To find the equation of the line \( y = mx \) that goes through the origin based on the given points \((5, 10)\), \((4, 8)\), and \((2, 4)\), we need to determine the slope \( m \).
The slope \( m \) can be found by using any two points. Let's take the points \((2, 4)\) and \((4, 8)\):
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Calculate the difference in the y-values: \[ 8 - 4 = 4 \]
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Calculate the difference in the x-values: \[ 4 - 2 = 2 \]
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Now, compute the slope \( m \): \[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{4}{2} = 2 \]
Therefore, the equation of the line is: \[ y = 2x \]
Thus, the correct answer is: B. \( y = 2x \)