To determine the relationships between the distance and the sum and the distance and the difference, we'll analyze the given data in detail.
Here is the table provided:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
a & b & a + b & a - b & \text{Distance} \\
\hline
1 & 2 & 3 & -1 & 1 \, \text{unit} \\
\hline
4 & -1 & 3 & 5 & 5 \, \text{units} \\
\hline
-6 & -3 & -9 & -3 & 3 \, \text{units} \\
\hline
\end{array}
\][/tex]
### Relationship between Distance and Sum
Let's examine the relationship between the 'sum' ([tex]\(a + b\)[/tex]) and the 'distance':
- For [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex]:
- [tex]\(a + b = 3\)[/tex]
- Distance = 1 unit
- Relationship of sum and distance: (3, 1 unit)
- For [tex]\(a = 4\)[/tex] and [tex]\(b = -1\)[/tex]:
- [tex]\(a + b = 3\)[/tex]
- Distance = 5 units
- Relationship of sum and distance: (3, 5 units)
- For [tex]\(a = -6\)[/tex] and [tex]\(b = -3\)[/tex]:
- [tex]\(a + b = -9\)[/tex]
- Distance = 3 units
- Relationship of sum and distance: (-9, 3 units)
From these relationships, we observe that:
- When the sum is 3, the distances are 1 unit and 5 units.
- When the sum is -9, the distance is 3 units.
There doesn't appear to be a direct or clear relationship between the distance and the sum, as different sums can correspond to different distances.
### Relationship between Distance and Difference
Let’s now examine the relationship between the 'difference' ([tex]\(a - b\)[/tex]) and the 'distance':
- For [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex]:
- [tex]\(a - b = -1\)[/tex]
- Distance = 1 unit
- Relationship of difference and distance: (-1, 1 unit)
- For [tex]\(a = 4\)[/tex] and [tex]\(b = -1\)[/tex]:
- [tex]\(a - b = 5\)[/tex]
- Distance = 5 units
- Relationship of difference and distance: (5, 5 units)
- For [tex]\(a = -6\)[/tex] and [tex]\(b = -3\)[/tex]:
- [tex]\(a - b = -3\)[/tex]
- Distance = 3 units
- Relationship of difference and distance: (-3, 3 units)
From these relationships, we observe that:
- When the difference is -1, the distance is 1 unit.
- When the difference is 5, the distance is 5 units.
- When the difference is -3, the distance is 3 units.
The analysis shows a more direct match between each difference and its corresponding distance. This indicates that there is likely a clearer and more consistent relationship between the difference and the distance compared to the relationship between the sum and the distance.
### Conclusion
Based on the provided data:
- There is no clear or direct relationship between the sum ([tex]\(a + b\)[/tex]) and the distance.
- There appears to be a more consistent and direct relationship between the difference ([tex]\(a - b\)[/tex]) and the distance.
