The Distributive Property
The distributive property states that multiplying two or more addends by a number produces the same result as distributing the multiplier, multiplying each addend separately, and adding the product. The same holds true when we are subtracting instead of adding.
For Sum
- Algebraic application:
The distributive property of multiplication can be algebraically expressed as:
(a + b)\(\times\)c = ac + bc
- Numerical application:
Let us consider three numbers, 1, 9, and 2, to verify the above property.
So, ( 1 + 9 )\(\times\)2 = 1\(\times\)2 + 9 \(\times\)2
= 2 + 18
= 20
For Difference
- Algebraic application:
The distributive property of multiplication can be algebraically expressed as:
( a – b )\(\times\)c = ac – bc
- Numerical application:
Let us consider three numbers, 1, 9, and 2, to verify the above property.
So, ( 9 – 1 )\(\times\)2 = 9\(\times\)2 – 1\(\times\)2
= 18 – 2
= 16
How can we Solve Problems Based on the Distributive Property
The steps to solve problems based on the distributive property are:
Step 1: Multiply (distribute) the number or algebraic term which is outside the parentheses by both the terms or numbers inside the parentheses. Keep in mind that the sign inside the parentheses will change according to the multiplication.
Step 2: Group the like terms and simplify.
Solved Examples
Example 1: Solve the given algebraic expression using the distributive property.
(a) 6(x + 2)
(b) 3(2a – 3)
(c) – 2(3a + 5b)
(d) 4(3 + x + 5)
Solution:
Part (a)
We have: 6(x + 2)
6(x + 2) = 6(x) + 6(2) Using the distributive property
= 6x + 12
Part (b)
We have: 3(2a – 3)
3(2a – 3) = 3(2a) – 3(3) Using the distributive property
= 6a – 9
Part (c)
We have: – 2(3a + 5b)
– 2(3a + 5b) = – 2(3a) + (- 2)(5b) Using the distributive property
= – 6a – 10b
Part (d)
We have: 4(3 + x + 5)
4(3 + x + 5) = 4(3) + 4(x) + 4(5) Using the distributive property
= 12 + 4x + 20
= 4x + 12 + 20 Using the commutative property of addition
Addition
= 4x + 12 + 20 Using the associative property of addition
Addition
= 4x + 32
Example 2: Solve the given algebraic expression using the distributive property by combining like terms.
(a) 2 a + 9 + 4a – 5
(b) a + a + a
(c) – 6x + 3(x – 5y)
Solution:
Part (a)
We have: 2a + 9 + 4a – 5
2a + 9 + 4a – 5
= 2a + 4a + 9 – 5 Using the commutative property of addition
= (2 + 4)a + 9 – 5
= 6a + 4
Part (b)
We have: a + a + a
a + a + a = 1a + 1a + 1a Using the multiplication property of 1
= (1 + 1 + 1)a Using the distributive property
= 3a
Part (c)
We have: – 6x + 3(x – 5y)
= – 6x + 3(x) – 3(5y) Using the distributive property
= – 6x + 3x – 15y
= (- 6 + 3)x – 15y Using the distributive property
= 9x – 15y
Example 3: John is an x-year-old boy. John’s older brother, Dean, is two years his senior. Dean’s aunt, Maria, is three times his age. Write an expression that represents Maria’s age in years and simplify it.
Solution:
We have to create a table to organize the given information and write the required expression that represents a person’s age in years.
Name | Description | Expression |
|---|---|---|
John | He is x year old | x |
Dean | Dean is two years older than John. | x + 2 |
Maria | Maria is three times older than Dean | 3(x + 2) |
Simplify the final expression of Maria to know her age:
We have: 3(x + 2)
= 3(x) + 3(2) Using the distributive property
= 3x + 6
Hence, Maria’s age is 3x + 6.
The four properties of whole number addition are:
- The additive identity property
- The associative property
- The distributive property
- The closure property
Multiplication has the distributive, the commutative, the associative, the removing a common factor and the neutral element properties.
The grouping property of addition states that the order in which addends are grouped has no effect on the result of the addition.