Question

identifying equivalent expressions
determine whether each pair of expressions is equivalent. show your work.
1 ( 2(x - y) ) and ( 2x - 2y )
2 ( 4(x + y) ) and ( 4y + 4x )
3 ( 4p + 3c ) and ( (c + 2p)(2) )
4 ( 21q - 7p ) and ( (3q - p)(7) )
5 ( 4(2a - 3v) ) and ( 8a + 6v )
6 ( 8(3x + c) - 1 ) and ( 8c + 24x - 1 )

Explanation:

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Pair 1: $2(x - y)$ and $2x - 2y$

Step1: Distribute the 2

$2(x - y) = 2 \cdot x - 2 \cdot y$

Step2: Simplify the expression

$2 \cdot x - 2 \cdot y = 2x - 2y$

Step3: Compare to the second expression

$2x - 2y = 2x - 2y$

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Pair 2: $4(x + y)$ and $4y + 4x$

Step1: Distribute the 4

$4(x + y) = 4 \cdot x + 4 \cdot y$

Step2: Simplify the expression

$4 \cdot x + 4 \cdot y = 4x + 4y$

Step3: Use commutative property

$4x + 4y = 4y + 4x$

Step4: Compare to the second expression

$4y + 4x = 4y + 4x$

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Pair 3: $4p + 3c$ and $(c + 2p)(2)$

Step1: Distribute the 2

$(c + 2p)(2) = 2 \cdot c + 2 \cdot 2p$

Step2: Simplify the expression

$2 \cdot c + 2 \cdot 2p = 2c + 4p$

Step3: Compare to the first expression

$4p + 3c
eq 2c + 4p$

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Pair 4: $21q - 7p$ and $(3q - p)(7)$

Step1: Distribute the 7

$(3q - p)(7) = 7 \cdot 3q - 7 \cdot p$

Step2: Simplify the expression

$7 \cdot 3q - 7 \cdot p = 21q - 7p$

Step3: Compare to the first expression

$21q - 7p = 21q - 7p$

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Pair 5: $4(2a - 3v)$ and $8a + 6v$

Step1: Distribute the 4

$4(2a - 3v) = 4 \cdot 2a - 4 \cdot 3v$

Step2: Simplify the expression

$4 \cdot 2a - 4 \cdot 3v = 8a - 12v$

Step3: Compare to the second expression

$8a - 12v
eq 8a + 6v$

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Pair 6: $8(3x + c) - 1$ and $8c + 24x - 1$

Step1: Distribute the 8

$8(3x + c) - 1 = 8 \cdot 3x + 8 \cdot c - 1$

Step2: Simplify the expression

$8 \cdot 3x + 8 \cdot c - 1 = 24x + 8c - 1$

Step3: Use commutative property

$24x + 8c - 1 = 8c + 24x - 1$

Step4: Compare to the second expression

$8c + 24x - 1 = 8c + 24x - 1$

Answer:

1. $2(x - y)$ and $2x - 2y$: Equivalent
2. $4(x + y)$ and $4y + 4x$: Equivalent
3. $4p + 3c$ and $(c + 2p)(2)$: Not Equivalent
4. $21q - 7p$ and $(3q - p)(7)$: Equivalent
5. $4(2a - 3v)$ and $8a + 6v$: Not Equivalent
6. $8(3x + c) - 1$ and $8c + 24x - 1$: Equivalent