What is an Example of the Distributive Property of Equality?

The distributive property of equality is a fundamental mathematical principle that demonstrates how multiplying a number by a sum or difference is equivalent to multiplying that number by each term individually and then combining the results, all while maintaining the equality of the expression.

Understanding the Distributive Property of Equality

This property is a cornerstone of algebra, essential for simplifying expressions and solving equations. It states that for any numbers a, b, and c, the multiplication operation distributes over addition or subtraction as follows:

• a(b + c) = ab + ac
• a(b - c) = ab - ac

When this property is applied, the transformed expression remains mathematically equal to the original one. This characteristic of preserving equality is why it's often referred to as the "distributive property of equality," as it ensures that the value of the expression or the balance of an equation is maintained throughout the process.

Practical Example of the Distributive Property

A straightforward example effectively illustrates how a factor outside the parentheses is distributed to each term within them, yielding an equivalent expression.

The Core Example

Consider the mathematical expression 3(x + 4). We can use the distributive property of equality to simplify this expression as follows:

Step-by-Step Breakdown

Let's meticulously break down the application of the distributive property in this example:

1. Identify the Multiplier: In the expression 3(x + 4), the number 3 is the factor (multiplier) outside the parentheses.
2. Identify the Terms: Inside the parentheses, the terms being added are x and 4.
3. Distribute the Multiplier: Apply the property by multiplying the 3 by each term inside the parentheses:
   • Multiply 3 by x, which results in 3x.
   • Multiply 3 by 4, which results in 12.
4. Combine the Products: The individual products are then combined using the operation (addition in this case) that was between the terms inside the parentheses. This yields the simplified expression: 3x + 12.

This entire process clearly demonstrates that the original expression 3(x + 4) is precisely equal to 3x + 12. Both forms represent the same mathematical value, highlighting the core principle of equality maintained by the distributive property.

Why is This Property Important?

The distributive property is foundational in algebra and beyond, enabling various mathematical operations:

• Simplifying Expressions: It transforms complex expressions into more manageable forms.
• Solving Equations: It's a critical step in isolating variables by removing parentheses from equations.
• Combining Like Terms: It often precedes or facilitates the combination of like terms in an expression.

Mastering this property is crucial for building a strong understanding of algebraic manipulation and solving a wide range of mathematical problems.