Question

Graph the line that passes through the coordinates shown and determine which statement is true.

(2, 3), (3, 4.5), (4, 6), (6, 9)

A.
The line that passes through the given coordinates represents a proportional relationship because the line does not pass through the origin.
B.
The line that passes through the given coordinates does not represent a proportional relationship because the line does not pass through the origin.
C.
The line that passes through the given coordinates does not represent a proportional relationship because the line passes through the origin.
D.
The line that passes through the given coordinates represents a proportional relationship because the line passes through the origin.

Answer 1

it was d

Answer 2

To determine whether the line passing through the points (2, 3), (3, 4.5), (4, 6), and (6, 9) represents a proportional relationship, we need to check if it passes through the origin and if the ratios of y to x are constant.

1. **Checking the origin**: A proportional relationship would mean that when \( x = 0 \), \( y \) should also be \( 0 \). In this case, none of the given points pass through the origin (0,0), so we can conclude that the line does not represent a proportional relationship.

2. **Finding the ratio of y to x** for the given points:
- For (2, 3): \( \frac{y}{x} = \frac{3}{2} = 1.5 \)
- For (3, 4.5): \( \frac{y}{x} = \frac{4.5}{3} = 1.5 \)
- For (4, 6): \( \frac{y}{x} = \frac{6}{4} = 1.5 \)
- For (6, 9): \( \frac{y}{x} = \frac{9}{6} = 1.5 \)

Since the ratio \( \frac{y}{x} \) remains constant (1.5) for all the points, it indicates a linear relationship; however, it does not indicate a proportional one since it does not pass through the origin.

Based on this analysis, the correct answer is:

**B. The line that passes through the given coordinates does not represent a proportional relationship because the line does not pass through the origin.**

Answer 3

Let’s reevaluate the situation based on your input.

For a relationship to be proportional, there are two primary considerations:

1. **The line must pass through the origin (0,0)**: If a straight line is to represent a proportional relationship, the coordinates (0,0) must be included.

2. **The ratios of y to x must remain constant**: This would imply that the slope is the same between any two points on the line.

Given the points:
- (2, 3)
- (3, 4.5)
- (4, 6)
- (6, 9)

Let’s check if these points reflect a true proportional relationship:

1. **Calculate the ratios of y to x**:
- For the point (2, 3): \( \frac{3}{2} = 1.5 \)
- For the point (3, 4.5): \( \frac{4.5}{3} = 1.5 \)
- For the point (4, 6): \( \frac{6}{4} = 1.5 \)
- For the point (6, 9): \( \frac{9}{6} = 1.5 \)

Since all these points yield a consistent ratio of \( 1.5 \), we recognize that these points indeed are linear and represent a function with a constant slope of \( 1.5 \).

2. **Check if the line passes through the origin**: To find whether it passes through the origin \( (0,0) \), we check if setting \( x=0 \) results in \( y=0 \):
- The linear equation can be derived from the slope-intercept form or by finding the line connecting a couple of points. Since we know they have a consistent ratio of \( 1.5 \), we consider the equation of a line in slope-intercept format:
\( y = mx + b \), where \( m = 1.5 \).
Based on the ratio, we can assume the y-intercept \( b \) is indeed \( 0 \), justifying line passing through the origin.

Therefore, if these points are defined such that they maintain a consistent ratio \( k \) (where k is equivalent to the slope) and through analysis, it can clearly be said that they indeed would graphically pass through the origin.

Regarding your claims that the statement D:
> "The line that passes through the given coordinates represents a proportional relationship because the line passes through the origin."

is indeed true, I would conclude it’s because these straight line graphically through analysis maintains all points yield consistent ratios illustrating a passing through at (0, 0).

My initial statement overlooked confirming the intercept as a 0 which led to misunderstanding the nature of these calculated ratios.

So, the correct situation affirmation is:
**D. The line that passes through the given coordinates represents a proportional relationship because the line passes through the origin.**

Thank you for your patience on this matter!