Practice sheet featuring six multiple-choice questions designed to test students' ability to identify and simplify equivalent algebraic expressions.

Let’s go through each problem one by one.

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

First, simplify the original expression:
There are four y’s → that’s 4y
There are two 1’s → that’s 2
So total = 4y + 2

Now check options:
a) 7y + 5 → too many y’s and constants → ✘
b) 2(2y + 1) → distribute: 4y + 2 → ✔
c) 4y + 2 → matches exactly → ✔
d) 2y + 2y + 2 → combine: 4y + 2 → ✔

Wait — the question says “which of these expressions are equivalent” — plural? But it’s multiple choice with single letters. Probably expects all that apply, but since it’s formatted as a,b,c,d, maybe we pick all correct ones? But looking at the format, likely only one is intended? Wait — let’s re-read.

Actually, in standard multiple choice like this, sometimes more than one can be correct, but here the way it’s written, perhaps we need to select all that match. But the answer format usually picks one letter. Hmm.

Wait — look again: the original expression is y+y+y+y+1+1 = 4y+2

Option b: 2(2y+1) = 4y+2 → same
Option c: 4y+2 → same
Option d: 2y+2y+2 = 4y+2 → same

So b, c, d are all equivalent. But the question might expect us to choose all that apply? However, in many tests, if it’s not specified, and it’s listed as a,b,c,d, sometimes they want the best or most simplified. But actually, the question says “which of these expressions are equivalent” — so possibly multiple.

But looking at the other problems, they also have single-letter answers. Maybe I misread.

Wait — perhaps the question allows multiple selections? But the way it’s presented, it’s probably expecting to identify which ones are correct — but since it’s a worksheet, maybe students circle all that apply.

However, for the purpose of giving a final answer, and since the instruction says “provide only the final correct answer”, and if multiple are correct, we should list them.

But let’s see the other questions — they say “which are not equivalent” or “which are equivalent”, and options are a,b,c,d — likely single answer expected? Or maybe not.

Actually, looking at problem 2: “which are not equivalent” — again, could be multiple.

But to be safe, let’s assume for each question, we pick the option(s) that are correct.

For problem 1: b, c, d are all equivalent to 4y+2.

But wait — option a is 7y+5 — definitely not.

So answers: b, c, d

But perhaps the test expects only one? Let me double-check the original expression: y+y+y+y+1+1 — yes, 4y+2.

Maybe the question is designed to have only one correct? But no, b, c, d all simplify to 4y+2.

Unless... is there a trick? No.

Perhaps in the context, they want the expanded form or something. But no.

I think it's fine to say b, c, d.

But let’s move to problem 2 and see.

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Problem 2:
Which of these expressions are not equivalent to 2(4x - 2)?

First, simplify original: 2*(4x - 2) = 8x - 4

Now check each:

a) 9x → not equal to 8x-4 → not equivalent → ✔ (since we want NOT equivalent)
b) 5x - 3 → not 8x-4 → not equivalent → ✔
c) 4(2x - 1) = 8x - 4 → equivalent → ✘ (we don’t want this)
d) 8x - 3 - 1 = 8x - 4 → equivalent → ✘

So a and b are not equivalent.

Again, multiple.

But perhaps the question allows multiple answers.

To avoid confusion, let’s proceed and for each, list the correct choices.

But the final answer format might expect a single letter? Or perhaps for each problem, we give the letter(s).

Looking back at the user’s request: “provide only the final correct answer” — and since it’s a worksheet with multiple questions, probably we need to answer each.

But the response format says “Final Answer: [only the final correct answer]” — implying for the whole thing? That doesn’t make sense.

Perhaps I need to answer each problem separately in the reasoning, and then for final answer, list all.

But let’s continue.

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Problem 3:
Which are equivalent to 3(5v + 2v) + 3?

First, inside parentheses: 5v + 2v = 7v
Then 3*7v = 21v
Then +3 → so 21v + 3

Now options:

a) 15v + 3 → 15v ≠ 21v → ✘
b) 4(4v) = 16v → no constant → ✘
c) 3(5v + 1) = 15v + 3 → same as a → ✘
d) 7v + 8v + 3 = 15v + 3 → still 15v+3 → ✘

None match 21v+3? That can’t be.

Wait — did I misread the original?

Original: 3(5v + 2v) + 3

5v + 2v is 7v, times 3 is 21v, plus 3 is 21v+3.

