Damian uses 2 cups of flour for each batch of scones and 3 cups of flour for each batch of muffins. He makes 5 times as many batches of muffins as scones, and he uses a total of 34 cups of flour. Write a system of equations for this situation with $$ x $$ representing the number of batches of scones and $$ y $$ representing the number of batches of muffins. Answer Solve the system of equations using the substitution method. Then, interpret the solution of the system in terms of this situation. Answer ( Damian makes batches of scones and batches of muffins. bit

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**Step 1: Define the Variables**

Let:
- $$ x $$ = number of batches of scones
- $$ y $$ = number of batches of muffins

**Step 2: Write the System of Equations**

From the problem statement, we have two pieces of information:

1. **Flour Usage:**
   - Each batch of scones uses 2 cups of flour.
   - Each batch of muffins uses 3 cups of flour.
   - Total flour used is 34 cups.

This translates to the equation:
   $$
   2x + 3y = 34
   $$

2. **Batch Relationship:**
   - Damian makes 5 times as many batches of muffins as scones.

This translates to the equation:
   $$
   y = 5x
   $$

**System of Equations:**
$$
\begin{cases}
2x + 3y = 34 \
y = 5x
\end{cases}
$$

---

**Step 3: Solve the System Using the Substitution Method**

1. **Substitute $$ y = 5x $$ into the first equation:**
   $$
   2x + 3(5x) = 34
   $$
   $$
   2x + 15x = 34
   $$
   $$
   17x = 34
   $$
   $$
   x = \frac{34}{17} = 2
   $$

2. **Find $$ y $$ using $$ y = 5x $$:**
   $$
   y = 5(2) = 10
   $$

---

**Step 4: Interpret the Solution**

Damian makes:
- $$ \mathbf{2} $$ **batches of scones**
- $$ \mathbf{10} $$ **batches of muffins**

**Verification:**

- Flour used for scones: $$ 2 \times 2 = 4 $$ cups
- Flour used for muffins: $$ 3 \times 10 = 30 $$ cups
- **Total flour used:** $$ 4 + 30 = 34 $$ cups (matches the given total)
Damian makes 2 batches of scones and 10 batches of muffins.

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