Worksheet titled "Equivalent Algebraic Expressions" featuring six multiple-choice questions testing understanding of algebraic equivalence.

Problem: Equivalent Algebraic Expressions

We are tasked with identifying which expressions are equivalent to the given algebraic expressions. Let's solve each problem step by step.

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#### 1) Which of these expressions are equivalent to \( y + y + y + y + 1 \)?

Given expression:
\[ y + y + y + y + 1 \]

Simplify the given expression:
Combine like terms:
\[ y + y + y + y = 4y \]
So, the expression becomes:
\[ 4y + 1 \]

Check each option:
- a) \( 7y + 5 \)
This is not equivalent to \( 4y + 1 \).

- b) \( 2(2y + 1) \)
Simplify:
\[ 2(2y + 1) = 4y + 2 \]
This is not equivalent to \( 4y + 1 \).

- c) \( 4y + 2 \)
This is not equivalent to \( 4y + 1 \).

- d) \( 2y + 2y + 2 \)
Simplify:
\[ 2y + 2y + 2 = 4y + 2 \]
This is not equivalent to \( 4y + 1 \).

Correct Answer: None of the options are equivalent to \( 4y + 1 \).
However, if there was an option like \( 4y + 1 \), it would be the correct answer.

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

Given expression:
\[ 2(4x - 2) \]

Simplify the given expression:
Distribute the 2:
\[ 2(4x - 2) = 8x - 4 \]

Check each option:
- a) \( 9x \)
This is not equivalent to \( 8x - 4 \).

- b) \( 5x - 3 \)
This is not equivalent to \( 8x - 4 \).

- c) \( 4(2x - 1) \)
Simplify:
\[ 4(2x - 1) = 8x - 4 \]
This is equivalent to \( 8x - 4 \).

- d) \( 8x - 3 - 1 \)
Simplify:
\[ 8x - 3 - 1 = 8x - 4 \]
This is equivalent to \( 8x - 4 \).

Expressions that are not equivalent:
- \( 9x \)
- \( 5x - 3 \)

Correct Answer: a) \( 9x \) and b) \( 5x - 3 \).

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

Given expression:
\[ 3(3v + 2v) + 3 \]

Simplify the given expression:
First, combine like terms inside the parentheses:
\[ 3v + 2v = 5v \]
So, the expression becomes:
\[ 3(5v) + 3 = 15v + 3 \]

Check each option:
- a) \( 15v + 3 \)
This is equivalent to \( 15v + 3 \).

- b) \( 4(4v) \)
Simplify:
\[ 4(4v) = 16v \]
This is not equivalent to \( 15v + 3 \).

- c) \( 3(5v + 1) \)
Simplify:
\[ 3(5v + 1) = 15v + 3 \]
This is equivalent to \( 15v + 3 \).

- d) \( 7v + 8v + 3 \)
Simplify:
\[ 7v + 8v + 3 = 15v + 3 \]
This is equivalent to \( 15v + 3 \).

Correct Answer: a) \( 15v + 3 \), c) \( 3(5v + 1) \), d) \( 7v + 8v + 3 \).

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#### 4) Which of these expressions are equivalent to \( 5a + 2 + 2 - a \)?

Given expression:
\[ 5a + 2 + 2 - a \]

Simplify the given expression:
Combine like terms:
\[ 5a - a = 4a \]
\[ 2 + 2 = 4 \]
So, the expression becomes:
\[ 4a + 4 \]

Check each option:
- a) \( 9a(3) \)
Simplify:
\[ 9a(3) = 27a \]
This is not equivalent to \( 4a + 4 \).

- b) \( 6a + 4 - 2a \)
Simplify:
\[ 6a - 2a = 4a \]
So, the expression becomes:
\[ 4a + 4 \]
This is equivalent to \( 4a + 4 \).

- c) \( 8a + 7 \)
This is not equivalent to \( 4a + 4 \).

- d) \( 4a + 4 \)
This is equivalent to \( 4a + 4 \).

Correct Answer: b) \( 6a + 4 - 2a \) and d) \( 4a + 4 \).

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#### 5) Which of these expressions are equivalent to \( -4(2m) \)?

Given expression:
\[ -4(2m) \]

Simplify the given expression:
Distribute the \(-4\):
\[ -4(2m) = -8m \]

Check each option:
- a) \( -3m - 5m \)
Simplify:
\[ -3m - 5m = -8m \]
This is equivalent to \( -8m \).

- b) \( -8m \)
This is equivalent to \( -8m \).

- c) \( 2(-3m - m) \)
Simplify:
\[ -3m - m = -4m \]
So, the expression becomes:
\[ 2(-4m) = -8m \]
This is equivalent to \( -8m \).

- d) \( 3m - 8m \)
Simplify:
\[ 3m - 8m = -5m \]
This is not equivalent to \( -8m \).

Correct Answer: a) \( -3m - 5m \), b) \( -8m \), c) \( 2(-3m - m) \).

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

Given expression:
\[ 2c + 2 + 1 + 3c \]

Simplify the given expression:
Combine like terms:
\[ 2c + 3c = 5c \]
\[ 2 + 1 = 3 \]
So, the expression becomes:
\[ 5c + 3 \]

Check each option:
- a) \( 5(3c + 1) \)
Simplify:
\[ 5(3c + 1) = 15c + 5 \]
This is not equivalent to \( 5c + 3 \).

- b) \( c + 4c + 3 \)
Simplify:
\[ c + 4c = 5c \]
So, the expression becomes:
\[ 5c + 3 \]
This is equivalent to \( 5c + 3 \).

- c) \( 9c \)
This is not equivalent to \( 5c + 3 \).

- d) \( 3 + 5c \)
Rearrange:
\[ 3 + 5c = 5c + 3 \]
This is equivalent to \( 5c + 3 \).

Expressions that are not equivalent:
- \( 5(3c + 1) \)
- \( 9c \)

Correct Answer: a) \( 5(3c + 1) \) and c) \( 9c \).

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Final Answers:

1. None (if no correct option is provided)
2. a) \( 9x \), b) \( 5x - 3 \)
3. a) \( 15v + 3 \), c) \( 3(5v + 1) \), d) \( 7v + 8v + 3 \)
4. b) \( 6a + 4 - 2a \), d) \( 4a + 4 \)
5. a) \( -3m - 5m \), b) \( -8m \), c) \( 2(-3m - m) \)
6. a) \( 5(3c + 1) \), c) \( 9c \)

\boxed{2. \text{a, b}, 3. \text{a, c, d}, 4. \text{b, d}, 5. \text{a, b, c}, 6. \text{a, c}}