To determine if the relationship shown by the data is linear, we can check if the differences in \( Y \) values are consistent for equal changes in \( X \) values. If they are, the data can be modeled with a linear equation of the form:
\[ Y = mX + b \]
where \( m \) is the slope and \( b \) is the y-intercept.
Step 1: Calculate the slope \( m \)
To find the slope \( m \), we can use any two points from the dataset. Let's take the first two points: \((-9, -2)\) and \((-5, -7)\).
The formula for the slope \( m \) between two points \((X_1, Y_1)\) and \((X_2, Y_2)\) is:
\[ m = \frac{Y_2 - Y_1}{X_2 - X_1} \]
Substituting in our points:
- \((X_1, Y_1) = (-9, -2)\)
- \((X_2, Y_2) = (-5, -7)\)
Calculating \( m \):
\[ m = \frac{-7 - (-2)}{-5 - (-9)} = \frac{-7 + 2}{-5 + 9} = \frac{-5}{4} = -\frac{5}{4} \]
Step 2: Find the y-intercept \( b \)
We can use one of the points to find \( b \). Let's use the point \((-9, -2)\):
Substituting into the line equation \( Y = mX + b \):
\[ -2 = -\frac{5}{4}(-9) + b \]
Calculating \(-\frac{5}{4}(-9)\):
\[ -\frac{5}{4} \times -9 = \frac{45}{4} \]
Substituting that back gives:
\[ -2 = \frac{45}{4} + b \]
To isolate \( b \):
\[ b = -2 - \frac{45}{4} = -\frac{8}{4} - \frac{45}{4} = -\frac{53}{4} \]
Step 3: Formulate the equation
Now, we can write the linear equation using the slope and y-intercept we found:
\[ Y = -\frac{5}{4}X - \frac{53}{4} \]
Conclusion
The relationship shown by the data is linear, and it can be modeled with the equation:
\[ Y = -\frac{5}{4}X - \frac{53}{4} \]