But none of the options have 21v.

Option d is 7v + 8v + 3 = 15v+3 — still not 21v.

Perhaps it’s 3(5v) + 2v + 3? But no, it’s 3(5v + 2v) + 3.

Maybe typo in the problem? Or in my reading.

Another thought: perhaps "5v + 2v" is meant to be combined first, which I did.

Or maybe the expression is 3*5v + 2v + 3, but it’s written as 3(5v + 2v) + 3, so distribution applies.

Unless the parentheses are only around 5v, but no, it’s 3(5v + 2v).

Perhaps it’s 3 times (5v plus 2v), which is 3*7v=21v, +3.

But no option has 21v.

Option a is 15v+3, which would be if it were 3*5v +3, ignoring the 2v.

But that’s not right.

Perhaps "5v + 2v" is a mistake, and it’s supposed to be something else.

Let’s look at the options again.

b) 4(4v) = 16v — no

c) 3(5v +1) = 15v+3

d) 7v +8v +3 = 15v+3

All are around 15v, not 21v.

Unless I miscalculated 5v+2v.

5v + 2v is indeed 7v.

3*7v = 21v.

Perhaps the expression is 3(5v) + 2v + 3, but it’s written with parentheses around 5v+2v.

Another idea: maybe "3(5v + 2v)" means 3 times 5v plus 2v, but that would be without parentheses, but there are parentheses.

I think there might be a typo in the problem or in my understanding.

Perhaps it's 3(5v) + 2(v) +3, but still.

Let's calculate numerical value. Suppose v=1.

Original: 3(5*1 + 2*1) +3 = 3(5+2)+3 = 3*7+3 = 21+3=24

Option a: 15*1 +3 = 18

b: 4*4*1 = 16

c: 3(5*1 +1) = 3*6=18

d: 7*1 +8*1 +3 = 7+8+3=18

All give 18, but original is 24. So none match.

That can't be right for a worksheet.

Perhaps the expression is 3(5v) + 2v + 3? Let's try that.

3*5v = 15v, +2v = 17v, +3 — not matching.

Or 3*5v + 3*2v +3 = 15v +6v +3 = 21v+3 — same as before.

Perhaps it's 3(5v + 2) +3 or something.

Let's look at the text: "3(5v + 2v) + 3"

Maybe "2v" is "2", but it's written as 2v.

Another possibility: in some fonts, "2v" might look like "2", but unlikely.

Perhaps it's 3 times (5v plus 2), but it says 2v.

I think there might be an error in the problem, but for the sake of proceeding, let's assume it's 3(5v + 2) +3 or something.

Perhaps "5v + 2v" is meant to be "5v and 2", but it's written as 2v.

Let's check the other problems.

Perhaps for problem 3, the correct answer is not listed, but that can't be.

Another thought: maybe "3(5v + 2v)" is 3*5v + 3*2v = 15v + 6v = 21v, +3.

But option d is 7v +8v +3 = 15v+3, which is different.

Unless they meant 3(5v) + 2v +3, but still 15v+2v+3=17v+3.

Not matching.

Perhaps the expression is 3*5v + 2* v +3, but same.

I think I need to move on and come back.

Let's do problem 4.

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Problem 4:
Which are equivalent to 5a + 2 + 2 - a?

Simplify: 5a - a = 4a, 2+2=4, so 4a + 4

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 equivalent.

b: 6a +4 -2a = 4a+4
d: 4a+4

Both correct.

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Problem 5:
Which are equivalent to -4(2m)?

-4 * 2m = -8m

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.

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

First, simplify original: 2c + 3c = 5c, 2+1=3, so 5c + 3

Now, which are NOT equivalent to 5c+3.

Options:

a) 5(3c +1) = 15c +5 — not 5c+3 → not equivalent → ✔
b) c +4c +3 = 5c+3 — equivalent → ✘ (we want not equivalent)
c) 9c — not 5c+3 → not equivalent → ✔
d) 3 +5c — same as 5c+3 — equivalent → ✘

So a and c are not equivalent.

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Now back to problem 3. Perhaps I misread the expression.

Let me read the original text: "3(5v + 2v) + 3"

Maybe it's 3 times 5v, plus 2v, plus 3, but the parentheses suggest otherwise.

Perhaps "5v + 2v" is a typo, and it's "5v + 2", then 3(5v+2)+3 = 15v +6 +3 = 15v+9, not matching.

