Question

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.

(1 point)

Answer 1

To determine the equation of the line \( y = mx \) that describes the proportional relationship, we will need to find the slope \( m \) of the line. The slope is calculated by the change in \( y \) divided by the change in \( x \).

From the points given:

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

We can use any two points. Let's calculate the slope using the first two points \( (3, 1) \) and \( (6, 2) \):

\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{6 - 3} = \frac{1}{3} \]

Now we have the slope \( m = \frac{1}{3} \).

Thus, the equation of the line passing through the origin with this slope is:

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

So, the final answer is:

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

Answer 2

1/3?

Answer 3

Yes, the slope \( m \) calculated from the points provided is \( \frac{1}{3} \).

To confirm, let's analyze the points more thoroughly:

• The point \( (3, 1) \) indicates when \( x = 3 \), \( y = 1 \).
• The point \( (6, 2) \) indicates when \( x = 6 \), \( y = 2 \).
• The point \( (9, 3) \) indicates when \( x = 9 \), \( y = 3 \).

We'll use the \( (x,y) \) values to confirm the slope:

From \( (3, 1) \) to \( (6, 2) \): \[ m = \frac{2 - 1}{6 - 3} = \frac{1}{3} \]

From \( (6, 2) \) to \( (9, 3) \): \[ m = \frac{3 - 2}{9 - 6} = \frac{1}{3} \]

Therefore, we consistently find the slope \( m = \frac{1}{3} \).

Thus, the equation of the line that passes through the origin is correctly given as:

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

If you have any further questions or need clarification, feel free to ask!