Sure! Let's prove that [tex]\((a + b) + c = a + (b + c)\)[/tex] and identify the name of the property. Given the values [tex]\(a = 5\)[/tex], [tex]\(b = -7\)[/tex], and [tex]\(c = 3\)[/tex]:
1. First, we calculate the left-hand side of the equation: [tex]\((a + b) + c\)[/tex].
- Calculate [tex]\(a + b\)[/tex]:
[tex]\[
5 + (-7) = -2
\][/tex]
- Now, add [tex]\(c\)[/tex] to the result:
[tex]\[
-2 + 3 = 1
\][/tex]
Hence, the left-hand side [tex]\((a + b) + c = 1\)[/tex].
2. Next, we calculate the right-hand side of the equation: [tex]\(a + (b + c)\)[/tex].
- Calculate [tex]\(b + c\)[/tex]:
[tex]\[
-7 + 3 = -4
\][/tex]
- Now, add [tex]\(a\)[/tex] to the result:
[tex]\[
5 + (-4) = 1
\][/tex]
Hence, the right-hand side [tex]\(a + (b + c) = 1\)[/tex].
Since [tex]\((a + b) + c = 1\)[/tex] and [tex]\(a + (b + c) = 1\)[/tex], we have shown that:
[tex]\[
(a + b) + c = a + (b + c)
\][/tex]
The property demonstrated here is called the Associative Property of addition. This property states that the way in which numbers are grouped when being added does not change their sum.
