The distributive property is a fundamental principle in algebra that relates to the multiplication of a number by a sum or difference. It states that:
\[ a(b + c) = ab + ac \] \[ a(b - c) = ab - ac \]
In words, this means that when you multiply a number (or variable) by a sum (or difference), you can distribute the multiplication to each term inside the parentheses and then perform the addition (or subtraction).
How to Use the Distributive Property
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Identify the expression you want to simplify or rewrite, which typically looks like \( a(b + c) \) or \( a(b - c) \).
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Multiply the term outside the parentheses by each term inside the parentheses:
- For a sum: \( a(b + c) \) becomes \( ab + ac \).
- For a difference: \( a(b - c) \) becomes \( ab - ac \).
Example 1: Using the Distributive Property with Addition
Suppose you have the expression:
\[ 3(x + 4) \]
You can apply the distributive property as follows:
\[ 3(x + 4) = 3x + 3 \cdot 4 = 3x + 12 \]
Example 2: Using the Distributive Property with Subtraction
Now, consider the expression:
\[ 5(y - 2) \]
Using the distributive property, you would write:
\[ 5(y - 2) = 5y - 5 \cdot 2 = 5y - 10 \]
Creating Equivalent Expressions
The distributive property allows you to write equivalent expressions. For example, from earlier:
- The expression \( 3(x + 4) \) can be rewritten as \( 3x + 12 \).
- The expression \( 5(y - 2) \) can be rewritten as \( 5y - 10 \).
These are equivalent because both expressions represent the same value for respective values of \( x \) or \( y \).
Summary
Using the distributive property is a powerful way to simplify expressions and make calculations easier. It helps you manipulate algebraic expressions, leading to new forms that are often easier to work with while maintaining their original value.