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An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 2 comma 4 right parenthesis, left parenthesis 4 comma 8 right parenthesis, and left parenthesis 5 comma 10 right parenthesis.

The proportional relationship is shown on the graph. Derive the equation of the line y=mx
through the origin.

1 answer

To determine the equation of the line \( y = mx \) that passes through the origin and reflects the relationship demonstrated by the points given, we first need to find the slope \( m \) of the line using two of the points.

Taking the points \( (2, 4) \) and \( (4, 8) \):

  1. Calculate the slope \( m \): \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points \( (2, 4) \) and \( (4, 8) \): \[ m = \frac{8 - 4}{4 - 2} = \frac{4}{2} = 2 \]

Now that we have the slope, we can substitute \( m \) into the equation \( y = mx \):

\[ y = 2x \]

To confirm that this line represents the relationship of the points \( (5, 10) \) as well, we can check if it lies on the line:

  1. Check if \( (5, 10) \) satisfies \( y = 2x \): \[ y = 2(5) = 10 \]

Since the point \( (5, 10) \) also lies on the line \( y = 2x \), we have verified our equation.

Thus, the equation of the line is:

\[ y = 2x \]