In a proportional relationship represented by a graph that passes through the origin (0,0), the equation takes the form of \( y = mx \), where \( m \) is the slope of the line.
To derive the correct equation from the options provided, we can analyze each one:
- y = 5x: This is in the form \( y = mx \) with a slope \( m = 5 \), passing through the origin.
- y = x + 5: This has a y-intercept of 5, meaning it does not pass through the origin.
- y = -5x: This is also in the form \( y = mx \) with a slope \( m = -5 \), passing through the origin.
- y = 5x + 5: Similar to option 2, this has a y-intercept of 5, meaning it does not pass through the origin.
Since the question states the line must go through the origin, the valid options that meet this criterion are:
- \( y = 5x \)
- \( y = -5x \)
Without additional context about the graph (such as the steepness or direction of the line), we can't definitively choose one over the other. However, if the graph represents a positive proportional relationship (like a direct increase), then \( y = 5x \) is the correct answer.
If you are instructed to derive the equation based on a positive slope, the correct answer would be:
y = 5x