**Identifying equivalent expressions using properties :**

The properties of operations like commutative property of addition, commutative property of multiplication, distributive property etc. which we can use to identify equivalent expressions.

**1. Commutative Property of Addition : **

When adding, changing the order of the numbers does not change the sum.

**Example : **

3 + 4 = 4 + 3

**2. Commutative Property of Multiplication : **

When multiplying, changing the order of the numbers does not change the product.

**Example : **

2 x 4 = 4 x 2

**3. Associative Property of Addition : **

When adding more than twonumbers, the grouping of the numbers does not change the sum.

**Example : **

(3 + 4) + 5 = 3 + (4 + 5)

**4. Associative Property of Multiplication : **

When multiplying more than twonumbers, the grouping of the numbers does not change the product.

**Example : **

(2 × 4) × 3 = 2 × (4 × 3)

**5. Distributive Property : **

Multiplying a number by a sum or difference is the same as multiplying by each number in the sum or difference and then adding or subtracting.

**Examples : **

6(2 + 4) = 6(2) + 6(4)

8(5 - 3) = 8(5) - 8(3)

**6. ****Identity Property of Addition :**

Adding zero to a number does not change its value.

**Examples : **

9 + 0 = 0 + 9 = 9

**7. ****Identity Property of Multiplication :**

Multiplying a number by one does not change its value.

**Examples : **

1 x 7 = 7 x 1 = 7

**Example 1 :**

Use the properties of operations to determine if the expressions are equivalent.

3(x - 2) and 3x - 6

**Solution :**

Let us apply distributive property in the first expression 3(x - 2).

3(x - 2) = 3(x) - 3(2)

3(x - 2) = 3x - 6 ----- (1)

The second expression is

3x - 6 ----- (2)

From (1) and (2), the given two expressions are equivalent.

**Example 2 :**

Use the properties of operations to determine if the expressions are equivalent.

6x - 8 and 2(3x - 5)

**Solution :**

The first expression is

6x - 8 ----- (1)

Let us apply distributive property in the second expression 2(3x - 5).

2(3x - 5) = 2(3x) - 2(5)

2(3x - 5) = 6x - 10 ----- (2)

From (1) and (2), the given two expressions are not equivalent.

**Example 3 :**

Use the properties of operations to determine if the expressions are equivalent.

3x + (y + 2) and (3x + y) + 2

**Solution :**

The first expression is

3x + (y + 2) ----- (1)

Let us apply associative property of addition in the second expression.

(3x + y) + 2 = 3x + (y + 2) ----- (2)

From (1) and (2), the given two expressions are equivalent.

**Example 4 :**

Use the properties of operations to determine if the expressions are equivalent.

2 - 2 + 5x and 5x

**Solution :**

Let us simplify the first expression

2 - 2 + 5x = 5x ----- (1)

The second expression is

5x ----- (2)

From (1) and (2), the given two expressions are equivalent.

**Example 5 :**

Use the properties of operations to determine if the expressions are equivalent.

2 + x and (1/2)(4 + x)

**Solution :**

The first expression is

2 + x ----- (1)

Let us apply distributive property in the second expression (1/2)(4 + x).

(1/2)(4 + x) = (1/2)(4) - (1/2)(x)

(1/2)(4 + x) = 2 - x/2 ----- (2)

From (1) and (2), the given two expressions are not equivalent.

**Example 6 :**

Jamal bought 2 packs of stickers and 8 individual stickers. Use x to represent the number of stickers in a pack of stickers and write an expression to represent the number of stickers Jamal bought. Is the expression equivalent to 2(4+x) ? Check your answer with algebra tile models.

**Solution :**

Given : There are "x" number of stickers in pack.

Then, the number of stickers in 2 packs is

2x

Jamal bought 2 packs of stickers and 8 individual stickers.

The expression which represents the above situation is

2x + 8

Let us apply distributive property in the expression 2x + 8.

2x + 8 = 2(x) + 2(4)

2x + 8 = 2(x + 4)

Let us apply commutative property of addition inside the parenthesis of 2(x + 4)

2(x + 4) = **2(4 + x)**

**Yes, the expression which represents the number of stickers Jamal bought is equivalent to 2(4 +x).**

**Using algebra tiles :**

2(4 + x) ----- (1)

Apply distributive property in the expression 2(4+x)

2(4 + x) = 2(4) + 2(x)

2(4 + x) = 8 + 2x

Apply commutative property of addition in 8 +2x

2(4 + x) = 2x + 8 ----- (2)

Now, let us use algebra tiles to model the two expressions 2(4 + x) and 2x + 8.

The expressions 2(4 + x) has two "x' tiles and eight "1" tiles. The expressions 2x + 8 also has two "x' tiles and eight "1" tiles.

Hence, 2(4 + x) and 2x + 8 are equivalent.

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