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!