Problem 33 - College Algebra Essentials

Question

In Exercises \(31-42,\) solve by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. $$ \left\\{\begin{array}{l} y=3 x-5 \\ 21 x-35=7 y \end{array}\right. $$

Step-by-Step Solution

Answer

The system of equations has infinitely many solutions represented as \(\{(x, y)∣y=3x−5\}\)

Step 1: Identify System

First, let's identify the system of equations. We have two equations here: \(y = 3x - 5\) and \(21x - 35 = 7y\).

Step 2: Write in Standard Form

The next step is to put both equations in the same form for easy comparison. The first equation is already in the slope-intercept form, \(y = mx + c\). Therefore, we will convert the second equation to the same form. We can rearrange the second equation into the form \(y = mx + c\) by isolating \(y\) on one side of the equation, which results in \(y = 3x - 5\).

Step 3: Compare Equations

Next, we compare the two equations. We see that they both have the same slope (\(m = 3\)) and y-intercept (\(c = -5\)). This indicates the two lines are the same and so the system of equations also has the same line. Therefore, we have infinitely many solutions.

Step 4: Write Solution in Set Notation

The final step is to express the solution in set notation. Since both lines are identical, any point on the line is a solution. Therefore, the set of solutions is \(\{(x, y)∣y=3x−5\}\)

Key Concepts

Infinitely Many Solutions

In a system of equations, having infinitely many solutions means that the equations describe the same line on a graph. This can occur in a system of linear equations when both equations share the same slope and y-intercept. For the given pair of equations, we converted both to the form \(y = mx + c\), and noticed they were identical: \(y = 3x - 5\).

When graphed, these equations produce the same line, indicating that every point on this line is a solution. Here's how you know if a system like this has infinitely many solutions:

• After rearranging, both equations appear as the same line equation.
• They share the same slope and y-intercept.

This means all solutions that satisfy one equation will satisfy the other, due to this overlap.

Set Notation

Set notation is a method used to represent an entire set of answers or solutions clearly and concisely. In the context of systems of linear equations, when we find the solution set for such a system, it often needs to be expressed in set notation to depict all possible solutions.

For our system, which resulted in infinitely many solutions, the set notation would be written as \( \{(x, y) \,|\, y = 3x - 5\} \). Here's a breakdown of this notation:

• The curly braces \(\{ \}\) indicate that we are describing a set.
• The "|" or sometimes a ":" is read as "such that."
• The expression inside specifies the condition, meaning all \((x, y)\) pairs where \(y = 3x - 5\) are part of this set.

This notation ensures clarity by explicitly showing the condition that defines all elements of the solution set.

Linear Equations

Linear equations are fundamental building blocks in algebra. These equations describe a straight line when plotted on a graph, usually in the form \(y = mx + c\), where \(m\) is the slope, and \(c\) is the y-intercept. Linear equations can help us model, understand, and solve real-world phenomena that exhibit constant rates of change.

For the system of equations given, both were already linear. The forms of these equations, as \(y = 3x - 5\), confirm this because:

• The parameter \(m = 3\) describes how steep the line is.
• The parameter \(c = -5\) informs you where the line intercepts the y-axis.

Linear equations form systems that can be analyzed for solutions, like intersections or overlaps, which we've examined in this exercise. Understanding linear equations involves recognizing their structure and using it to identify solutions efficiently.