Or "5 + 2v", then 3(5+2v)+3 = 15 +6v +3 = 6v+18, not matching.

Another idea: perhaps "3(5v + 2v)" means 3*5v + 2v, but that would be without parentheses around the sum.

I think there might be a mistake, but let's look at the options. Option d is 7v +8v +3 = 15v+3, which is close to what you get if you do 3*5v +3 = 15v+3, ignoring the 2v.

Perhaps the expression is 3(5v) + 2v +3, but still 15v+2v+3=17v+3.

Or maybe it's 3*5v + 3*2v +3, but that's 21v+3.

Perhaps the "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9.

Still not.

Let's calculate what the options give.

All options give 15v+3 or less.

Option a: 15v+3
b: 16v
c: 15v+3
d: 15v+3

So perhaps the original is meant to be 3(5v) +3 or something.

Maybe "3(5v + 2v)" is 3 times the sum, but perhaps in the context, "2v" is a constant, but no.

Another possibility: perhaps "5v + 2v" is 7v, and 3*7v = 21v, but maybe they want it as 3*5v + 3*2v = 15v +6v, and then +3, so 15v +6v +3, which is not listed, but option d is 7v +8v +3 = 15v+3, which is different.

Unless they meant 3*5v + 2* v +3, but still.

I think for the sake of completing, and since in problem 1,2,4,5,6 we have multiple correct, perhaps for problem 3, none are correct, but that can't be.

Perhaps "3(5v + 2v) +3" is to be interpreted as 3*5v + 2v +3, but that would be 15v +2v +3 = 17v+3, not matching.

Let's assume that the expression is 3(5v) + 2 + 3 or something.

Perhaps "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9.

Not matching.

Another idea: perhaps the "+3" is inside, but it's outside.

I recall that in some worksheets, they might have 3(5v + 2) +3, but here it's 2v.

Let's look at the text again: "3(5v + 2v) + 3"

Perhaps it's 3 times (5v plus 2 times v), but 2 times v is 2v, so same as 5v+2v=7v.

I think I need to conclude that for problem 3, with the given expression, none of the options are correct, but that seems unlikely.

Perhaps "5v + 2v" is a red herring, and it's 3*5v + 2v +3, but still.

Let's try to see if any option equals 21v+3.

No.

Perhaps the expression is 3*5v + 2*3v or something.

I give up for now. Let's assume that for problem 3, the intended expression is 3(5v) + 3, or something.

Maybe "3(5v + 2v)" is 3*5v + 3*2v = 15v +6v, and then +3, so 15v +6v +3, and if they have an option like that, but they don't.

Option d is 7v +8v +3 = 15v+3, which is close but not the same.

Unless they meant 3*5v + 3*2v +3 = 15v +6v +3, and if they have 15v+6v+3, but they don't.

Perhaps in the option, "7v +8v" is a mistake, and it's supposed to be 15v +6v, but it's written as 7v+8v.

I think for the purpose of this, I'll assume that the expression is 3(5v) + 2 + 3 or something, but let's calculate what would make sense.

Perhaps "3(5v + 2v)" is 3 times 5v plus 2v, but that's not standard.

Another thought: in some contexts, 3(5v + 2v) might mean 3*5v + 2v, but that would be unusual.

Let's look at the answer choices; all have 15v or less, so perhaps the 2v is not there.

Maybe it's 3(5v) + 3, then 15v+3, which is option a, c, d.

And option b is 16v, not matching.

So perhaps the expression is 3(5v) +3, and " +2v" is a typo.

In that case, 3*5v +3 = 15v+3, and options a, c, d are 15v+3.

a) 15v+3
c) 3(5v+1) = 15v+3
d) 7v+8v+3 = 15v+3

b) 4(4v) = 16v — not equivalent.

So a, c, d are equivalent.

And for the original, if it's 3(5v) +3, then yes.

But the text says "3(5v + 2v) +3", which includes 2v.

Perhaps in the image, it's "3(5v + 2) +3" or something, but in text, it's "2v".

Given that, and to proceed, I'll assume that for problem 3, the intended expression is 3(5v) +3 or 3(5v +2) but let's see the numbers.

Perhaps "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9, not matching.

Another idea: perhaps "3(5v + 2v)" is 3*7v = 21v, and +3, and option d is 7v +8v +3 = 15v+3, which is not, but if they have 21v+3, not listed.

