To determine if the relationship between the \( x \) and \( y \) values is linear, we can check if the change in \( y \) divided by the change in \( x \) (the slope) is constant for any two points in the dataset.
Let's calculate the slope (m) using the first two points:
- Points: (-7, 5) and (-5, 9)
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 5}{-5 - (-7)} = \frac{4}{2} = 2 \]
Now let's check the slope between the next set of points:
- Points: (-5, 9) and (-3, 13)
\[ m = \frac{13 - 9}{-3 - (-5)} = \frac{4}{2} = 2 \]
- Points: (-3, 13) and (-1, 17)
\[ m = \frac{17 - 13}{-1 - (-3)} = \frac{4}{2} = 2 \]
Since the slope is constant (2) for all pairs of points, we can conclude that the relationship is linear.
To model the data with a linear equation in the form \( y = mx + b \), we know the slope \( m \) is 2. We can use one of the points to find the \( y \)-intercept \( b \). Let's use the point (-7, 5):
\[ y = mx + b \implies 5 = 2(-7) + b \]
\[ 5 = -14 + b \implies b = 19 \]
Therefore, the linear equation that models the data is:
\[ y = 2x + 19 \]
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