To determine if the given points form a linear relationship, we can look at the change in \( y \) values with respect to the change in \( x \) values. A relationship is linear if the slope between any two points remains constant.
Let's analyze the points you provided:
- \( (-2, -7) \)
- \( (-1, -3) \)
- \( (0, 1) \)
- \( (1, 5) \)
- \( (2, 9) \)
- \( (2, 13) \)
First, we will calculate the slope between consecutive points. The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Slope calculations
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Between \( (-2, -7) \) and \( (-1, -3) \): \[ m = \frac{-3 - (-7)}{-1 - (-2)} = \frac{4}{1} = 4 \]
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Between \( (-1, -3) \) and \( (0, 1) \): \[ m = \frac{1 - (-3)}{0 - (-1)} = \frac{4}{1} = 4 \]
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Between \( (0, 1) \) and \( (1, 5) \): \[ m = \frac{5 - 1}{1 - 0} = \frac{4}{1} = 4 \]
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Between \( (1, 5) \) and \( (2, 9) \): \[ m = \frac{9 - 5}{2 - 1} = \frac{4}{1} = 4 \]
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Between \( (2, 9) \) and \( (2, 13) \): \[ m = \frac{13 - 9}{2 - 2} = \frac{4}{0} \] This is undefined, indicating that these two points are vertical and do not contribute to a linear relationship.
Conclusion
All calculated slopes except for the last pair (where \( x \) is the same) are equal to 4, indicating a linear relationship within the main sequence of points. However, the presence of the point \( (2, 13) \) (which shares the same \( x \)-coordinate as \( (2, 9) \) but has a different \( y \)-coordinate) breaks the linearity.
Thus, the set of points does not form a linear relationship overall due to the duplicate \( x \) value with different \( y \) values.
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