Problem 71 - Algebra 2

Question

Name the property illustrated by each statement. \(5 y(4 a-6 b)=20 a y-30 b y\)

Step-by-Step Solution

Answer

The distributive property.

Step 1: Identify the Expression Type

We start by identifying the type of mathematical expression presented. The given equation is structured as an algebraic expression with multiplication and distributive properties applied on one side.

Step 2: Identify the Mathematical Operation

The key operation involved here is distribution, where a term outside the parenthesis is multiplied with terms inside the parenthesis. The expression on one side, \(5y(4a - 6b)\), expands to \(20ay - 30by\) on the other side.

Step 3: Define the Property Involved

The equality of \(5y(4a - 6b) = 20ay - 30by\) is achieved by applying the Distributive Property of multiplication over subtraction. Specifically, the expression illustrates that multiplying each term inside the parenthesis by the term outside the parenthesis yields equivalent expressions.

Key Concepts

Algebraic Expression

Algebraic expressions in mathematics are combinations of numbers, variables, and operators presented in a structured form. They are a fundamental aspect of algebra and can be adjusted or simplified using various properties and operations. In the exercise, the expression is structured as \(5y(4a - 6b)\), which includes:

• The number 5 and variable \(y\) multiplying the terms inside the bracket.
• The term \(4a\), indicating 4 times the variable \(a\).
• The term \(-6b\), indicating negative 6 times the variable \(b\).

Understanding the structure of algebraic expressions helps in identifying which operations and properties to use for simplification or evaluation. Here, these components interact through multiplication within the brackets, making it necessary to apply the distributive property to simplify.

Multiplication

Multiplication in algebra extends the basic concept of repeated addition and applies it to variables and numbers within expressions. It's a core operation that allows us to combine like terms and factors of algebraic expressions. In the example problem, multiplication is used to expand the expression \(5y(4a - 6b)\) as follows:

• The number outside the parenthesis, 5, multiplies each term inside the parenthesis separately. This results in \(5y \cdot 4a\) and \(5y \cdot -6b\).

This operation is called distributing, a crucial part of applying the distributive property. The primary goal when multiplying in expressions is to simplify or expand them to a more workable form, making it easier to understand and process.

Properties of Operations

Properties of operations are rules that help us to perform arithmetic and algebraic calculations more efficiently. They guide how we manipulate expressions while preserving equality. One of the main properties applied often in algebra is the Distributive Property.The Distributive Property states that for any numbers or variables \(a, b,\) and \(c\), the expression \(a(b + c)\) can be expanded to \(ab + ac\). In the original exercise, this property is used to simplify the expression from \(5y(4a - 6b)\) to \(20ay - 30by\). By distribute, we mean multiplying the term outside the parenthesis by each term inside:

• \(5y \times 4a = 20ay\)
• \(5y \times (-6b) = -30by\)

This demonstrates how the Distributive Property is a powerful tool in algebra, providing a method to transform and simplify expressions seamlessly.