https://file.aiquickdraw.com/mathbot/file/1730305271474kiejlxj1.jpg

This problem asks you to **match each equation to its corresponding graph**. Each equation given is a linear function of the form \( y = mx \), where \( m \) is the slope of the line. Let's go through the process of identifying the correct matches by observing the **slopes** from both the equations and the graphs.

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### Step 1: Equation Slopes
Below are the equations and their slopes:
- \( a. \quad y = 2x \quad (m = 2) \)
- \( b. \quad y = \frac{4}{5}x \quad (m = \frac{4}{5}) \)
- \( c. \quad y = \frac{1}{4}x \quad (m = \frac{1}{4}) \)
- \( d. \quad y = \frac{2}{3}x \quad (m = \frac{2}{3}) \)
- \( e. \quad y = \frac{3}{4}x \quad (m = \frac{3}{4}) \)
- \( f. \quad y = \frac{3}{2}x \quad (m = \frac{3}{2}) \)

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### Step 2: Identify Graph Slopes
Now, we need to determine the **rise over run** (slope) from each graph.

1. **Graph 1:** Rise = 5, Run = 4 → Slope \( m = \frac{5}{4} \). (This matches **none** of the given equations perfectly since no equation has \( \frac{5}{4}x \)).
2. **Graph 2:** Rise = 2, Run = 3 → Slope \( m = \frac{2}{3} \). (Matches **d. \( y = \frac{2}{3}x \)**).
3. **Graph 3:** Rise = 3, Run = 2 → Slope \( m = \frac{3}{2} \). (Matches **f. \( y = \frac{3}{2}x \)**).
4. **Graph 4:** Rise = 1, Run = 4 → Slope \( m = \frac{1}{4} \). (Matches **c. \( y = \frac{1}{4}x \)**).
5. **Graph 5:** Rise = 3, Run = 4 → Slope \( m = \frac{3}{4} \). (Matches **e. \( y = \frac{3}{4}x \)**).
6. **Graph 6:** Rise = 4, Run = 5 → Slope \( m = \frac{4}{5} \). (Matches **b. \( y = \frac{4}{5}x \)**).

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### Step 3: Matching Results
Based on the above analysis:

- **Graph 2** → \( d. \quad y = \frac{2}{3}x \)
- **Graph 3** → \( f. \quad y = \frac{3}{2}x \)
- **Graph 4** → \( c. \quad y = \frac{1}{4}x \)
- **Graph 5** → \( e. \quad y = \frac{3}{4}x \)
- **Graph 6** → \( b. \quad y = \frac{4}{5}x \)

**Note:** Graph 1 has a slope of \( \frac{5}{4} \), which does not directly match any equation. However, it seems the closest matching equation might be \( a. \quad y = 2x \), though the slope is not exactly aligned.

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### Final Matches:

- **Graph 1**: \( a. \quad y = 2x \) (approximate match)
- **Graph 2**: \( d. \quad y = \frac{2}{3}x \)
- **Graph 3**: \( f. \quad y = \frac{3}{2}x \)
- **Graph 4**: \( c. \quad y = \frac{1}{4}x \)
- **Graph 5**: \( e. \quad y = \frac{3}{4}x \)
- **Graph 6**: \( b. \quad y = \frac{4}{5}x \)

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Do you need any further clarification or help?  
Here are five follow-up questions you might consider:  
1. How do you calculate the slope between two points on a graph?  
2. What would the equation look like if the slope were negative?  
3. How can we determine the y-intercept of a line from a graph?  
4. Why does a greater slope result in a steeper line?  
5. What happens to the graph if we multiply the slope by a negative number?

**Tip:** To quickly estimate the slope from a graph, pick two points on the line, and use \( m = \frac{\text{rise}}{\text{run}} \).

This exercise requires matching linear equations to their corresponding graphs by analyzing slopes. Learn how to calculate the slope of a line, understand linear equations, and connect equations to graphs. Suitable for students in Grades 7-9.

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