To determine if the relationship shown by the data is linear, we can first examine the pairs of data points:
- (3, 1)
- (7, 2)
- (11, 3)
- (18, 5)
Next, we can calculate the change in the Y-values in relation to the changes in the X-values to check for a constant rate of change.
Calculation of slope:
-
From (3, 1) to (7, 2):
- Change in Y: 2 - 1 = 1
- Change in X: 7 - 3 = 4
- Slope: \( \frac{1}{4} = 0.25 \)
-
From (7, 2) to (11, 3):
- Change in Y: 3 - 2 = 1
- Change in X: 11 - 7 = 4
- Slope: \( \frac{1}{4} = 0.25 \)
-
From (11, 3) to (18, 5):
- Change in Y: 5 - 3 = 2
- Change in X: 18 - 11 = 7
- Slope: \( \frac{2}{7} \approx 0.2857 \)
Analysis:
The first two intervals have a consistent slope of \(0.25\), but the last interval has a different slope of approximately \(0.2857\). This indicates that the relationship between the X and Y values is not linear across all points because the rate of change is not constant.
Conclusion:
Given this analysis, the data does not exhibit a linear relationship throughout. Thus, we cannot successfully model the data with a single linear equation. If you want to explore other types of models (such as polynomial or exponential), please let me know!