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

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