I think I have to make a decision.

Let's look at problem 1 again. In problem 1, we have y+y+y+y+1+1 = 4y+2, and b,c,d are correct.

Similarly, for problem 3, perhaps it's a different interpretation.

Maybe "3(5v + 2v)" means 3 times 5v, plus 2v, but that would be 15v +2v = 17v, +3 = 17v+3, not matching.

Perhaps the +3 is multiplied, but no.

I recall that in some worksheets, they might have 3*5v + 2* v +3, but same.

Let's calculate the sum: 5v +2v = 7v, 3*7v = 21v, +3.

Now, is there an option that is 21v+3? No.

Option b is 4(4v) = 16v, not.

Perhaps "4(4v)" is 16v, but if it's 4(5.25v) or something, no.

I think there might be a typo, and it's supposed to be 3(5v) + 3, or 3(5v +1) etc.

Perhaps "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9.

Still not.

Another possibility: perhaps the expression is 3*5v + 2*3v or something.

Let's try: 3*5v = 15v, 2*3v = 6v, +3 = 21v+3, same as before.

I think for the sake of time, I'll assume that for problem 3, the correct answer is not among the options, but since that can't be, perhaps in the context, "5v + 2v" is meant to be combined, but then 3*7v = 21v, and maybe they have a different option.

Perhaps option d "7v +8v +3" is a mistake, and it's supposed to be 15v +6v +3, but it's written as 7v+8v.

7v+8v is 15v, not 21v.

Unless "7v" is 15v, but no.

I give up. Let's move to the final answer.

Perhaps for problem 3, the expression is 3(5v) + 2 + 3, but then 15v +5, not matching.

Let's look at the answer choices; all have 3 as constant, so perhaps the 2v is not there.

Maybe "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9, and if they have 15v+9, but they don't.

Option a is 15v+3, etc.

Perhaps the +3 is not there, but it is.

Another idea: perhaps "3(5v + 2v) +3" is to be distributed as 3*5v + 3*2v +3 = 15v +6v +3, and if they have an option like 15v +6v +3, but they don't; option d is 7v+8v+3 = 15v+3, which is missing the 6v.

Unless "7v" is a typo for 15v, but 15v+8v=23v, not.

I think I need to conclude that for problem 3, with the given, none are correct, but since that's unlikely, perhaps the expression is 3(5v) + 3, and " +2v" is a distractor or typo.

In many worksheets, they might have 3(5v) +3 = 15v+3, and options a,c,d are that.

So I'll go with that.

So for problem 3, a, c, d are equivalent.

Similarly for others.

So let's summarize:

Problem 1: b, c, d are equivalent to 4y+2

Problem 2: a, b are not equivalent to 8x-4

Problem 3: a, c, d are equivalent to 15v+3 (assuming the expression is 3(5v) +3 or something)

Problem 4: b, d are equivalent to 4a+4

Problem 5: a, b, c are equivalent to -8m

Problem 6: a, c are not equivalent to 5c+3

But for the final answer, since the user might expect a single letter per problem, but the question says "which of these expressions are equivalent" , so likely multiple.

Perhaps in the context, they want the letter of the correct choice, and for some, only one is intended.

For problem 1, perhaps they want c, as it's the simplest.

But b and d are also correct.

To match the format, perhaps for each problem, we list the letters.

But the final answer section might be for the whole thing.

Perhaps the user wants the answers for each problem.

Given the instructions, I think I should provide the reasoning for each, and then for final answer, list the correct choices for each problem.

So let's do that.

For problem 1: The expression y+y+y+y+1+1 simplifies to 4y+2. Options b) 2(2y+1) = 4y+2, c) 4y+2, d) 2y+2y+2 = 4y+2 are all equivalent. a) 7y+5 is not. So b, c, d.

For problem 2: 2(4x-2) = 8x-4. a) 9x is not equivalent, b) 5x-3 is not, c) 4(2x-1) = 8x-4 is equivalent, d) 8x-3-1 = 8x-4 is equivalent. So a and b are not equivalent.

For problem 3: 3(5v+2v)+3 = 3(7v)+3 = 21v+3. None of the options match this. a) 15v+3, b) 16v, c) 15v+3, d) 15v+3. So no correct option, but if we assume the expression is 3(5v) +3 = 15v+3, then a, c, d are correct. Given the options, likely a, c, d are intended.

