indicating that every point on one line is also on the other.
What makes dependent systems fascinating is that they have infinitely many solutions. Every pair of \( (x, y) \) values that satisfy one equation will automatically satisfy the other, because there’s essentially only one line being described. The equivalent equations might look different, like \( y = 3x + 2 \) and \( 6x - 2y = -4 \), but a closer inspection will show that they represent the same line.
Recognizing a dependent system is crucial when working with linear systems because it prevents unnecessary computations, allowing you to understand that all valid solutions for one equation are valid for all.

Equal Slopes

In a linear system, the slope of a line tells us how steep the line is. When two lines have equal slopes, they either never intersect, or they are the same line. This is where understanding the concept of equal slopes comes into play.
Equal slopes mean that the lines either parallel or coinciding. The distinction between these two outcomes is determined by their y-intercepts:

• If the y-intercepts are different, the lines are parallel but distinct, leading to an inconsistent system.
• If the y-intercepts are the same, the lines coincide, indicating a dependent system.

Grasping the meaning of equal slopes helps in predicting the behavior of the lines when graphing them and checking for solutions to the equations. It is a fundamental aspect when analyzing linear systems and determining the nature of their solutions.