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Let’s work through each problem step-by-step.

### Problem 1
- **Question**: Write an equation for the graph of a proportional relationship that passes through the point (25, 15). Explain your thinking.
- **Solution**: 
  Since it’s a proportional relationship, the equation will be in the form $ y = kx $, where $ k $ is the constant of proportionality (slope).
  
  To find $ k $, use the point (25, 15):
  \[
  k = \frac{y}{x} = \frac{15}{25} = \frac{3}{5}
  \]
  So, the equation is:
  \[
  y = \frac{3}{5}x
  \]

### Problem 2
- **Question**: Write an equation of the line graphed. Use $ y = mx $ form, where $ m $ represents the slope of the line.
- **Solution**:
  Looking at the graph, pick two points to find the slope $ m $. Let’s assume points (10, 5) and (20, 10) are on the line.
  \[
  m = \frac{10 - 5}{20 - 10} = \frac{5}{10} = \frac{1}{2}
  \]
  Thus, the equation is:
  \[
  y = \frac{1}{2}x
  \]

### Problem 3
- **Question**: Write an equation to represent this relationship, where $ x $ is the number of drinks sold at a concession stand and $ y $ is the amount of money collected in dollars.
- **Solution**:
  Given the table with values for 1 drink sold at $220 and 1500 drinks sold, we find the constant of proportionality.
  \[
  k = \frac{y}{x} = \frac{220}{1} = 220
  \]
  So, the equation is:
  \[
  y = 220x
  \]

### Problem 4
- **Question**: Use your equation to complete this table.
  - **Drinks Sold**: 220, 1, 1500
  - **Money (Dollars)**: Complete based on the equation $ y = 220x $
- **Solution**:
  - For $ x = 220 $: $ y = 220 \times 220 = 48400 $
  - For $ x = 1 $: $ y = 220 $
  - For $ x = 1500 $: $ y = 220 \times 1500 = 330000 $
  
  So, the completed table is:
  | Drinks Sold | Money (Dollars) |
  |-------------|-----------------|
  | 220         | 48400           |
  | 1           | 220             |
  | 1500        | 330000          |

### Problem 5
- **Question**: Does Graph 2 represent the same relationship as Graph 1? Explain your thinking.
- **Solution**:
  In Graph 1, the point (3, 4) suggests a slope of $ \frac{4}{3} $. For Graph 2, using the point (6, 4), the slope is $ \frac{4}{6} = \frac{2}{3} $. Since the slopes are different, **Graph 2 does not represent the same relationship as Graph 1**.

### Problem 6
- **Question**: Does Graph 3 represent the same relationship as Graph 1? Explain your thinking.
- **Solution**:
  In Graph 3, using the point (6, 8), the slope is $ \frac{8}{6} = \frac{4}{3} $, which is the same as Graph 1. Therefore, **Graph 3 represents the same relationship as Graph 1**.

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Would you like further details on any of these problems or have specific questions?

### Related Questions
1. How do we interpret the constant of proportionality in a proportional relationship?
2. How can we determine if two graphs represent the same proportional relationship?
3. What is the significance of the slope in a proportional relationship graph?
4. Why is it important for a relationship to pass through the origin to be proportional?
5. How can we verify if two equations represent the same line?

### Tip
When determining if two graphs represent the same relationship, check if their slopes are equal; for proportional relationships, also ensure both lines pass through the origin.

Solve various problems involving proportional relationships, including writing equations for graphs, calculating slope, and comparing relationships between graphs.

Learn how to solve proportional relationship problems involving slope, writing equations, and comparing graphs. Step-by-step guidance for students in Grades 7-8, covering key concepts like slope, linear equations, and graph interpretation.

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