Andrea and Carter are going on a picnic. To prepare for the picnic, Andrea buys 4 bags of granola and 3 bags of dried fruit and spends $$ \$ 51.50 $$. Carter buys 2 bags of granola and 4 bags of dried fruit and spends $$ \$ 45.50 $$. A. Write and solve a system of equations to determine the cost of a bag of granola and the cost of a bag of dried fruit. What is the cost of a bag of granola? What is the cost of a bag of dried fruit? Wher

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Let's solve the problem step by step to determine the cost of a bag of granola and a bag of dried fruit.

### **Step 1: Define the Variables**

- Let $$ g $$ represent the cost of one bag of granola (in dollars).
- Let $$ f $$ represent the cost of one bag of dried fruit (in dollars).

### **Step 2: Translate the Problem into a System of Equations**

From the information provided:

1. **Andrea's Purchase:**
   - 4 bags of granola and 3 bags of dried fruit cost \$51.50.
   $$
   4g + 3f = 51.50 \quad \text{(Equation 1)}
   $$

2. **Carter's Purchase:**
   - 2 bags of granola and 4 bags of dried fruit cost \$45.50.
   $$
   2g + 4f = 45.50 \quad \text{(Equation 2)}
   $$

### **Step 3: Solve the System of Equations**

We'll use the **Elimination Method** to solve the system.

**a. Align the Equations:**
$$
\begin{cases}
4g + 3f = 51.50 \quad \text{(Equation 1)} \
2g + 4f = 45.50 \quad \text{(Equation 2)}
\end{cases}
$$

**b. Make the Coefficients of $$ g $$ the Same:**

Multiply **Equation 2** by 2 to match the coefficient of $$ g $$ in **Equation 1**.

$$
2 \times (2g + 4f) = 2 \times 45.50
$$
$$
4g + 8f = 91.00 \quad \text{(Equation 3)}
$$

**c. Subtract Equation 1 from Equation 3:**

$$
(4g + 8f) - (4g + 3f) = 91.00 - 51.50
$$
$$
4g - 4g + 8f - 3f = 39.50
$$
$$
5f = 39.50
$$

**d. Solve for $$ f $$:**

$$
f = \frac{39.50}{5} = 7.90
$$

So, **one bag of dried fruit costs \$7.90**.

**e. Substitute $$ f $$ back into Equation 1 to find $$ g $$:**

$$
4g + 3(7.90) = 51.50
$$
$$
4g + 23.70 = 51.50
$$
$$
4g = 51.50 - 23.70
$$
$$
4g = 27.80
$$
$$
g = \frac{27.80}{4} = 6.95
$$

So, **one bag of granola costs \$6.95**.

### **Conclusion**

- **Cost of a bag of granola ($$ g $$):** \$6.95
- **Cost of a bag of dried fruit ($$ f $$):** \$7.90
A bag of granola costs \$6.95 and a bag of dried fruit costs \$7.90.

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