The question asks for the line with the greatest rate of change. In a linear equation of the form $y = mx$, the coefficient $m$ represents the slope, which is also the rate of change.

Let's examine the options:

1. $y = 5x$: Here, the slope (rate of change) is $m = 5$.
2. $y = 3x$: Here, the slope is $m = 3$.
3. $y = 0.3x$: Here, the slope is $m = 0.3$.
4. $y = 0.5x$: Here, the slope is $m = 0.5$.

The line with the greatest rate of change will have the largest slope. Therefore, the answer is:

a. $y = 5x$, since 5 is the largest slope among the choices.

Would you like more details on this solution or have additional questions?

Here are five related questions for further exploration:

1. How is the slope of a line related to its steepness?
2. What does it mean if the slope of a line is negative?
3. How would you calculate the slope if given two points on a line?
4. What is the significance of the y-intercept in the equation $y = mx + b$?
5. How does the slope affect the direction of a line on a graph?

Tip: For any line equation $y = mx + b$, the slope $m$ determines how much $y$ changes with each unit increase in $x$.