To correctly rewrite an equation using the distributive property, you apply a fundamental algebraic principle that allows you to multiply a single term by two or more terms within a set of parentheses. This process simplifies expressions, making them easier to manage or solve.
Understanding the Distributive Property
The distributive property states that multiplying a sum or difference by a number is the same as multiplying each addend or subtrahend by the number and then adding or subtracting the products. In simpler terms, it "distributes" the multiplication across addition or subtraction.
General Formulas:
- For addition:
a(b + c) = ab + ac - For subtraction:
a(b - c) = ab - ac
Here, a is the multiplier outside the parentheses, and b and c are the terms inside.
Steps to Apply the Distributive Property
Applying the distributive property involves a straightforward, three-step process to transform the expression from its factored form to an expanded form.
-
Step 1: Distribute the Multiplier
Multiply the number or term outside the parentheses by each term inside the parentheses. This creates new individual multiplication problems. -
Step 2: Find the Individual Products
Perform each of the multiplications from Step 1. This will give you a set of new terms. -
Step 3: Add or Subtract
Combine the individual products obtained in Step 2, using the original operation (addition or subtraction) that was between the terms inside the parentheses.
Example Application of the Distributive Property
Let's illustrate how to rewrite an equation using the distributive property with a practical example: Rewrite the expression 4(x + 5) using the distributive property.
| Step | Description | Example: 4(x + 5) |
|---|---|---|
| Step 1: Distribute the Multiplier | Multiply the term outside (4) by each term inside (x and 5). |
4 * x and 4 * 5 |
| Step 2: Find the Individual Products | Perform the individual multiplications. | 4x and 20 |
| Step 3: Add or Subtract | Combine the products using the operation from the original parentheses. | 4x + 20 |
Therefore, 4(x + 5) can be correctly rewritten as 4x + 20 using the distributive property.
Why is the Distributive Property Important?
The distributive property is a cornerstone of algebra, crucial for:
- Simplifying Expressions: It helps break down complex expressions into simpler, more manageable parts, making them easier to work with.
- Solving Equations: It's often the first step in solving equations where a term is multiplied by a sum or difference, allowing you to isolate variables.
- Factoring: Understanding distribution is key to its inverse operation, factoring, where you pull out a common multiplier from an expression.
Mastering this property enhances your ability to manipulate algebraic expressions efficiently and accurately.
