To determine if the given points \((-2, -7)\), \((-1, -3)\), \((0, 2)\), \((1, 8)\), \((2, 15)\), and \((3, 23)\) represent a linear relationship, we need to check if the differences in the \(y\)-values correspond to the differences in the \(x\)-values consistently.
We can calculate the differences in the \(y\)-values (\(y_2 - y_1\)) and the differences in the \(x\)-values (\(x_2 - x_1\)) for consecutive points:
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Between points \((-2, -7)\) and \((-1, -3)\):
- \( x: -1 - (-2) = 1 \)
- \( y: -3 - (-7) = 4 \)
- Slope = \( \frac{y}{x} = \frac{4}{1} = 4 \)
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Between points \((-1, -3)\) and \((0, 2)\):
- \( x: 0 - (-1) = 1 \)
- \( y: 2 - (-3) = 5 \)
- Slope = \( \frac{y}{x} = \frac{5}{1} = 5 \)
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Between points \((0, 2)\) and \((1, 8)\):
- \( x: 1 - 0 = 1 \)
- \( y: 8 - 2 = 6 \)
- Slope = \( \frac{y}{x} = \frac{6}{1} = 6 \)
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Between points \((1, 8)\) and \((2, 15)\):
- \( x: 2 - 1 = 1 \)
- \( y: 15 - 8 = 7 \)
- Slope = \( \frac{y}{x} = \frac{7}{1} = 7 \)
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Between points \((2, 15)\) and \((3, 23)\):
- \( x: 3 - 2 = 1 \)
- \( y: 23 - 15 = 8 \)
- Slope = \( \frac{y}{x} = \frac{8}{1} = 8 \)
The slopes from one pair of points to the next are \(4\), \(5\), \(6\), \(7\), and \(8\). Since the slopes are not constant, these points do not form a linear relationship.
Thus, the relationship among these points is not linear.
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