To determine which data set can reasonably be modeled by a linear relationship, we need to analyze the patterns in each set of coordinates. A linear relationship suggests that, as one variable increases, the other variable changes at a constant rate.
Analysis of Each Data Set:
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Data Set A: (0,2), (1,4), (2,8), (3,16), (4,32), (5,64)
- Here, the y-values (2, 4, 8, 16, 32, 64) show exponential growth rather than a linear pattern. The differences between consecutive y-values are increasing, indicating a nonlinear relationship.
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Data Set B: (3,4), (4,6), (5,8), (6,10), (7,12)
- This data set shows a consistent increase in y-values as x-values increase. The differences between y (6-4=2, 8-6=2, 10-8=2, 12-10=2) are constant, indicating a linear relationship.
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Data Set C: (0,1), (3,3), (6,9), (9,27), (12,81)
- The y-values show rapid growth (not linear) as the x-values increase. The differences between y-values are not consistent, demonstrating a nonlinear relationship.
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Data Set D: (2,4), (−1,2), (7,16), (−4,32), (1,64)
- The y-values do not follow a consistent pattern as x-values change. Some y-values increase and others decrease significantly, indicating a nonlinear relationship.
Conclusion:
The best candidate for a linear relationship among the data sets is Data Set B, since it exhibits consistent increments in y-values with corresponding increases in x-values.
Answer: Data Set B