Problem 26 - Elementary and Intermediate Algebra

Question

Use the commutative law of multiplication to write an equivalent expression. $$ 9(x+5) $$

Step-by-Step Solution

Answer

(x + 5) \times 9

Step 1: Identify the Commutative Law of Multiplication

The commutative law of multiplication states that changing the order of factors in a multiplication operation does not change the product. In other words, if you have any two numbers or expressions, say 'a' and 'b', then \(a \times b = b \times a\).

Step 2: Apply the Commutative Law to the Given Expression

Given the expression \(9(x+5)\), you can identify the two factors: 9 and \(x+5\). According to the commutative law, you can switch their order: \(9(x+5) = (x+5) \times 9\).

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of letters and numbers linked by mathematical operations like addition, subtraction, multiplication, and division. These expressions do not equate to a specific value unless the variable is given a specific number.

For example, in the expression \(4a + 3b\), both \(4a\) and \(3b\) are terms. The letters \(a\) and \(b\) represent variables, while the numbers 4 and 3 are coefficients. Together, they create an algebraic expression that can change according to the values of the variables.

Algebraic expressions are crucial in mathematics because they allow us to represent and solve real-world problems numerically. Understanding the rules and properties that apply to these expressions, like the commutative law of multiplication, helps simplify and manipulate them accurately.

Multiplication Properties

One of the foundational properties in multiplication is the commutative property. This property states that the order in which two numbers or expressions are multiplied does not affect the product.

In mathematical terms, this means that for any two expressions \(a\) and \(b\), the equation \(a \times b = b \times a\) holds true. This can be quite useful when dealing with complex algebraic expressions, as it provides flexibility in rearranging terms to simplify calculations.

Other important multiplication properties include:

• Associative Property: The grouping of factors does not change the product. For example, \[ (a \times b) \times c = a \times (b \times c) \]
• Distributive Property: A single term multiplied by a sum is the same as multiplying each addend separately and then adding the results. For example, \[ a \times (b + c) = a \times b + a \times c \]

Knowing these properties helps in rewriting and solving algebraic expressions more effectively, as shown in the given exercise.

Mathematical Equivalence

Mathematical equivalence means that two expressions are equal in value, even if they look different. It's a core concept that allows us to transform and solve equations more flexibly.

In the given exercise, we were asked to use the commutative law of multiplication to write an equivalent expression for \(9(x + 5)\). By applying the commutative property, we rearrange the factors from \((9)\) and \((x + 5)\) to \((x + 5) \times 9\). Both of these expressions are mathematically equivalent, meaning they result in the same product when evaluated for any value of \(x\).

Understanding equivalence helps simplify complex expressions and solve for variables more efficiently. It's important to always remember that the transformation must maintain the equality of the original expression. This concept is widely used in algebra, calculus, and other advanced mathematical fields to manipulate, solve, and prove equations and expressions.