Properties of Equality: Applying the Commutative, Associative, and Distributive

In mathematics, certain rules always hold true no matter which numbers we use. These rules, called properties of equality, describe how numbers can be rearranged, grouped, or expanded without changing their value. The three most common are the commutative, associative, and distributive properties. They are the foundation for simplifying expressions and solving equations in arithmetic and algebra.

Commutative Property

We can swap the order of numbers in addition or multiplication without changing the result.

Commutative Property of Addition

???? + ???? = ???? + ????

Example:

Commutative Property of Multiplication

???? × ???? = ???? × ????

Example:

Subtraction has NO Commutative Property

???? − ???? ≠ ???? − ????

Example:

10 − 6 = 4 , but 6 − 10 ≠ −4

Division has NO Commutative Property

???? ÷ ???? ≠ ???? ÷ ????

Example:

10 ÷ 5 = 2,

but

\[\small 5 \div 10 = \frac{5}{10} = \frac{1}{2} = 0.5 \ne 2\]

Associative Property

When we add or multiply numbers, the way we group them with parentheses does not change the result.

Associative Property of Addition

???? + (???? + ????) = (???? + ????) + ????

Example:

Associative Property of Multiplication

???? × (???? × ????) = (???? × ????) × ????

Example:

Subtraction has NO Associative Property

???? − (???? − ????) ≠ (???? − ????) − ????

Example:

10 − (5 − 3) = 10 − 2 = 8

(10 − 5) − 3 = 5 − 3 = 2

Since 8 ≠ 2, 10 − (5 − 3) ≠ (10 − 5) − 3.

Division has NO Associative Property

???? ÷ (???? ÷ ????) ≠ (???? ÷ ????) ÷ ????

Example:

40 ÷ (10 ÷ 2) = 40 ÷5 = 8

(40 ÷ 10) ÷ 2 = 4 ÷ 2 = 2

Since 8 ≠ 2, 40 ÷ (10 ÷ 2) ≠ (40 ÷ 10) ÷ 2.

Distributive Property

It connects two operations. It allows us to distribute one operation over another. The most common type is multiplication over addition or subtraction, but there are also cases where division works in a distributive way.

Distributive Property of Multiplication over Addition

Left-Distributivity (number outside on the left)

???? × (???? + ????) = (???? × ????) + (???? × ????)

Example:

Right-Distributivity (number outside on the right)

(???? + ????) × ???? = (???? × ????) + (???? × ????)

Example:

Distributive Property of Multiplication over Subtraction

Left-Distributivity (number outside on the left)

???? × (???? − ????) = (???? × ????) − (???? × ????)

Example:

Right-Distributivity (number outside on the right)

(???? − ????) × ???? = (???? × ????) − (???? × ????)

Example:

Summary

???? × (???? ± ????) = (???? × ????) ± (???? × ????)
(???? ± ????) × ???? = (???? × ????) ± (???? × ????)

Distributive Property of Division over Addition

Right-Distributivity

(???? + ????) ÷ ???? = (???? ÷ ????) + (???? ÷ ????)

Example:

Division over Addition has NO Left-Distributive Property 

???? ÷ (???? + ????) ≠ (???? ÷ ????) + (???? ÷ ????)

Example:

24 ÷ (2 + 4) = 24 ÷ 6 = 4

24 ÷ 2 + 24 ÷ 4 = 12 + 6 = 18 ≠ 4

Distributive Property of Division over Subtraction

Right-Distributivity

(???? − ????) ÷ ???? = (???? ÷ ????) − (???? ÷ ????)

Example:

Division over Subtraction has NO Left-Distributive Property

???? ÷ (???? − ????) ≠ (???? ÷ ????) − (???? ÷ ????)

Example:

24 ÷ (4 − 2) = 24 ÷ 2 = 12
24 ÷ 4 − 24 ÷ 2 = 6 − 12 ≠ 12

Summary

(???? ± ????) ÷ ???? = (???? ÷ ????) ± (???? ÷ ????)

Example Problems: Commutative, Associative, and Distributive Property

Example 1

Use the Distributive Property to calculate 19 × 8 + 19 × 2.

Solution:

Distributive Property of Multiplication over Addition: ???? × (???? + ????) = (???? × ????) + (???? × ????)
∴ 19 × 8 + 19 × 2 = 19 × (8 + 2) = 19 × 10 = 190
The final result is 190

Example 2

Which expression is equivalent to (4·????) + 7 ?
A. 7(4·????)
B. 7 + (????·4)
C. 4 + 7·????
D. 4(???? + 7)

Solution:

Commutative property of multiplication: ???? × ???? = ???? × ????
∴ 4·???? = ????·4
Commutative property of addition: ???? + ???? = ???? + ????
∴ (4·????) + 7 = 7 + (4·????)
The answer is B

Summary: Commutative, Associative, and Distributive Property.

• Commutative Property
  • ???? + ???? = ???? + ????
  • ???? × ???? = ???? × ????
• Associative Property
  • ???? + (???? + ????) = (???? + ????) + ????
  • ???? × (???? × ????) = (???? × ????) × ????
• Distributive Property
  • ???? × (???? ± ????) = (???? × ????) ± (???? × ????)
  • (???? ± ????) × ???? = (???? × ????) ± (???? × ????)
  • (???? ± ????) ÷ ???? = (???? ÷ ????) ± (???? ÷ ????)

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