Question

unit 13
lesson 12 homework period
write an equivalent expression using the associative, commutative, or distributive
property. state the property used.

1. $7x - \frac{3}{7}(14x - 21)$
2. $(16x + 13) - 2x + 5$
3. $\left(-\frac{2}{3}y + 8\

ight) \cdot 9$

1. $-8\left(\frac{3}{4}x + 2\

ight) + (4 - 2x)$

1. if expressions are equivalent, then they will have the same terms when they are simplified using properties of operatio

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Explanation:

1. Problem 1

Step1: Apply distributive property

$\frac{3}{7}(14x - 21) = \frac{3}{7} \cdot 14x - \frac{3}{7} \cdot 21$

Step2: Simplify the products

$\frac{3}{7} \cdot 14x = 6x$, $\frac{3}{7} \cdot 21 = 9$

Step3: Substitute back and combine terms

$7x - 6x + 9 = x + 9$
Property used: Distributive Property

2. Problem 2

Step1: Remove parentheses (associative)

$16x + 13 - 2x + 5$

Step2: Group like terms (commutative)

$16x - 2x + 13 + 5$

Step3: Combine like terms

$14x + 18$
Property used: Commutative & Associative Properties

3. Problem 3

Step1: Apply distributive property

$-\frac{2}{3}y \cdot 9 + 8 \cdot 9$

Step2: Simplify the products

$-\frac{2}{3} \cdot 9y = -6y$, $8 \cdot 9 = 72$

Step3: Write final expression

$-6y + 72$
Property used: Distributive Property

4. Problem 4

Step1: Apply distributive property

$-8 \cdot \frac{3}{4}x + (-8) \cdot 2 + 4 - 2x$

Step2: Simplify the products

$-8 \cdot \frac{3}{4}x = -6x$, $(-8) \cdot 2 = -16$

Step3: Combine like terms

$-6x - 16 + 4 - 2x = -8x - 12$
Property used: Distributive Property

5. Problem 5

Step1: Analyze equivalent expressions

Equivalent expressions simplify to identical results, so their simplified terms are always the same.

Answer:

1. $x + 9$, Property: Distributive Property
2. $14x + 18$, Property: Commutative & Associative Properties
3. $-6y + 72$, Property: Distributive Property
4. $-8x - 12$, Property: Distributive Property
5. always