To determine if the given input-output pairs represent a linear relationship, we can look for a consistent change in the output values that corresponds to a consistent change in the input values. In other words, in a linear relationship, the change in output (y) should be proportional to the change in input (x).
Let's take a look at the provided data:
Input (x) | Output (y)
------------------------
-2 | -7
-1 | -3
0 | 2
1 | 8
2 | 15
3 | 23
Now, let's calculate the differences in output as we move through the inputs:
-
From -2 to -1:
- Change in x = -1 - (-2) = 1
- Change in y = -3 - (-7) = 4
-
From -1 to 0:
- Change in x = 0 - (-1) = 1
- Change in y = 2 - (-3) = 5
-
From 0 to 1:
- Change in x = 1 - 0 = 1
- Change in y = 8 - 2 = 6
-
From 1 to 2:
- Change in x = 2 - 1 = 1
- Change in y = 15 - 8 = 7
-
From 2 to 3:
- Change in x = 3 - 2 = 1
- Change in y = 23 - 15 = 8
Now, let's summarize the changes in y:
- From -7 to -3: increase of 4
- From -3 to 2: increase of 5
- From 2 to 8: increase of 6
- From 8 to 15: increase of 7
- From 15 to 23: increase of 8
We can see that the changes in y are not consistent. They increase by 4, 5, 6, 7, and 8, which means the output does not change at a constant rate relative to the input. This means the relationship is not linear.
Thus, the answer is these pairs do not represent a linear relationship.