Use the image to answer the 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 3 comma 1 right parenthesis, left parenthesis 6 comma 2 right parenthesis, and left parenthesis 9 comma 3 right parenthesis. Write the equation of the line y=mx through the origin that describes the proportional relationship shown in the graph.

Question

Use the image to answer the 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 3 comma 1 right parenthesis, left parenthesis 6 comma 2 right parenthesis, and left parenthesis 9 comma 3 right parenthesis.  Write the equation of the line  y=mx   through the origin that describes the proportional relationship shown in the graph.

GPT-4o mini · Answer

To find the equation of the line that describes the proportional relationship shown in the graph, we start by determining the slope (m) of the line that passes through the given points.

From the points provided:

(3, 1)
(6, 2)
(9, 3)

We can calculate the slope using the formula:

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

Using the points (3, 1) and (6, 2):

\[ m = \frac{2 - 1}{6 - 3} = \frac{1}{3} \]

Now, using the equation of a line through the origin, which can be written as:

\[ y = mx \]

We substitute the value of \(m\):

\[ y = \frac{1}{3}x \]

Thus, the equation of the line that describes the proportional relationship shown in the graph is:

\[ y = \frac{1}{3}x \]