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The problem asks for the constant of proportionality for each line, given that both lines represent proportional relationships. Proportional relationships have a constant ratio between the values of \(y\) and \(x\), and this constant is the slope of the line.

From the graph, we see two lines labeled \(a\) (blue) and \(b\) (red). To find the constant of proportionality, we can calculate the slope for each line using the formula:

\[
\text{slope} = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}
\]

### Line \(a\) (blue line):
- The line passes through points (0,0) and (1,4).
- The slope (constant of proportionality) is:
  \[
  \frac{4 - 0}{1 - 0} = 4
  \]
  So, the constant of proportionality for line \(a\) is 4.

### Line \(b\) (red line):
- The line passes through points (0,0) and (2,3).
- The slope (constant of proportionality) is:
  \[
  \frac{3 - 0}{2 - 0} = 1.5
  \]
  So, the constant of proportionality for line \(b\) is 1.5.

### Conclusion:
- The constant of proportionality for line \(a\) is 4.
- The constant of proportionality for line \(b\) is 1.5.

Let me know if you'd like more details or have any further questions.

Here are some related questions:
1. How do we interpret the constant of proportionality in a real-world context?
2. What happens to the graph of a proportional relationship if the constant changes?
3. How do you know that both lines represent proportional relationships?
4. Can a line that doesn't go through the origin represent a proportional relationship?
5. How can we find the constant of proportionality from a table instead of a graph?

**Tip:** Remember, for proportional relationships, the line must pass through the origin!

Both lines on the graph represent proportional relationships. What is the constant of proportionality for each relationship? Explain how you know.

Learn how to determine the constant of proportionality for two proportional relationships represented on a graph. Understand the slope formula and how it applies to proportional relationships in algebra. Suitable for students in Grades 7-8.

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