Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Use the commutative property of addition to write an equivalent algebraic expression. $$10 x+9 y$$

Step-by-Step Solution

Answer

The equivalent expression of \(10x + 9y\) using the commutative property of addition is \(9y + 10x\).

Step 1: Identify the Terms

In the expression \(10x + 9y\), \(10x\) and \(9y\) are the terms. The commutative property of addition applies to these terms.

Step 2: Apply the Commutative Property of Addition

According to the commutative property of addition, changing the order of the terms does not affect the sum. Therefore, reorder the terms in the given expression. This involves writing the term \(9y\) before the term \(10x\).

Step 3: Write the Equivalent Expression

The equivalent expression of \(10x + 9y\) using the commutative property of addition is \(9y + 10x\).

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical statement that includes numbers, variables (such as x or y), and operations (like addition, multiplication). Unlike an equation, an algebraic expression doesn't have an equality sign (=); instead, it's a combination of terms. In the expression 10x + 9y, we see two terms: 10x, which is the coefficient 10 multiplied by the variable x; and 9y, the coefficient 9 multiplied by the variable y.

These terms can be combined using operation symbols to form an algebraic expression, expressing a particular relationship between numbers and variables. Algebraic expressions are fundamental in algebra and are used to describe patterns, relationships, and changes.

Equivalent Expression

An equivalent expression in algebra means a different expression that has the same value as the original one, regardless of what values we substitute for its variables. For example, if we have an algebraic expression like 10x + 9y, we can apply various algebraic properties to transform it without changing its inherent value.

One such property is the commutative property of addition, which states that the order in which two terms are added doesn't affect their sum. So, 10x + 9y is equivalent to 9y + 10x. Through this transformation, we haven't changed the meaning of the expression, just how it's presented. Understanding how to form equivalent expressions is essential for simplifying algebraic expressions and solving equations.

Terms in Algebra

In algebra, a term is a single mathematical entity that can be a number, a variable, or a combination of numbers and variables multiplied together. For instance, in the expression 10x + 9y, 10x and 9y are each considered a term. Here, 10x consists of the coefficient 10 and variable x, while 9y is composed of the coefficient 9 and variable y.

Understanding terms is crucial when applying algebraic properties, such as the commutative property of addition. By correctly identifying each term, we can apply these properties to rearrange and simplify complex algebraic expressions, making it easier to work with the expressions to solve problems.