# 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.

## Properties of Operations

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 twonumbers, 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 twonumbers, 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

## Solved Examples

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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