To derive the equation of a line \(y = mx\) that passes through the origin and is proportional to the given points \((1, 5)\) and \((2, 10)\), we first need to find the slope \(m\).
Step 1: Calculate the slope between the points.
The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points \((1, 5)\) and \((2, 10)\):
- Let \((x_1, y_1) = (1, 5)\)
- Let \((x_2, y_2) = (2, 10)\)
Substituting these values into the slope formula:
\[ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \]
Step 2: Write the equation of the line through the origin.
Since the line is proportional and passes through the origin, the equation can be expressed in the form:
\[ y = mx \]
Substituting in the slope we calculated:
\[ y = 5x \]
Conclusion
The equation of the line in the form \(y = mx\) that passes through the origin and describes the proportional relationship is:
\[ \boxed{y = 5x} \]