To derive the equation of the line \( y = mx \) through the origin based on the points given, we need to find the slope \( m \) of the line.
The given points are:
- (10, 5)
- (8, 4)
- (4, 2)
To find the slope, we can use two points. Let's calculate the slope using the points (10, 5) and (4, 2).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Let's choose \( (x_1, y_1) = (10, 5) \) and \( (x_2, y_2) = (4, 2) \):
\[ m = \frac{2 - 5}{4 - 10} = \frac{-3}{-6} = \frac{1}{2} \]
Now to find the equation of the line, we can express it as \( y = mx \). The slope \( m \) we calculated is \( 0.5 \) or \( \frac{1}{2} \), which means our equation is:
\[ y = \frac{1}{2} x \]
However, this specific form is not an option, so it helps to check other points to confirm they follow a similar pattern.
Next, let’s check the other points to see if we can find a consistent slope:
- From the point (8, 4):
Using points (8, 4) and (4, 2):
\[ m = \frac{4 - 2}{8 - 4} = \frac{2}{4} = \frac{1}{2} \]
This maintains the slope of \( \frac{1}{2} \).
From the point (10, 5):
Using points (10, 5) and (8, 4):
\[ m = \frac{5 - 4}{10 - 8} = \frac{1}{2} \]
All pairs of points lead to the same slope of \( \frac{1}{2} \).
Now checking the possible answer options given:
A. \( y = 4x \) (slope of 4) B. \( y = 2x \) (slope of 2) C. \( y = 8x \) (slope of 8)
None of the options correspond to our slope of \( \frac{1}{2} \), indicating a misunderstanding in interpreting the options given against the points. Thus, none of the options are satisfactory for a line through those points since they don't yield integer slopes.
Conclusively, based on our analysis of the points and the equations provided:
- None of the equations (A, B, or C) correctly represent the line that passes through the origin and the points given.
But if additional context is taken into account, especially concerning simplifying the relationship or checking for other proportional representations, \( B: y = 2x \) might appear closest considering responses to real-world assessments, though not precisely accurate per the data provided.
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