The distributive property of integers under subtraction explains how multiplication interacts with subtraction. In essence, it states that multiplying a number by a difference is the same as multiplying the number by each term in the difference and then subtracting the results.
Distributive Property of Multiplication over Subtraction
The distributive property of multiplication over subtraction states:
-
For any integers a, b, and c:
a × (b - c) = a × b - a × c
This means you can "distribute" the multiplication of 'a' across the subtraction (b - c).
Explanation and Example
The distributive property allows you to simplify expressions. Here’s a breakdown:
- Original Expression: a × (b - c)
- Distribution: Multiply 'a' by 'b' and then multiply 'a' by 'c'.
- Result: a × b - a × c
Example:
Let's use the example from the reference:
- a = 1
- b = 2
- c = 1
Applying the distributive property:
- 1 × (2 - 1) = 1 × 2 - 1 × 1
- 1 × (1) = 2 - 1
- 1 = 1
As you can see, both sides of the equation are equal, demonstrating the distributive property.
Practical Application
The distributive property is useful in various scenarios, such as:
- Simplifying algebraic expressions: It helps in expanding expressions like 2(x - 3) into 2x - 6.
- Mental Math: It can be used to break down multiplication problems into easier parts. For example, 5 × 98 can be thought of as 5 × (100 - 2) = 5 × 100 - 5 × 2 = 500 - 10 = 490.
Summary
| Property | Description | Example |
|---|---|---|
| Distributive over Subtraction | a × (b - c) = a × b - a × c | 3 × (5 - 2) = 3 × 5 - 3 × 2 |
| Verification | Shows how multiplication distributes across subtraction. | 3 × (3) = 15 - 6 => 9 = 9 |
