Question 1198769
To analyze the situation mathematically, we can represent it as a linear relationship where the balance on the gift card decreases as the number of milkshakes purchased increases.

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### Step 1: Define Variables
- **Independent Variable**: The number of milkshakes ordered (denoted as \( x \)).
- **Dependent Variable**: The remaining balance on the gift card (denoted as \( y \)).

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### Step 2: Identify Two Points
We are given:
1. After Connor and one friend (2 milkshakes) ordered, the balance was $22.45. This gives the point \((2, 22.45)\).
2. After Connor and four friends (5 milkshakes) ordered, the balance was $6.70. This gives the point \((5, 6.70)\).

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### Step 3: Find the Rate of Change (Slope)
The slope represents the rate at which the balance decreases per milkshake:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the points \((2, 22.45)\) and \((5, 6.70)\):
\[
m = \frac{6.70 - 22.45}{5 - 2} = \frac{-15.75}{3} = -5.25
\]
The slope is \( -5.25 \), meaning each milkshake costs **$5.25**.

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### Step 4: Equation of the Line
Using the point-slope form:
\[
y - y_1 = m(x - x_1)
\]
Substitute \( m = -5.25 \) and \((x_1, y_1) = (2, 22.45)\):
\[
y - 22.45 = -5.25(x - 2)
\]
Simplify:
\[
y = -5.25x + 10.50 + 22.45
\]
\[
y = -5.25x + 32.95
\]

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### Step 5: Answers
1. **Two points**: \((2, 22.45)\) and \((5, 6.70)\).
2. **Rate of change or slope**: \( -5.25 \), meaning the cost of each milkshake is **$5.25**.
3. **Dependent variable**: The remaining balance on the gift card (\( y \)).
4. **Independent variable**: The number of milkshakes ordered (\( x \)).

Let me know if you'd like further clarification!