Worksheet on identifying equivalent algebraic expressions with multiple-choice questions.

Let’s solve each problem step by step.

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Problem 1: Which expressions are equivalent to y + y + y + y + 1 + 1?

We have:
- Four y’s → that’s 4y
- Two 1’s → that’s 2

So total = 4y + 2

Now check options:

a) 7y + 5 → no
b) 2(2y + 1) → distribute: 4y + 2 → YES
c) 4y + 2 → YES
d) 2y + 2y + 2 → combine: 4y + 2 → YES

✔ So b, c, d are correct.

Wait — the question says “which of these expressions” — plural — so likely multiple answers. But let’s double-check if it’s single or multiple choice. Looking at format, probably multiple can be selected. But in many such worksheets, sometimes only one is expected? Let’s read again.

Actually, looking at the wording: “Which of these expressions are equivalent...” — uses “are”, so likely more than one.

But let’s verify each:

Original: y+y+y+y+1+1 = 4y + 2

a) 7y+5 → not same
b) 2(2y+1) = 4y+2 → same
c) 4y+2 → same
d) 2y+2y+2 = 4y+2 → same

So b, c, d are all equivalent.

But wait — maybe the test expects us to pick all that apply? Since it's listed as a), b), c), d) without saying "choose one", we should list all correct ones.

However, in some contexts, they might expect you to select all that match. But since this is homework help, and user didn’t specify, I’ll assume we list all correct letters.

But let’s hold on — perhaps I misread. Let me check problem 2 — it says “not equivalent”, so for problem 1, it’s “equivalent”.

I think for accuracy, I’ll note which ones are correct.

But final answer must be clear. Maybe the worksheet allows multiple selections. Since it’s not specified, I’ll go with mathematically correct.

So for Problem 1: b, c, d

But let’s move to next problems and come back if needed.

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Problem 2: Which expressions are NOT equivalent to 2(4x - 2)?

First, simplify 2(4x - 2) = 8x - 4

Now check each option:

a) 9x → not 8x-4 → NOT equivalent → so this is a candidate for “not equivalent”

b) 5x - 3 → not 8x-4 → NOT equivalent

c) 4(2x - 1) = 8x - 4 → EQUIVALENT → so not this one

d) 8x - 3 - 1 = 8x - 4 → EQUIVALENT

So NOT equivalent are: a and b

Question asks: “which of these expressions are not equivalent” — so again, possibly multiple.

Options: a, b, c, d

c and d are equivalent → so not them

a and b are not → so answer is a and b

But let’s confirm:

a) 9x vs 8x-4 → different
b) 5x-3 vs 8x-4 → different
c) 4(2x-1)=8x-4 → same
d) 8x-3-1=8x-4 → same

Yes.

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Problem 3: Equivalent to 3(3v + 2v) + 3?

First, inside parentheses: 3v + 2v = 5v

Then: 3*(5v) + 3 = 15v + 3

Now check options:

a) 15v + 3 → YES
b) 4(4v) = 16v → NO
c) 3(5v + 1) = 15v + 3 → YES
d) 7v + 8v + 3 = 15v + 3 → YES

So a, c, d are equivalent.

b is not.

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Problem 4: Equivalent to 5a + 2 + 2 - a?

Combine like terms:

5a - a = 4a
2 + 2 = 4
So total: 4a + 4

Check options:

a) 9a(3) = 27a → no
b) 6a + 4 - 2a = (6a - 2a) + 4 = 4a + 4 → YES
c) 8a + 7 → no
d) 4a + 4 → YES

So b and d are correct.

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Problem 5: Equivalent to -4(2m)?

-4 * 2m = -8m

Check options:

a) -3m -5m = -8m → YES
b) -8m → YES
c) 2(-3m - m) = 2*(-4m) = -8m → YES
d) 3m -8m = -5m → NO

So a, b, c are equivalent; d is not.

But question is: “which are equivalent” — so a, b, c

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Problem 6: Which are NOT equivalent to 2c + 2 + 1 + 3c?

Simplify: 2c + 3c = 5c; 2 + 1 = 3 → so 5c + 3

Now check:

a) 5(3c + 1) = 15c + 5 → NOT equivalent → so this is a candidate
b) c + 4c + 3 = 5c + 3 → EQUIVALENT
c) 9c → NOT equivalent
d) 3 + 5c = 5c + 3 → EQUIVALENT

So NOT equivalent: a and c

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Now, compiling final answers per problem.

Since the user did not specify whether to list all correct or just one, but based on standard interpretation and the fact that questions use “which... are” (plural), I will provide all correct choices for each.

But let’s make sure for each:

Problem 1: b, c, d
Problem 2: a, b
Problem 3: a, c, d
Problem 4: b, d
Problem 5: a, b, c
Problem 6: a, c

But wait — in Problem 5, option d is 3m -8m = -5m, which is not -8m, so not equivalent — but the question is “which are equivalent”, so we don’t include d.

Similarly, for others.

Now, for Final Answer, since it’s multiple problems, I need to present answers clearly.

Perhaps list each problem’s answer.

Given the format, I think:

Final Answer:
1) b, c, d
2) a, b
3) a, c, d
4) b, d
5) a, b, c
6) a, c

But let me double-check Problem 4:

Original: 5a + 2 + 2 - a = (5a - a) + (2+2) = 4a + 4

Option b: 6a + 4 - 2a = 4a + 4 → yes
Option d: 4a + 4 → yes
Others no.

Problem 5: -4(2m) = -8m

a) -3m -5m = -8m → yes
b) -8m → yes
c) 2(-3m - m) = 2(-4m) = -8m → yes
d) 3m -8m = -5m → no → so not included

Problem 6: 2c + 2 + 1 + 3c = 5c + 3

a) 5(3c+1)=15c+5 ≠ 5c+3 → not equivalent → so include in "not equivalent"
b) c+4c+3=5c+3 → equivalent → exclude
c) 9c ≠ 5c+3 → not equivalent → include
d) 3+5c=5c+3 → equivalent → exclude

So not equivalent: a, c → correct.

All seem verified.

Now, for Final Answer section, since it’s multiple parts, I’ll write them as above.

But the instruction says: “Provide only the final correct answer” — and since there are six problems, I need to give all.

Perhaps format as:

1) b, c, d
2) a, b
3) a, c, d
4) b, d
5) a, b, c
6) a, c

Yes.

I think that’s accurate.

One last check on Problem 1: original is y+y+y+y+1+1 = 4y+2

b) 2(2y+1)=4y+2 → ok
c) 4y+2 → ok
d) 2y+2y+2=4y+2 → ok
a) 7y+5 → no

Good.

Problem 2: 2(4x-2)=8x-4

a)9x≠8x-4 → not equiv
b)5x-3≠8x-4 → not equiv
c)4(2x-1)=8x-4 → equiv
d)8x-3-1=8x-4 → equiv
So not equiv: a,b → correct.

All good.

Final Answer:
1) b, c, d
2) a, b
3) a, c, d
4) b, d
5) a, b, c
6) a, c