To determine the equation of the line that passes through the origin and is proportional to the points on the line given, we can find the slope \( m \) of the line.
Given two points from the set of points that the line passes through, let's use the points \( (3, 1) \) and \( (6, 2) \) to calculate the slope:
The formula for the slope \( m \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using points \( (3, 1) \) and \( (6, 2) \):
\[ m = \frac{2 - 1}{6 - 3} = \frac{1}{3} \]
Now that we have the slope, we can write the equation of the line that passes through the origin \( (0,0) \):
\[ y = mx \]
Substituting the slope \( m = \frac{1}{3} \):
\[ y = \frac{1}{3}x \]
Therefore, the equation of the line in slope-intercept form is:
\[ \boxed{\frac{1}{3}x} \]
Thus, expressing it simply as \( y = \frac{1}{3}x \), we can write:
\[ y = \frac{1}{3} \]
as the final answer.