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Answer :
Sure! Let's solve the problem step by step to find the expressions that are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex].
### Step 1: Simplify the Original Expression
1. Start with the expression:
[tex]\[
-\frac{3}{4}(32 + 24e - 4f)
\][/tex]
2. Distribute [tex]\(-\frac{3}{4}\)[/tex] across each term in the parentheses:
- [tex]\(-\frac{3}{4} \times 32 = -24\)[/tex]
- [tex]\(-\frac{3}{4} \times 24e = -18e\)[/tex]
- [tex]\(-\frac{3}{4} \times -4f = 3f\)[/tex]
3. Combine the terms to get the simplified expression:
[tex]\[
-18e + 3f - 24
\][/tex]
### Step 2: Compare with the Given Expressions
Now, let's compare this expression [tex]\(-18e + 3f - 24\)[/tex] with each of the given expressions to see which ones are equivalent.
1. Expression: [tex]\(24 + 18e - 3f\)[/tex]
Check: This is not equivalent because the terms and signs differ.
2. Expression: [tex]\(-18e + 3f - 24\)[/tex]
Check: This matches exactly, so it is equivalent.
3. Expression: [tex]\(-24 - 18e + 3f\)[/tex]
Check: Rearranging terms, it matches [tex]\(-18e + 3f - 24\)[/tex], so it is equivalent.
4. Expression: [tex]\(8 - 6e - f\)[/tex]
Check: This does not match either in terms or values.
5. Expression: [tex]\(3(-8 - 6e + f)\)[/tex]
Check: Distributing the 3 gives [tex]\(-24 - 18e + 3f\)[/tex], which matches, so it is equivalent.
### Conclusion
The expressions that are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex] are:
- [tex]\(-18e + 3f - 24\)[/tex]
- [tex]\(-24 - 18e + 3f\)[/tex]
- [tex]\(3(-8 - 6e + f)\)[/tex]
### Step 1: Simplify the Original Expression
1. Start with the expression:
[tex]\[
-\frac{3}{4}(32 + 24e - 4f)
\][/tex]
2. Distribute [tex]\(-\frac{3}{4}\)[/tex] across each term in the parentheses:
- [tex]\(-\frac{3}{4} \times 32 = -24\)[/tex]
- [tex]\(-\frac{3}{4} \times 24e = -18e\)[/tex]
- [tex]\(-\frac{3}{4} \times -4f = 3f\)[/tex]
3. Combine the terms to get the simplified expression:
[tex]\[
-18e + 3f - 24
\][/tex]
### Step 2: Compare with the Given Expressions
Now, let's compare this expression [tex]\(-18e + 3f - 24\)[/tex] with each of the given expressions to see which ones are equivalent.
1. Expression: [tex]\(24 + 18e - 3f\)[/tex]
Check: This is not equivalent because the terms and signs differ.
2. Expression: [tex]\(-18e + 3f - 24\)[/tex]
Check: This matches exactly, so it is equivalent.
3. Expression: [tex]\(-24 - 18e + 3f\)[/tex]
Check: Rearranging terms, it matches [tex]\(-18e + 3f - 24\)[/tex], so it is equivalent.
4. Expression: [tex]\(8 - 6e - f\)[/tex]
Check: This does not match either in terms or values.
5. Expression: [tex]\(3(-8 - 6e + f)\)[/tex]
Check: Distributing the 3 gives [tex]\(-24 - 18e + 3f\)[/tex], which matches, so it is equivalent.
### Conclusion
The expressions that are equivalent to [tex]\(-\frac{3}{4}(32 + 24e - 4f)\)[/tex] are:
- [tex]\(-18e + 3f - 24\)[/tex]
- [tex]\(-24 - 18e + 3f\)[/tex]
- [tex]\(3(-8 - 6e + f)\)[/tex]
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