Cody Has $$ 7$. He Wants To Buy At Least 4 Snacks. Hot Dogs ( $x$ ) Are $$ 2$ Each, And Peanuts ( $y$ ) Are $$ 1$ Each. Which Ordered Pair Is A Solution? [ \begin{array}{r} x +
Introduction
Cody has a budget of $7 and wants to buy at least 4 snacks. He has two options: hot dogs, which cost $2 each, and peanuts, which cost $1 each. In this problem, we will find the ordered pair that represents the number of hot dogs and peanuts Cody can buy within his budget.
The Problem
Cody's budget is $7, and he wants to buy at least 4 snacks. The cost of hot dogs is $2 each, and the cost of peanuts is $1 each. We need to find the ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget.
Mathematical Representation
Let's represent the number of hot dogs as x and the number of peanuts as y. The total cost of the snacks can be represented by the equation:
2x + y = 7
We also know that Cody wants to buy at least 4 snacks, so we can represent this as an inequality:
x + y ≥ 4
Solving the System of Equations
To find the ordered pair (x, y), we need to solve the system of equations:
2x + y = 7 x + y ≥ 4
We can start by solving the first equation for y:
y = 7 - 2x
Now, we can substitute this expression for y into the second inequality:
x + (7 - 2x) ≥ 4
Simplifying the inequality, we get:
-x + 7 ≥ 4
Subtracting 7 from both sides, we get:
-x ≥ -3
Multiplying both sides by -1, we get:
x ≤ 3
Now, we can substitute this expression for x into the first equation:
2(3) + y = 7
Simplifying the equation, we get:
6 + y = 7
Subtracting 6 from both sides, we get:
y = 1
Finding the Ordered Pair
Now that we have found the values of x and y, we can find the ordered pair (x, y). We have:
x = 3 y = 1
So, the ordered pair that represents the number of hot dogs and peanuts Cody can buy within his budget is (3, 1).
Conclusion
In this problem, we found the ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget. We used a system of equations and inequalities to solve the problem. The ordered pair (3, 1) represents the number of hot dogs and peanuts Cody can buy within his budget.
Discussion
This problem is a classic example of a linear programming problem. Linear programming is a method of optimization that is used to find the best solution to a problem that involves linear equations and inequalities. In this problem, we used linear programming to find the ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget.
Real-World Applications
This problem has many real-world applications. For example, in a restaurant, the owner may want to know how many hot dogs and peanuts to buy within a certain budget. The owner can use linear programming to find the optimal solution. Similarly, in a store, the manager may want to know how many products to stock within a certain budget. The manager can use linear programming to find the optimal solution.
Future Research
This problem can be extended in many ways. For example, we can add more constraints to the problem, such as a limit on the number of hot dogs and peanuts that can be bought. We can also add more variables to the problem, such as the cost of other snacks. We can use linear programming to find the optimal solution to these extended problems.
References
- [1] Chvatal, V. (1983). Linear Programming. W.H. Freeman and Company.
- [2] Hillier, F. S., & Lieberman, G. J. (2015). Introduction to Operations Research. McGraw-Hill Education.
Appendix
The following is a list of the variables and constants used in this problem:
- x: the number of hot dogs
- y: the number of peanuts
- 2: the cost of a hot dog
- 1: the cost of a peanut
- 7: Cody's budget
- 4: the minimum number of snacks Cody wants to buy
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 find the ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget of $7.
Q: What are the constraints of the problem?
A: The constraints of the problem are:
- Cody's budget is $7
- The cost of hot dogs is $2 each
- The cost of peanuts is $1 each
- Cody wants to buy at least 4 snacks
Q: How do we represent the number of hot dogs and peanuts mathematically?
A: We represent the number of hot dogs as x and the number of peanuts as y. The total cost of the snacks can be represented by the equation:
2x + y = 7
Q: How do we find the ordered pair (x, y)?
A: We find the ordered pair (x, y) by solving the system of equations:
2x + y = 7 x + y ≥ 4
We can start by solving the first equation for y:
y = 7 - 2x
Now, we can substitute this expression for y into the second inequality:
x + (7 - 2x) ≥ 4
Simplifying the inequality, we get:
-x + 7 ≥ 4
Subtracting 7 from both sides, we get:
-x ≥ -3
Multiplying both sides by -1, we get:
x ≤ 3
Now, we can substitute this expression for x into the first equation:
2(3) + y = 7
Simplifying the equation, we get:
6 + y = 7
Subtracting 6 from both sides, we get:
y = 1
Q: What is the ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget?
A: The ordered pair (x, y) that represents the number of hot dogs and peanuts Cody can buy within his budget is (3, 1).
Q: What is the significance of this problem in real-world applications?
A: This problem has many real-world applications, such as:
- Finding the optimal solution to a problem that involves linear equations and inequalities
- Determining the number of products to stock within a certain budget
- Optimizing the production of goods and services
Q: Can this problem be extended in any way?
A: Yes, this problem can be extended in many ways, such as:
- Adding more constraints to the problem, such as a limit on the number of hot dogs and peanuts that can be bought
- Adding more variables to the problem, such as the cost of other snacks
- Using different optimization techniques, such as linear programming or dynamic programming
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include:
- Not considering all the constraints of the problem
- Not using the correct optimization technique
- Not checking the solution for feasibility
Q: What are some tips for solving this problem?
A: Some tips for solving this problem include:
- Breaking down the problem into smaller sub-problems
- Using a systematic approach to solve the problem
- Checking the solution for feasibility and optimality
Q: What are some resources for learning more about this problem?
A: Some resources for learning more about this problem include:
- Textbooks on linear programming and optimization
- Online courses and tutorials on linear programming and optimization
- Research papers and articles on linear programming and optimization