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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