Answer :
To determine which expressions are equivalent to the given expression [tex]\(8.9x + 6.2 + 8.7\)[/tex], we'll follow these steps:
1. Understand the Original Expression:
The original expression is [tex]\(8.9x + 6.2 + 8.7\)[/tex].
2. Rearrange and Simplify:
We can rearrange the terms in any order due to the commutative property of addition:
[tex]\[
= 8.9x + (6.2 + 8.7)
\][/tex]
Simplifying the numerical constant:
[tex]\[
= 8.9x + 14.9
\][/tex]
3. Analyze Each Given Expression:
To check which expressions are equivalent, we need to verify if rearranging or combining terms results in the same expression as [tex]\(8.9x + 14.9\)[/tex]. Let's analyze each option:
- Expression 1: [tex]\(9x + 8 + 9\)[/tex]
- This expression translates to [tex]\(9x + 17\)[/tex] when combined, which is not equal to [tex]\(8.9x + 14.9\)[/tex].
- Expression 2: [tex]\(8.9 + 6.2 + 8.7x\)[/tex]
- This can be rearranged to [tex]\(8.7x + (8.9 + 6.2) = 8.7x + 15.1\)[/tex], which is not equal to [tex]\(8.9x + 14.9\)[/tex].
- Expression 3: [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- This is [tex]\(8.9x + (8.7 + 6.2)\)[/tex], leading to [tex]\(8.9x + 14.9\)[/tex], which is equivalent to the original expression.
- Expression 4: [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- This can be rearranged to [tex]\(8.9x + (8.7 + 6.2)\)[/tex] or [tex]\(8.9x + 14.9\)[/tex], which matches the simplified original expression.
- Expression 5: [tex]\(6.2 + 8.7 + 8.9\)[/tex]
- This simplifies to [tex]\(6.2 + 8.7 + 8.9 = 23.8\)[/tex], which does not include [tex]\(x\)[/tex] and is not equivalent to [tex]\(8.9x + 14.9\)[/tex].
- Expression 6: [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
- This can be rearranged to [tex]\(8.9x + (6.2 + 8.7)\)[/tex], leading to [tex]\(8.9x + 14.9\)[/tex], which is correct.
- Expression 7: [tex]\(8.9 + 6.2x + 8.7\)[/tex]
- This expression becomes [tex]\(8.9 + 8.7 + 6.2x = 17.6 + 6.2x\)[/tex], which is not equivalent to [tex]\(8.9x + 14.9\)[/tex].
Therefore, the expressions that are equivalent to [tex]\(8.9x + 6.2 + 8.7\)[/tex] are:
- Expression 3: [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- Expression 4: [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- Expression 6: [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
These correspond to the indices [3, 4, 6].
1. Understand the Original Expression:
The original expression is [tex]\(8.9x + 6.2 + 8.7\)[/tex].
2. Rearrange and Simplify:
We can rearrange the terms in any order due to the commutative property of addition:
[tex]\[
= 8.9x + (6.2 + 8.7)
\][/tex]
Simplifying the numerical constant:
[tex]\[
= 8.9x + 14.9
\][/tex]
3. Analyze Each Given Expression:
To check which expressions are equivalent, we need to verify if rearranging or combining terms results in the same expression as [tex]\(8.9x + 14.9\)[/tex]. Let's analyze each option:
- Expression 1: [tex]\(9x + 8 + 9\)[/tex]
- This expression translates to [tex]\(9x + 17\)[/tex] when combined, which is not equal to [tex]\(8.9x + 14.9\)[/tex].
- Expression 2: [tex]\(8.9 + 6.2 + 8.7x\)[/tex]
- This can be rearranged to [tex]\(8.7x + (8.9 + 6.2) = 8.7x + 15.1\)[/tex], which is not equal to [tex]\(8.9x + 14.9\)[/tex].
- Expression 3: [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- This is [tex]\(8.9x + (8.7 + 6.2)\)[/tex], leading to [tex]\(8.9x + 14.9\)[/tex], which is equivalent to the original expression.
- Expression 4: [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- This can be rearranged to [tex]\(8.9x + (8.7 + 6.2)\)[/tex] or [tex]\(8.9x + 14.9\)[/tex], which matches the simplified original expression.
- Expression 5: [tex]\(6.2 + 8.7 + 8.9\)[/tex]
- This simplifies to [tex]\(6.2 + 8.7 + 8.9 = 23.8\)[/tex], which does not include [tex]\(x\)[/tex] and is not equivalent to [tex]\(8.9x + 14.9\)[/tex].
- Expression 6: [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
- This can be rearranged to [tex]\(8.9x + (6.2 + 8.7)\)[/tex], leading to [tex]\(8.9x + 14.9\)[/tex], which is correct.
- Expression 7: [tex]\(8.9 + 6.2x + 8.7\)[/tex]
- This expression becomes [tex]\(8.9 + 8.7 + 6.2x = 17.6 + 6.2x\)[/tex], which is not equivalent to [tex]\(8.9x + 14.9\)[/tex].
Therefore, the expressions that are equivalent to [tex]\(8.9x + 6.2 + 8.7\)[/tex] are:
- Expression 3: [tex]\(8.9x + 8.7 + 6.2\)[/tex]
- Expression 4: [tex]\(8.7 + 8.9x + 6.2\)[/tex]
- Expression 6: [tex]\(6.2 + 8.7 + 8.9x\)[/tex]
These correspond to the indices [3, 4, 6].
