Use the image to answer the question.

Question

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=

Answer 1

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

The formula for the slope \( m \) is:

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

Substituting the coordinates of the points:

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

Now that we have the slope, we can write the equation of the line through the origin in the form:

\[ y = mx \]

Substituting \( m \):

\[ y = 5x \]

Therefore, the equation of the line is:

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