Problem 16 - Beginning and Intermediate Algebra

Question

Use a commutative or an associative property to complete each statement. State which property is used. \(6+(-2)=-2+\) ____

Step-by-Step Solution

Answer

6. Property: Commutative.

Step 1 - Identify the Properties

The commutative property of addition states that changing the order of the numbers being added does not change their sum. In other words, for any numbers a and b, a + b = b + a.

Step 2 - Apply the Commutative Property

In this case, we need to complete the statement by recognizing it follows the format of the commutative property. Given the expression: 6 + (-2), we can rewrite it as: -2 + 6, since the order of addition does not affect the sum.

Step 3 - Complete the Statement

To complete the statement, replace the blank (___) with 6: 6 + (-2) = -2 + 6.

Key Concepts

Understanding Addition Properties

In mathematics, properties of addition are fundamental concepts that help to simplify and solve equations. The main properties include the commutative property, associative property, identity property, and distributive property.
Understanding these properties can make working with algebraic expressions and equations easier. Let's focus on the commutative property of addition.

Commutative Property of Addition

The commutative property of addition states that the order in which numbers are added does not change their sum. For any two numbers, a and b, the following is always true:

• \( a + b = b + a \)

This means if you are adding 6 and -2, you can write it as 6 + (-2) or -2 + 6. The result is the same in both cases.

In the example 6 + (-2) = -2 + 6, the commutative property ensures that rearranging the numbers does not change the sum. This property is particularly useful when solving algebraic expressions, making calculations easier and more flexible.

Simplifying Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols. Simplifying algebraic expressions involves combining like terms and using mathematical properties like the commutative property of addition to make expressions easier to work with.

For instance, suppose we have the algebraic expression:

• \( x + y + z \)

Using the commutative property, we can switch the order of the terms without changing the sum:

• \( y + z + x \)

This flexibility allows us to group terms more effectively and simplify complex calculations.

Mastering the commutative property and other addition properties is vital for solving algebra problems efficiently and accurately.