Question

The graph shows the proportional relationship (1, 5) (2,10)
Derive the equation of the line y = mx through the origin

Answer 1

To derive the equation of the line \( y = mx \) that passes through the origin and represents the proportional relationship shown by the points (1, 5) and (2, 10), we need to first determine the slope \( m \).

The slope \( m \) is calculated using the formula:

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

For the points (1, 5) and (2, 10):

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

Now, substituting these values into the slope formula:

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

Now that we have the slope \( m = 5 \), we can write the equation of the line that passes through the origin.

The equation of the line in slope-intercept form (where the y-intercept is 0) is:

\[
y = mx
\]

Substituting \( m \) into the equation gives:

\[
y = 5x
\]

Thus, the equation of the line is:

\[
\boxed{y = 5x}
\]