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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 \( x \).

When graphed, which line will have the greatest rate of change?
a. y = 5x
b. y = 3x
c. y = 0.3x
d. y = 0.5x

Determine which line among y = 5x, y = 3x, y = 0.3x, and y = 0.5x has the greatest rate of change by analyzing the slope. This problem focuses on understanding the concept of slope in linear equations, suitable for Grades 7-9.

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