The Distributive Property is a fundamental principle in algebra that states you can multiply a number or expression by a sum or difference by distributing the multiplication across each term inside the parentheses. It can be expressed mathematically as:
\[ a(b + c) = ab + ac \] \[ a(b - c) = ab - ac \]
Examples of the Distributive Property
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Using Addition: If you have \( 3(x + 4) \), you can apply the distributive property as follows: \[ 3(x + 4) = 3x + 12 \]
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Using Subtraction: For \( 5(2y - 3) \), you can also distribute: \[ 5(2y - 3) = 10y - 15 \]
Combining Like Terms
After using the Distributive Property, you might need to simplify the expression further by combining like terms. For example:
- From the previous example: if you have \( 10y - 15 + 2y \), you can combine \( 10y \) and \( 2y \) to get: \[ 12y - 15 \]
Application in Algebraic Expressions
The Distributive Property is particularly useful when simplifying algebraic expressions or solving equations. It allows you to expand expressions, which can make them easier to work with. For example:
Example 1: Simplifying an expression
- Simplify \( 2(x + 3) + 4(x + 5) \): \[ = 2x + 6 + 4x + 20 \] \[ = (2x + 4x) + (6 + 20) \] \[ = 6x + 26 \]
Example 2: Solving an equation
- Solve for \( x \) in the equation \( 2(x + 1) = 14 \): \[ 2x + 2 = 14 \] \[ 2x = 14 - 2 \] \[ 2x = 12 \] \[ x = 6 \]
Conclusion
The Distributive Property is essential for expanding, simplifying, and solving algebraic expressions. Mastery of this property will enhance your ability to work with a variety of algebraic problems effectively.
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