Cody Has $7. He Wants To Buy At Least 4 Snacks. Hot Dogs (x) Are $ 2 E A C H . P E A N U T S ( Y ) A R E $ 1 E A C H . W H I C H O R D E R E D P A I R I S A S O L U T I O N ? 2 Each. Peanuts (y) Are \$1 Each. Which Ordered Pair Is A Solution? 2 E A C H . P E An U T S ( Y ) A Re $1 E A C H . Whi C H Or D Ere D P Ai R I S A So L U T I O N ? { \begin{align*} x + Y & \geq 4 \\ 2x + Y & \leq 7 \end{align*} \} A. { (3,2)$}$B.

Introduction

Cody has a budget of $7 and wants to buy at least 4 snacks. He is considering two types of snacks: hot dogs and peanuts. Hot dogs cost $2 each, while peanuts cost $1 each. To determine which ordered pair of hot dogs and peanuts Cody can buy, we need to solve a system of linear inequalities.

The System of Linear Inequalities

The system of linear inequalities is given by:

$$\begin{align*} x + y & \geq 4 \\ 2x + y & \leq 7 \end{align*}$$

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

Understanding the Inequalities

The first inequality, $x + y \geq 4$, states that the total number of snacks (hot dogs and peanuts) must be at least 4. This means that Cody must buy at least 4 snacks in total.

The second inequality, $2x + y \leq 7$, states that the total cost of the snacks (hot dogs and peanuts) must be less than or equal to $7. Since hot dogs cost $2 each and peanuts cost $1 each, this inequality ensures that Cody does not exceed his budget.

Finding the Ordered Pair Solution

To find the ordered pair solution, we need to find the values of $x$ and $y$ that satisfy both inequalities.

Let's start by analyzing the first inequality, $x + y \geq 4$. Since Cody wants to buy at least 4 snacks, we can start by assuming that he buys 4 snacks. This means that $x + y = 4$.

Now, let's substitute this value into the second inequality, $2x + y \leq 7$. We get:

$$2x + 4 \leq 7$$

Simplifying this inequality, we get:

$$2x \leq 3$$

Dividing both sides by 2, we get:

$$x \leq 1.5$$

Since $x$ represents the number of hot dogs, it must be an integer. Therefore, the maximum value of $x$ is 1.

Now, let's substitute $x = 1$ into the first inequality, $x + y \geq 4$. We get:

$$1 + y \geq 4$$

Simplifying this inequality, we get:

$$y \geq 3$$

Since $y$ represents the number of peanuts, it must be an integer. Therefore, the minimum value of $y$ is 3.

The Ordered Pair Solution

Based on our analysis, we have found that the ordered pair solution is:

$$(x, y) = (1, 3)$$

However, we need to check if this solution satisfies both inequalities.

Substituting $x = 1$ and $y = 3$ into the first inequality, $x + y \geq 4$, we get:

$$1 + 3 \geq 4$$

This inequality is true.

Substituting $x = 1$ and $y = 3$ into the second inequality, $2x + y \leq 7$, we get:

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

Simplifying this inequality, we get:

$$5 \leq 7$$

This inequality is also true.

Therefore, the ordered pair solution is indeed:

$$(x, y) = (1, 3)$$

Conclusion

In this problem, we used a system of linear inequalities to find the ordered pair solution that satisfies the conditions. We analyzed the inequalities, found the values of $x$ and $y$ that satisfy both inequalities, and determined the ordered pair solution.

Answer

The ordered pair solution is:

$$(x, y) = (1, 3)$$

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 solution that satisfies the conditions of buying at least 4 snacks and staying within a budget of $7.

Q: What are the two types of snacks that Cody is considering?

A: Cody is considering two types of snacks: hot dogs and peanuts. Hot dogs cost $2 each, while peanuts cost $1 each.

Q: What is the system of linear inequalities that Cody needs to satisfy?

A: The system of linear inequalities that Cody needs to satisfy is:

$$\begin{align*} x + y & \geq 4 \\ 2x + y & \leq 7 \end{align*}$$

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

Q: What does the first inequality, $x + y \geq 4$, represent?

A: The first inequality, $x + y \geq 4$, represents the condition that Cody must buy at least 4 snacks in total.

Q: What does the second inequality, $2x + y \leq 7$, represent?

A: The second inequality, $2x + y \leq 7$, represents the condition that Cody must stay within his budget of $7.

Q: How did you find the ordered pair solution?

A: To find the ordered pair solution, we analyzed the inequalities, found the values of $x$ and $y$ that satisfy both inequalities, and determined the ordered pair solution.

Q: What is the ordered pair solution that satisfies both inequalities?

A: The ordered pair solution that satisfies both inequalities is:

$$(x, y) = (1, 3)$$

This means that Cody can buy 1 hot dog and 3 peanuts, which satisfies both inequalities and stays within his budget of $7.

Q: Why is the ordered pair solution $(1, 3)$ the only solution that satisfies both inequalities?

A: The ordered pair solution $(1, 3)$ is the only solution that satisfies both inequalities because it is the only combination of hot dogs and peanuts that meets the conditions of buying at least 4 snacks and staying within a budget of $7.

Q: What is the significance of the ordered pair solution $(1, 3)$?

A: The ordered pair solution $(1, 3)$ is significant because it represents the minimum number of hot dogs and peanuts that Cody can buy to satisfy both inequalities.

Q: Can Cody buy any other combination of hot dogs and peanuts that satisfies both inequalities?

A: No, Cody cannot buy any other combination of hot dogs and peanuts that satisfies both inequalities. The ordered pair solution $(1, 3)$ is the only solution that meets the conditions.

Q: What is the main takeaway from Cody's snack problem?

A: The main takeaway from Cody's snack problem is that by analyzing the inequalities and finding the values of $x$ and $y$ that satisfy both inequalities, we can determine the ordered pair solution that satisfies the conditions.