To determine which expression illustrates the associative property of addition, we need to understand what the associative property of addition means. The associative property of addition states that the way in which numbers are grouped when being added does not change the sum. In other words, for any numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[
(a + b) + c = a + (b + c)
\][/tex]
Given the expressions:
1. [tex]\((3 + 19) - 12 = (3 + 12) - 19\)[/tex]
2. [tex]\(3 + (19 - 12) = 3 + (19 + 12)\)[/tex]
3. [tex]\((3 + 19) - 12 = 3 + (19 - 12)\)[/tex]
4. [tex]\(3 + (19 - 12) = 3 - (19 + 12)\)[/tex]
We need to determine if any of these expressions follow the associative property of addition.
### Analyzing Each Expression
1. [tex]\((3 + 19) - 12 = (3 + 12) - 19\)[/tex]
- This expression involves subtraction, not just addition. The associative property of addition cannot be applied here as one side is being subtracted, not simply grouped differently.
2. [tex]\(3 + (19 - 12) = 3 + (19 + 12)\)[/tex]
- On the left side, we have [tex]\(3 + (19 - 12)\)[/tex], which simplifies to [tex]\(3 + 7 = 10\)[/tex].
- On the right side, we have [tex]\(3 + (19 + 12)\)[/tex], which simplifies to [tex]\(3 + 31 = 34\)[/tex].
- Since [tex]\(10 \neq 34\)[/tex], this expression does not hold true and is not an example of the associative property.
3. [tex]\((3 + 19) - 12 = 3 + (19 - 12)\)[/tex]
- On the left side, we have [tex]\((3 + 19) - 12\)[/tex], which simplifies to [tex]\(22 - 12 = 10\)[/tex].
- On the right side, we have [tex]\(3 + (19 - 12)\)[/tex], which simplifies to [tex]\(3 + 7 = 10\)[/tex].
- Both sides simplify to 10, so this equality holds, but it again involves subtraction and is not an example of the associative property which applies to addition only, not subtraction.
4. [tex]\(3 + (19 - 12) = 3 - (19 + 12)\)[/tex]
- On the left side, we have [tex]\(3 + (19 - 12)\)[/tex], which simplifies to [tex]\(3 + 7 = 10\)[/tex].
- On the right side, we have [tex]\(3 - (19 + 12)\)[/tex], which simplifies to [tex]\(3 - 31 = -28\)[/tex].
- Since [tex]\(10 \neq -28\)[/tex], this expression does not hold true and does not illustrate the associative property.
### Conclusion
None of the provided expressions illustrate the associative property of addition.
Thus, the correct response, based on analyzing the given options, is that none of the expressions provided illustrate the associative property of addition. This aligns with the results (0, 0, 0, 0) indicating that none of the expressions met the criteria for demonstrating the associative property.
