Question

3,9, 2,6, 1,3,
Write the equation of the line y = mx through the origin that describes the proportional relationship shown in the graph.
__

Answer 1

To determine the equation of the line \( y = mx \) that describes the proportional relationship indicated by the points (3,9), (2,6), and (1,3), we need to find the slope \( m \).

The slope \( m \) can be calculated using two points from the data. Let's use the points (1,3) and (3,9).

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 points (1,3) and (3,9):

- \( (x_1, y_1) = (1, 3) \)
- \( (x_2, y_2) = (3, 9) \)

Plugging in the values:

\[
m = \frac{9 - 3}{3 - 1} = \frac{6}{2} = 3
\]

Now that we have the slope \( m = 3 \), the equation of the line through the origin is:

\[
y = 3x
\]

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

\[
y = 3x
\]

Answer 2

not correct

Answer 3

I apologize for any confusion. Let's analyze the data points you provided to determine the correct slope for the equation of the line \( y = mx \).

You mentioned the points:
- (3, 9)
- (2, 6)
- (1, 3)

To find the slope \( m \), we can choose any two points. Let’s calculate the slope using points (2, 6) and (1, 3):

Using the slope formula:

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

Choosing:
- \( (x_1, y_1) = (1, 3) \)
- \( (x_2, y_2) = (2, 6) \)

Calculating:

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

Now we can calculate using another pair, say (3, 9) and (1, 3):

Choosing:
- \( (x_1, y_1) = (1, 3) \)
- \( (x_2, y_2) = (3, 9) \)

Calculating:

\[
m = \frac{9 - 3}{3 - 1} = \frac{6}{2} = 3
\]

Since all points give us the same slope, we can confirm that:

### Final Equation
The line that represents the relationship of the points you provided, through the origin, is:

\[
y = 3x
\]

If this was not what you expected, could you please clarify or provide additional information about the relationship you're looking to describe?

Answer 4

not correct AGAIN

Answer 5

I apologize for the oversight. Let’s carefully inspect the points you provided and derive the correct relationship this time. The points you've given are:

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

It seems these points describe a linear relationship, and we want to express the relationship in the form \( y = mx \).

First, let’s look for a pattern in the data:

1. For \( x = 1 \), \( y = 3 \)
2. For \( x = 2 \), \( y = 6 \)
3. For \( x = 3 \), \( y = 9 \)

Now, we can find the ratio of \( y \) to \( x \) for each point:

- For \( (1, 3) \): \( \frac{y}{x} = \frac{3}{1} = 3 \)
- For \( (2, 6) \): \( \frac{y}{x} = \frac{6}{2} = 3 \)
- For \( (3, 9) \): \( \frac{y}{x} = \frac{9}{3} = 3 \)

Since the ratio \(\frac{y}{x}\) is consistently 3 for all the points, we can identify that the relationship is indeed proportional with:

\[
m = 3
\]

Therefore, the equation representing the proportional relationship is:

\[
y = 3x
\]

Please let me know if there are specific details or information you seek, or any additional context about the relationship that I might be missing!