Equivalent Algebraic Expressions Worksheets - Free Printable

Let's solve each of these problems step by step and explain the reasoning behind each answer.

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1) Which of these expressions are equivalent to $ y + y + y + y + 1 + 17 $?

First, simplify the given expression:

$$
y + y + y + y + 1 + 17 = 4y + 18
$$

Now check each option:

a) $ 7y + 5 $ → Not equivalent
b) $ 2(2y + 1) = 4y + 2 $ → Not equivalent
c) $ 4y + 2 $ → Not equivalent
d) $ 2y + 2y + 2 = 4y + 2 $ → Not equivalent

Wait — none of them match $ 4y + 18 $? Let's double-check.

But wait: Is there a typo? Let’s recheck the original:

> $ y + y + y + y + 1 + 17 $

That’s four y’s → $ 4y $, and $ 1 + 17 = 18 $, so $ 4y + 18 $

But none of the options is $ 4y + 18 $. Let’s look again at the choices:

a) $ 7y + 5 $ → No
b) $ 2(2y + 1) = 4y + 2 $ → No
c) $ 4y + 2 $ → No
d) $ 2y + 2y + 2 = 4y + 2 $ → No

None match $ 4y + 18 $. But maybe the original was meant to be $ y + y + y + y + 1 + 1 $? That would be $ 4y + 2 $. Then d) or c) would work.

But as written: $ y + y + y + y + 1 + 17 = 4y + 18 $

So none of the options are equivalent. But that can't be right for a multiple-choice question.

Wait — perhaps it's a typo in the problem or options.

But let’s assume the problem is correct. Then no option is correct, but since this is likely an error, let’s consider if maybe the expression was meant to be:

> $ y + y + y + y + 1 + 1 $ → $ 4y + 2 $

Then:

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

So b, c, d would be equivalent.

But based on the given: $ y + y + y + y + 1 + 17 = 4y + 18 $

No option equals $ 4y + 18 $

So unless there's a mistake in the problem, none are equivalent.

But since this is unlikely, perhaps the "17" is a typo and should be "1"?

Alternatively, maybe the expression is $ y + y + y + y + 1 + 1 $, which is $ 4y + 2 $

Then:
- b) $ 2(2y + 1) = 4y + 2 $
- c) $ 4y + 2 $
- d) $ 2y + 2y + 2 = 4y + 2 $

So b, c, d are equivalent.

But since the image says +1 + 17, it’s $ +18 $

So unless one of the options is $ 4y + 18 $, none match.

But none do.

So either:
- There's a typo in the problem, or
- We must go with what's written.

But let’s move on and come back.

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

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

Now check each option:

a) $ 9x $ → Not equal to $ 8x - 4 $ → Not equivalent

b) $ 5x - 3 $ → Not equal → Not equivalent

c) $ 4(2x - 1) = 8x - 4 $ → Equivalent

d) $ 8x - 3 - 1 = 8x - 4 $ → Equivalent

So the ones that are not equivalent are:

→ a) and b)

Answer: a and b

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3) Which of these expressions are equivalent to $ 3(3v + 2v) + 3 $?

Simplify inside first:

$ 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.

Answer: a, c, d

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4) Which of these expressions are equivalent to $ 5a + 2 + 2 - a $?

Simplify:

$ 5a - a + 2 + 2 = 4a + 4 $

Now check:

a) $ 9a(3) = 27a $ → No
b) $ 6a + 4 - 2a = 4a + 4 $ → Yes ✔
c) $ 8a + 7 $ → No
d) $ 4a + 4 $ → Yes ✔

So b and d are equivalent.

Answer: b, d

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5) Which of these expressions are 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.

Answer: a, b, c

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6) Which of these expressions are not equivalent to $ 2c + 2 + 1 + 3c $?

Simplify:

$ 2c + 3c + 2 + 1 = 5c + 3 $

Now check:

a) $ 5(3c + 1) = 15c + 5 $ → No → Not equivalent
b) $ c + 4c + 3 = 5c + 3 $ → Yes
c) $ 9c $ → No → Not equivalent
d) $ 3 + 5c $ → Same as $ 5c + 3 $ → Yes

So not equivalent: a and c

Answer: a and c

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Now back to Question 1

We had:

> $ y + y + y + y + 1 + 17 = 4y + 18 $

Options:

a) $ 7y + 5 $ → No
b) $ 2(2y + 1) = 4y + 2 $ → No
c) $ 4y + 2 $ → No
d) $ 2y + 2y + 2 = 4y + 2 $ → No

None are $ 4y + 18 $

But perhaps the original expression was meant to be:

> $ y + y + y + y + 1 + 1 $ → $ 4y + 2 $

Then:
- b) $ 2(2y + 1) = 4y + 2 $
- c) $ 4y + 2 $
- d) $ 2y + 2y + 2 = 4y + 2 $

So b, c, d would be correct.

Alternatively, maybe the "+17" is a typo and should be "+1", but even then, only if it's two 1s.

But as written, none of the options match.

However, looking at the pattern of the other questions, it's likely that the intended expression was:

> $ y + y + y + y + 1 + 1 = 4y + 2 $

So we'll assume that.

Thus, b, c, d are equivalent to $ 4y + 2 $

But only if the original expression is $ 4y + 2 $

But as written: $ +1 + 17 = 18 $

So unless there's a typo, no option is correct

But since the question asks “which of these expressions are equivalent”, and expects answers, it's likely that the expression was meant to be:

> $ y + y + y + y + 1 + 1 $ → $ 4y + 2 $

So final decision: Assume typo, and the expression is $ 4y + 2 $

Then:

b) $ 2(2y + 1) = 4y + 2 $
c) $ 4y + 2 $
d) $ 2y + 2y + 2 = 4y + 2 $

So b, c, d are equivalent.

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✔ Final Answers:

1) b, c, d (assuming typo: +1 +1 instead of +1 +17)
2) a and b
3) a, c, d
4) b and d
5) a, b, c
6) a and c

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📝 Explanation Summary:

- Always simplify the original expression first.
- Use distributive property: $ a(b + c) = ab + ac $
- Combine like terms: $ 5a - a = 4a $
- Be careful with signs: $ -4(2m) = -8m $
- Check each option by simplifying both sides.

Let me know if you want the corrected version of Question 1 with a typo fix!