Problem 67 - Algebra 2

Question

Name the property illustrated by each equation. $$ 6(9 a)=9 a(6) $$

Step-by-Step Solution

Answer

The equation illustrates the Commutative Property of Multiplication.

Step 1: Identify the Equation Structure

The given equation is \( 6(9a) = 9a(6) \). Let's break this down: the left side contains the term \( 6 \) multiplied by the product \( 9a \), while the right side swaps this order to \( 9a \) multiplied by \( 6 \).

Step 2: Recognize the Mathematical Operation

The equation involves multiplication since both sides are products of the terms \( 6 \) and \( 9a \). This indicates we are dealing with a multiplication scenario, which is crucial for identifying the property.

Step 3: Recall Commutative Property of Multiplication

In mathematics, the Commutative Property of Multiplication states that the order of factors does not affect the product. In other words, for any numbers \( a \) and \( b \), \( a \times b = b \times a \).

Step 4: Match the Equation to the Property

Comparing our equation \( 6 \times (9a) = 9a \times 6 \) with the Commutative Property \( a \times b = b \times a \), it matches perfectly, signifying that the equation illustrates this property.

Key Concepts

Commutative Property

The Commutative Property is one of the foundational properties in mathematics, especially in the realm of arithmetic and algebra. It's a simple yet powerful rule governing the order of operations in both addition and multiplication. When applied to multiplication, the commutative property tells us that we can multiply numbers in any order and still get the same product. For example, if you have two numbers, say 3 and 4, the commutative property assures us that \( 3 \times 4 = 4 \times 3 \). This property is not exclusive to simple numbers; it also applies to algebraic expressions, as seen in the equation \( 6 \times (9a) = 9a \times 6 \). Here, it demonstrates that rearranging the multiplicative order of a constant and an algebraic term doesn't affect the result. In practical terms, the commutative property provides flexibility in computation.

• Simplifying complex expressions by rearranging terms.
• Making mental math calculations easier when shifting the order of operations is advantageous.

It's important to note, though, this property does not apply to subtraction or division!

Multiplication

Multiplication is a fundamental operation in mathematics that involves combining groups of equal size. It is often described as repeated addition. For example, multiplying 5 by 3 means adding 5 three times: \( 5 + 5 + 5 = 15 \), or simply \( 5 \times 3 = 15 \). When it comes to algebra, multiplication extends beyond numbers to include variables and expressions. This expansion allows for operations such as \( 6 \times (9a) \), where you're multiplying a number by a term involving a variable.Multiplication has several important properties that make it unique and useful, such as the

• Commutative Property: Order does not matter.
• Associative Property: Grouping does not affect the product \( (a \times b) \times c = a \times (b \times c) \).
• Distributive Property: Helps to simplify expressions \( a \times (b + c) = a \times b + a \times c \).

Understanding these properties is crucial for solving complex algebraic expressions and equations in a systematic way.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and mathematical operations (like addition, subtraction, multiplication, and division). They form the backbone of algebra and are used to represent real-world situations in mathematical terms. An example of an algebraic expression is \( 9a + 6 \). In this expression, "9a" is a term where "9" is the coefficient (the numerical factor), and "a" is the variable.Working with algebraic expressions involves several key skills:

• Simplifying expressions: Combining like terms to make expressions more manageable.
• Evaluating expressions: Substituting numbers for variables to find the numerical value.
• Factoring expressions: Writing expressions as a product of their factors.

When presented with an equation such as \( 6 \cdot (9a) \), understanding these skills can help in manipulating the expression to solve or simplify the equation efficiently. Algebraic expressions are not just academic exercises. They are practical tools for solving real-world problems in engineering, science, and finance.