To determine if the relationship between the X and Y values is linear, we can first check if the differences in Y correspond linearly with the differences in X.
Let's calculate the differences for each data pair:
- For X = -9 and Y = -2
- For X = -5 and Y = -7
- For X = -1 and Y = -12
- For X = 3 and Y = -17
Now, we can find the changes in Y and X.
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From X = -9 to X = -5, Y changes from -2 to -7: \[ \Delta Y = -7 - (-2) = -5 \] \[ \Delta X = -5 - (-9) = 4 \]
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From X = -5 to X = -1, Y changes from -7 to -12: \[ \Delta Y = -12 - (-7) = -5 \] \[ \Delta X = -1 - (-5) = 4 \]
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From X = -1 to X = 3, Y changes from -12 to -17: \[ \Delta Y = -17 - (-12) = -5 \] \[ \Delta X = 3 - (-1) = 4 \]
Now let's summarize the changes:
- For each increment in X (4 units), Y consistently decreases by 5.
To determine if we have a linear relationship, we can calculate the slope \( m \) of the line using the formula: \[ m = \frac{\Delta Y}{\Delta X} = \frac{-5}{4} = -\frac{5}{4} \]
Since the change between each set of X and Y values yields the same slope, we can conclude that the relationship is indeed linear.
Now, to find the linear equation in the slope-intercept form \( Y = mX + b \): We need to find \( b \), the Y-intercept.
Using one of the points (for example, when \( X = -9 \) and \( Y = -2 \)): \[ -2 = -\frac{5}{4}(-9) + b \] \[ -2 = \frac{45}{4} + b \] \[ b = -2 - \frac{45}{4} = -\frac{8}{4} - \frac{45}{4} = -\frac{53}{4} \]
Thus, the equation modeling the relationship is: \[ Y = -\frac{5}{4}X - \frac{53}{4} \]
So, the relationship is linear, and the equation is: \[ Y = -\frac{5}{4}X - \frac{53}{4} \]