To determine if the given points \((-2,-17)\), \((-1,-11)\), \((0,-5)\), \((1,1)\), \((2,7)\), and \((3,13)\) represent a linear relationship, we can check if the slope between each pair of consecutive points is constant.
The formula for the slope \(m\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Now, let's calculate the slope between each pair of consecutive points:
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Between points \((-2, -17)\) and \((-1, -11)\): \[ m_1 = \frac{-11 - (-17)}{-1 - (-2)} = \frac{-11 + 17}{-1 + 2} = \frac{6}{1} = 6 \]
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Between points \((-1, -11)\) and \((0, -5)\): \[ m_2 = \frac{-5 - (-11)}{0 - (-1)} = \frac{-5 + 11}{0 + 1} = \frac{6}{1} = 6 \]
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Between points \((0, -5)\) and \((1, 1)\): \[ m_3 = \frac{1 - (-5)}{1 - 0} = \frac{1 + 5}{1} = \frac{6}{1} = 6 \]
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Between points \((1, 1)\) and \((2, 7)\): \[ m_4 = \frac{7 - 1}{2 - 1} = \frac{6}{1} = 6 \]
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Between points \((2, 7)\) and \((3, 13)\): \[ m_5 = \frac{13 - 7}{3 - 2} = \frac{6}{1} = 6 \]
Since the slope is constant at \(6\) for all consecutive pairs of points, we conclude that the relationship represented by the given points is indeed linear.
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