Graphical Representation of the Solution Set

Oct 13, 2025 by magicalceylontours.com 52 views

Introduction

In mathematics, systems of inequalities are a set of linear inequalities that involve multiple variables. These inequalities are used to describe the relationships between variables and can be used to model real-world problems. In this article, we will focus on solving systems of inequalities, specifically the given system of inequalities: $4x - 5y \leq 1$ and $\frac{1}{2} y - x \leq 3$ . We will show how to rewrite these inequalities in slope-intercept form and discuss the graphical representation of the solution set.

Rewriting Inequalities in Slope-Intercept Form

The slope-intercept form of a linear inequality is given by $y \leq mx + b$ , where $m$ is the slope and $b$ is the y-intercept. To rewrite the given inequalities in slope-intercept form, we need to isolate $y$ on one side of the inequality.

Inequality 1: $4x - 5y \leq 1$

To isolate $y$ , we need to subtract $4x$ from both sides of the inequality and then divide both sides by $-5$ . This gives us:

\begin{array}{l} -5y \leq -4x + 1 \\ y \geq \frac{4}{5}x - \frac{1}{5} \end{array}

Inequality 2: $\frac{1}{2} y - x \leq 3$

To isolate $y$ , we need to add $x$ to both sides of the inequality and then multiply both sides by $2$ . This gives us:

\begin{array}{l} \frac{1}{2} y \leq x + 3 \\ y \leq 2x + 6 \end{array}

Graphical Representation of the Solution Set

The solution set of a system of inequalities is the set of all points that satisfy both inequalities. To graph the solution set, we need to graph the two inequalities separately and then find the intersection of the two graphs.

Graphing Inequality 1: $y \geq \frac{4}{5}x - \frac{1}{5}$

The graph of this inequality is a line with a slope of $\frac{4}{5}$ and a y-intercept of $-\frac{1}{5}$ . The inequality is satisfied by all points on or above the line.

Graphing Inequality 2: $y \leq 2x + 6$

The graph of this inequality is a line with a slope of $2$ and a y-intercept of $6$ . The inequality is satisfied by all points on or below the line.

Finding the Intersection of the Two Graphs

To find the intersection of the two graphs, we need to find the point where the two lines intersect. We can do this by setting the two equations equal to each other and solving for $x$ .

\begin{array}{l} \frac{4}{5}x - \frac{1}{5} = 2x + 6 \\ \frac{4}{5}x - 2x = 6 + \frac{1}{5} \\ -\frac{6}{5}x = \frac{31}{5} \\ x = -\frac{31}{6} \end{array}

Now that we have found the value of $x$ , we can substitute it into one of the original equations to find the value of $y$ .

\begin{array}{l} y = \frac{4}{5}x - \frac{1}{5} \\ y = \frac{4}{5}(-\frac{31}{6}) - \frac{1}{5} \\ y = -\frac{124}{30} - \frac{6}{30} \\ y = -\frac{130}{30} \\ y = -\frac{13}{3} \end{array}

Therefore, the point of intersection is $(-\frac{31}{6}, -\frac{13}{3})$ .

Conclusion

In this article, we have shown how to rewrite the given system of inequalities in slope-intercept form and discussed the graphical representation of the solution set. We have also found the intersection of the two graphs, which is the point where the two lines intersect. This point represents the solution to the system of inequalities.

Final Answer

$The final answer is: $\boxed{(-\frac{31}{6}, -\frac{13}{3})}$<br/> **Solving Systems of Inequalities: A Q&A Guide**$

Introduction

In our previous article, we discussed how to solve systems of inequalities, specifically the given system of inequalities: $4x - 5y \leq 1$ and $\frac{1}{2} y - x \leq 3$ . We showed how to rewrite these inequalities in slope-intercept form and discussed the graphical representation of the solution set. In this article, we will answer some frequently asked questions about solving systems of inequalities.

Q&A

Q: What is a system of inequalities?

A: A system of inequalities is a set of linear inequalities that involve multiple variables. These inequalities are used to describe the relationships between variables and can be used to model real-world problems.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to rewrite each inequality in slope-intercept form and then graph the solution set. You can use the graphical method or the algebraic method to find the solution set.

Q: What is the graphical method?

A: The graphical method involves graphing the solution set of each inequality separately and then finding the intersection of the two graphs. This method is useful for visualizing the solution set and can be used to find the solution to a system of inequalities.

Q: What is the algebraic method?

A: The algebraic method involves solving the system of inequalities using algebraic techniques, such as substitution and elimination. This method is useful for finding the solution to a system of inequalities when the graphical method is not feasible.

Q: How do I find the intersection of two graphs?

A: To find the intersection of two graphs, you need to set the two equations equal to each other and solve for the variable. This will give you the point of intersection, which represents the solution to the system of inequalities.

Q: What is the point of intersection?

A: The point of intersection is the point where the two graphs intersect. This point represents the solution to the system of inequalities and can be used to find the values of the variables.

Q: How do I determine the solution set?

A: To determine the solution set, you need to graph the solution set of each inequality separately and then find the intersection of the two graphs. The solution set is the set of all points that satisfy both inequalities.

Q: What is the solution set?

A: The solution set is the set of all points that satisfy both inequalities. This set represents the solution to the system of inequalities and can be used to find the values of the variables.

Q: Can I use technology to solve systems of inequalities?

A: Yes, you can use technology, such as graphing calculators or computer software, to solve systems of inequalities. These tools can be used to graph the solution set and find the intersection of the two graphs.

Q: Are there any limitations to solving systems of inequalities?

A: Yes, there are limitations to solving systems of inequalities. For example, if the system of inequalities has no solution, then the solution set is empty. Additionally, if the system of inequalities has an infinite number of solutions, then the solution set is a line or a plane.

Conclusion

In this article, we have answered some frequently asked questions about solving systems of inequalities. We have discussed the graphical method and the algebraic method, as well as the importance of finding the intersection of two graphs. We have also discussed the limitations of solving systems of inequalities and the use of technology in solving these systems.

Final Answer

The final answer is: There is no final answer, as this article is a Q&A guide and does not have a specific numerical answer.