To determine the correct statement about the line that passes through the provided coordinates (1, 2), (3, 4), (5, 6), and (7, 8), we need to analyze the relationship represented by these points.
Step 1: Check if the points are on a straight line.
We can observe that the differences in y-coordinates and x-coordinates suggest a linear relationship. The changes between the points can be calculated:
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The difference between (1, 2) and (3, 4):
- Change in x: \(3 - 1 = 2\)
- Change in y: \(4 - 2 = 2\)
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The difference between (3, 4) and (5, 6):
- Change in x: \(5 - 3 = 2\)
- Change in y: \(6 - 4 = 2\)
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The difference between (5, 6) and (7, 8):
- Change in x: \(7 - 5 = 2\)
- Change in y: \(8 - 6 = 2\)
Since the change in y and change in x remains consistent (2/2 = 1), these points lie on a straight line.
Step 2: Determine if the line passes through the origin.
To see if the line passes through the origin (0, 0), we need to check if the equation obtained from the line can be satisfied when \( x = 0 \).
To find the slope:
- Use any two points, e.g., (1, 2) and (3, 4):
- Slope \( m = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1\).
Using point-slope form to find the line's equation:
- Starting from point (1,2):
- \(y - 2 = 1(x - 1)\)
- \(y - 2 = x - 1\)
- \(y = x + 1\)
Step 3: Check the y-intercept.
The y-intercept of the line is 1, which means the line does not pass through the origin.
Conclusion:
From this analysis, we can deduce:
- The points form a linear relationship.
- The line does not pass through the origin, confirming that it does not represent a proportional relationship.
Thus, the correct statement is:
A. The line that passes through the given coordinates does not represent a proportional relationship because the line does not pass through the origin.
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