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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=

1 answer

To derive the equation of the line in the form \(y = mx\) that passes through the origin (0, 0) and has a slope \(m\) derived from the points given, we need to calculate the slope using the two points provided: (1, 5) and (2, 10).

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:

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

Plugging in the values:

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

Now that we have the slope \(m\), we can write the equation of the line that goes through the origin:

\[ y = mx \implies y = 5x \]

So, the final equation of the line is:

\[ \boxed{5x} \]