Cody Has $ 7. H E W A N T S T O B U Y A T L E A S T 4 S N A C K S . H O T D O G S ( 7. He Wants To Buy At Least 4 Snacks. Hot Dogs ( 7. He W An T S T O B U Y A Tl E A S T 4 S Na C K S . Ho T D O G S ( {x}$) Are $ 2 E A C H . P E A N U T S ( 2 Each. Peanuts ( 2 E A C H . P E An U T S ( { Y\$} ) Are $1 Each. Which Ordered Pair Is A Solution?${ \begin{array}{r} x + Y \geq 4 \ 2x + Y \leq 7 \end{array} }$A.

Introduction

Cody has a budget of $7 and wants to buy at least 4 snacks. He is considering two options: hot dogs and peanuts. Hot dogs cost $2 each, while peanuts cost $1 each. In this article, we will explore the possible combinations of hot dogs and peanuts that Cody can buy within his budget, using a system of linear inequalities.

The Problem

Cody's problem can be represented by the following system of linear inequalities:

$$\begin{array}{r} x + y \geq 4 \\ 2x + y \leq 7 \end{array}$$

where $x$ is the number of hot dogs and $y$ is the number of peanuts.

Understanding the Inequalities

The first inequality, $x + y \geq 4$, represents the condition that Cody wants to buy at least 4 snacks. This means that the total number of hot dogs and peanuts must be greater than or equal to 4.

The second inequality, $2x + y \leq 7$, represents the condition that Cody's total expenditure must not exceed $7. Since hot dogs cost $2 each and peanuts cost $1 each, the total expenditure can be represented by the equation $2x + y \leq 7$.

Solving the System of Inequalities

To find the possible combinations of hot dogs and peanuts that Cody can buy, we need to solve the system of linear inequalities. We can start by graphing the two inequalities on a coordinate plane.

Graphing the Inequalities

The first inequality, $x + y \geq 4$, can be graphed by drawing a line with a slope of -1 and a y-intercept of 4. The region above this line represents the solutions to the inequality.

The second inequality, $2x + y \leq 7$, can be graphed by drawing a line with a slope of -2 and a y-intercept of 7. The region below this line represents the solutions to the inequality.

Finding the Intersection

To find the possible combinations of hot dogs and peanuts that Cody can buy, we need to find the intersection of the two regions. This can be done by finding the point of intersection between the two lines.

The point of intersection can be found by solving the system of linear equations:

$$\begin{array}{r} x + y = 4 \\ 2x + y = 7 \end{array}$$

Subtracting the first equation from the second equation, we get:

$$x = 3$$

Substituting this value into the first equation, we get:

$$y = 1$$

Therefore, the point of intersection is (3, 1).

Checking the Solutions

To check if the point (3, 1) is a solution to the system of linear inequalities, we need to plug in the values of x and y into both inequalities.

For the first inequality, $x + y \geq 4$, we get:

$$3 + 1 \geq 4$$

This is true, so the point (3, 1) satisfies the first inequality.

For the second inequality, $2x + y \leq 7$, we get:

$$2(3) + 1 \leq 7$$

This is also true, so the point (3, 1) satisfies the second inequality.

Conclusion

Therefore, the ordered pair (3, 1) is a solution to the system of linear inequalities. This means that Cody can buy 3 hot dogs and 1 peanut within his budget of $7.

Discussion

Cody's snack problem is a classic example of a system of linear inequalities. By graphing the inequalities and finding the intersection, we were able to determine the possible combinations of hot dogs and peanuts that Cody can buy within his budget.

This problem can be extended to more complex systems of linear inequalities, where there are multiple variables and constraints. In such cases, the solution can be found using techniques such as substitution and elimination.

Applications

Cody's snack problem has many real-world applications. For example, in business, a company may have a budget for marketing and advertising. The company may want to allocate a certain amount of money for each marketing channel, such as social media and print ads. By using a system of linear inequalities, the company can determine the optimal allocation of resources to meet its marketing goals.

In finance, a person may have a budget for saving and investing. The person may want to allocate a certain amount of money for each investment, such as stocks and bonds. By using a system of linear inequalities, the person can determine the optimal allocation of resources to meet their financial goals.

Conclusion

Introduction

In our previous article, we explored Cody's snack problem, a classic example of a system of linear inequalities. We determined that Cody can buy 3 hot dogs and 1 peanut within his budget of $7. In this article, we will answer some frequently asked questions about Cody's snack problem.

Q&A

Q: What is the main goal of Cody's snack problem?

A: The main goal of Cody's snack problem is to determine the possible combinations of hot dogs and peanuts that Cody can buy within his budget of $7.

Q: What are the two inequalities that represent Cody's snack problem?

A: The two inequalities that represent Cody's snack problem are:

$$\begin{array}{r} x + y \geq 4 \\ 2x + y \leq 7 \end{array}$$

where $x$ is the number of hot dogs and $y$ is the number of peanuts.

Q: How do we graph the inequalities?

A: We graph the inequalities by drawing a line with a slope of -1 and a y-intercept of 4 for the first inequality, and a line with a slope of -2 and a y-intercept of 7 for the second inequality.

Q: How do we find the intersection of the two regions?

A: We find the intersection of the two regions by solving the system of linear equations:

$$\begin{array}{r} x + y = 4 \\ 2x + y = 7 \end{array}$$

Q: What is the point of intersection?

A: The point of intersection is (3, 1).

Q: How do we check if the point (3, 1) is a solution to the system of linear inequalities?

A: We check if the point (3, 1) is a solution to the system of linear inequalities by plugging in the values of x and y into both inequalities.

Q: What is the solution to Cody's snack problem?

A: The solution to Cody's snack problem is that Cody can buy 3 hot dogs and 1 peanut within his budget of $7.

Q: What are some real-world applications of Cody's snack problem?

A: Some real-world applications of Cody's snack problem include:

• Business: A company may have a budget for marketing and advertising, and want to allocate a certain amount of money for each marketing channel.
• Finance: A person may have a budget for saving and investing, and want to allocate a certain amount of money for each investment.

Q: How can we extend Cody's snack problem to more complex systems of linear inequalities?

A: We can extend Cody's snack problem to more complex systems of linear inequalities by adding more variables and constraints. We can use techniques such as substitution and elimination to solve the system of linear inequalities.

Conclusion

In conclusion, Cody's snack problem is a classic example of a system of linear inequalities. By graphing the inequalities and finding the intersection, we were able to determine the possible combinations of hot dogs and peanuts that Cody can buy within his budget. This problem has many real-world applications, and can be extended to more complex systems of linear inequalities.

Additional Resources

For more information on systems of linear inequalities, please see the following resources:

• Khan Academy: Systems of Linear Inequalities
• Mathway: Systems of Linear Inequalities
• Wolfram Alpha: Systems of Linear Inequalities

Final Thoughts

Cody's snack problem is a fun and easy way to introduce students to the concept of systems of linear inequalities. By using real-world examples and visual aids, we can make math more accessible and engaging for students.