Problem 52 - Introductory Algebra for College Students

Question

In Exercises \(45-56,\) solve each system by the method of your choice. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. Explain why you selected one method over the other two. $$\left\\{\begin{array}{l} y=2 x+4 \\ y=2 x-1 \end{array}\right.$$

Step-by-Step Solution

Answer

The given system of equations has no solution. This can be represented in set notation as \(\emptyset\).

Step 1: Comparing the Slopes

The first step in this exercise requires comparing the slopes of the given linear equations. The slope of any linear equation in form \(y = mx + b\) is the coefficient of \(x\), represented by \(m\). Here in both equations, \(m=2.\)

Step 2: Comparing the y-Intercepts

After finding out the slopes, the next step is comparing the y-intercepts, represented by \(b\). The y-intercept is the point on the y-axis where the line crosses. In the given equations, human can see that the y-intercepts aren't the same. The first equation's y-intercept is +4 and the second's is -1.

Step 3: Concluding the Solution

As the system of equations has the same slope but different y-intercepts, it means that two lines are parallel to each other and will never intersect. As such, there is no solution to the system.

Step 4: Expressing in Set Notation

Set notation is a way of expressing the solution of the system. Here, as there is no solution, it can be represented as an empty set, denoted by \(\emptyset.\)

Step 5: Selection of the Method

The comparison method was chosen in this case because we already had both equations in slope-intercept form, which allows easy comparison of slopes and y-intercepts. As per the slopes and y-intercepts, it was clear that the system has no solution.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a powerful tool in algebra. It is expressed as \(y = mx + b\), where:

• \(m\) is the slope of the line, indicating its steepness and direction.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

This form is incredibly useful because it provides a clear picture of a line’s behavior. The slope \(m\) helps us understand how the line rises or falls, while the y-intercept \(b\) tells us where the line starts on the y-axis.
By rewriting an equation in this form, you can easily identify these two critical pieces of information, making it simpler to graph the line or compare it with others.
In the given exercise, both equations are already in slope-intercept form, which allows for straightforward comparison.

Parallel Lines

Parallel lines are an important concept in geometry and algebra. They have the following characteristics:

• They never meet, no matter how far extended.
• They have the same slope.

When two linear equations in slope-intercept form have the same slope but different y-intercepts, they describe parallel lines. This means the lines will run in the same direction but at different heights on the graph.
In the exercise, both lines have a slope of 2, but different y-intercepts (+4 and -1). This confirms that the lines are parallel. Because they never intersect, the system of equations has no solution. Understanding this concept aids in identifying when no solutions exist for a system.

Linear Equations

Linear equations are fundamental in mathematics, forming straight lines when graphed. They can be written in various forms, including the slope-intercept form, which is often the most intuitive.

• Linear equations model relationships with constant rates of change.
• They are widely used to solve real-world problems involving proportional relationships.

Each linear equation tells a story about how two quantities relate to each other, increasing or decreasing at a constant rate.
In the exercise, we worked with two linear equations to examine the relationship between their slopes and y-intercepts. Understanding how to manipulate and compare these equations is crucial in analyzing systems, determining intersections, and recognizing parallelism.