To determine if the relationship shown by the data is linear, we can check if the change in \(y\) is consistent with the change in \(x\). We need to calculate the differences in \(y\) for each change in \(x\).
Given the points:
- (-7, 5)
- (-5, 9)
- (-3, 13)
- (-1, 17)
Now let's calculate the differences:
-
From (-7, 5) to (-5, 9):
- Change in \(x\): \(-5 - (-7) = 2\)
- Change in \(y\): \(9 - 5 = 4\)
-
From (-5, 9) to (-3, 13):
- Change in \(x\): \(-3 - (-5) = 2\)
- Change in \(y\): \(13 - 9 = 4\)
-
From (-3, 13) to (-1, 17):
- Change in \(x\): \(-1 - (-3) = 2\)
- Change in \(y\): \(17 - 13 = 4\)
The change in \(y\) is consistently 4 for a change in \(x\) of 2, which suggests that the data is linear.
Now, we can find the slope \(m\) of the line: \[ m = \frac{\Delta y}{\Delta x} = \frac{4}{2} = 2 \]
Now, we can use the slope and one of the points to find the equation of the line in slope-intercept form \(y = mx + b\). We can use the point (-7, 5) for this calculation.
Substituting \(m = 2\): \[ y = 2x + b \]
Now, plug in the point (-7, 5): \[ 5 = 2(-7) + b \] \[ 5 = -14 + b \] \[ b = 5 + 14 = 19 \]
Thus, the equation of the linear model that represents the data is: \[ y = 2x + 19 \]
In conclusion, the relationship is linear, and the linear model is given by the equation \(y = 2x + 19\).