In a proportional relationship, the equation of the line through the origin can be expressed in the form \( y = mx \), where \( m \) is the slope of the line.
To find the slope \( m \), you can use any two points on the line. In this case, we have the points \( (1, 5) \) and \( (2, 10) \).
The slope \( m \) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting the coordinates of the points:
- Let \( (x_1, y_1) = (1, 5) \)
- Let \( (x_2, y_2) = (2, 10) \)
Now, substituting these values into the slope formula:
\[ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \]
Now that we have the slope \( m = 5 \), we can write the equation of the line in the form \( y = mx \):
\[ y = 5x \]
Thus, the derived equation of the line through the origin that represents the proportional relationship is:
\[ \boxed{y = 5x} \]
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