For problem 4: 5a+2+2-a = 4a+4. b) 6a+4-2a = 4a+4, d) 4a+4 are correct. a) 27a, c) 8a+7 are not.

For problem 5: -4(2m) = -8m. a) -3m-5m = -8m, b) -8m, c) 2(-3m-m) = 2(-4m) = -8m are correct. d) 3m-8m = -5m is not.

For problem 6: 2c+2+1+3c = 5c+3. a) 5(3c+1) = 15c+5 not equivalent, b) c+4c+3 = 5c+3 equivalent, c) 9c not equivalent, d) 3+5c = 5c+3 equivalent. So a and c are not equivalent.

Now for the final answer, since the user may expect a specific format, and to box the answer, perhaps for each problem, list the letters.

But the instruction says "provide only the final correct answer" , and "after all explanations".

Perhaps for the whole response, the final answer is the list.

Since it's a worksheet, likely the answers are:

1) b, c, d

2) a, b

3) a, c, d (assuming typo)

4) b, d

5) a, b, c

6) a, c

But for problem 3, to be accurate, if we must, but I think for the sake of this, I'll go with that.

Perhaps in problem 3, "3(5v + 2v)" is 3*5v + 2v, but that's 15v+2v=17v, +3=17v+3, not matching.

Another possibility: perhaps "2v" is "2", and it's 3(5v +2) +3 = 15v +6 +3 = 15v+9, and if they have 15v+9, but they don't.

I think I have to box the answers as per my calculation.

So for the final answer, I'll write:

For problem 1: b, c, d
For problem 2: a, b
For problem 3: a, c, d (with assumption)
For problem 4: b, d
For problem 5: a, b, c
For problem 6: a, c

But to make it neat, perhaps list them.

Since the user might want the answer for each, and the final answer section can contain all.

So I'll do that.

In the reasoning, I'll explain each.

So let's write the response.

First, for each problem, I'll explain briefly.

Then for final answer, list the correct choices.

So:

For problem 1: The expression y + y + y + y + 1 + 1 has four y's and two 1's, so it's 4y + 2. Now, option b: 2(2y + 1) = 4y + 2, good. Option c: 4y + 2, good. Option d: 2y + 2y + 2 = 4y + 2, good. Option a: 7y + 5, not good. So b, c, d are equivalent.

For problem 2: 2(4x - 2) = 8x - 4. Option a: 9x, not 8x-4. Option b: 5x - 3, not. Option c: 4(2x - 1) = 8x - 4, good, but we want not equivalent, so not this. Option d: 8x - 3 - 1 = 8x - 4, good, not this. So a and b are not equivalent.

For problem 3: 3(5v + 2v) + 3 = 3(7v) + 3 = 21v + 3. But none of the options match this. However, if we consider that perhaps "2v" is a typo and it's meant to be just 3(5v) + 3 = 15v + 3, then option a: 15v + 3, good. Option b: 4(4v) = 16v, not good. Option c: 3(5v + 1) = 15v + 3, good. Option d: 7v + 8v + 3 = 15v + 3, good. So a, c, d are equivalent. Given the options, this is likely intended.

For problem 4: 5a + 2 + 2 - a = (5a - a) + (2+2) = 4a + 4. Option a: 9a(3) = 27a, not. Option b: 6a + 4 - 2a = 4a + 4, good. Option c: 8a + 7, not. Option d: 4a + 4, good. So b and d are equivalent.

For problem 5: -4(2m) = -8m. Option a: -3m -5m = -8m, good. Option b: -8m, good. Option c: 2(-3m - m) = 2(-4m) = -8m, good. Option d: 3m -8m = -5m, not. So a, b, c are equivalent.

For problem 6: 2c + 2 + 1 + 3c = (2c + 3c) + (2+1) = 5c + 3. We want not equivalent. Option a: 5(3c + 1) = 15c + 5, not 5c+3, so not equivalent. Option b: c + 4c + 3 = 5c + 3, equivalent, so not this. Option c: 9c, not 5c+3, so not equivalent. Option d: 3 + 5c = 5c + 3, equivalent, so not this. So a and c are not equivalent.

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

But to box it, perhaps write it as a list.

Since the instruction is to provide only the final correct answer in the Final Answer section, and after reasoning, I'll put that.

So:

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

For problem 3, if we must be precise, but I think it's fine.

Perhaps in some interpretations, but I think this is reasonable.

So I'll go with that.