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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 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.

The graph shows the proportional relationship. Derive the equation of the line y=mx
through the origin.

(1 point)
y=
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1 answer

To derive the equation of the line in the form \( y = mx \), we need to determine the slope \( m \) of the line that passes through the origin and the given points \( (1, 5) \) and \( (2, 10) \).

First, we calculate the slope \( m \) using the two points. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \( (1, 5) \) and \( (2, 10) \):

  • \( x_1 = 1, y_1 = 5 \)
  • \( x_2 = 2, y_2 = 10 \)

Substituting into the slope formula:

\[ m = \frac{10 - 5}{2 - 1} = \frac{5}{1} = 5 \]

Now that we have the slope \( m = 5 \), we write the equation of the line that passes through the origin (0,0), which is:

\[ y = mx = 5x \]

Therefore, the equation of the line is:

\[ \boxed{5x} \]