Problem 28 - Elementary and Intermediate Algebra

Question

Use the associative law of addition to write an equivalent expression. $$ (5+m)+r\(5+(m+r)\) $$

Step-by-Step Solution

Answer

5 + (m + r)

Step 1: Identify the original expression

The original expression given is \( (5 + m) + r \). According to the problem, this expression needs to be rewritten using the associative law of addition.

Step 2: Understand the associative law of addition

The associative law of addition states that the way in which numbers are grouped in an addition operation does not change the sum. Mathematically, this is written as \((a + b) + c = a + (b + c)\).

Step 3: Apply the associative law

Apply the associative law to the given expression \((5 + m) + r\). By rearranging the parentheses according to the law, the expression becomes \(5 + (m + r)\).

Key Concepts

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. These symbols represent quantities without fixed values, known as variables.

In the given exercise, the variables are represented by letters such as 'm' and 'r'. Algebra helps us express and solve problems using these variables. We use algebraic techniques to find unknowns and understand the relationships between different quantities.

When working with expressions like \( (5 + m) + r \), it is crucial to understand how different properties of algebra, like the associative law of addition, help simplify and manipulate these expressions effectively.

Addition

Addition is one of the basic operations in arithmetic and algebra. It combines two or more quantities to get a total or sum. For example, in the expression \( (5 + m) + r \), we are adding three quantities: 5, m, and r.

The associative law of addition shows us that the way we group these quantities does not affect the sum. Whether we add 5 to m first and then add r, or add m to r first and then 5, the result is the same.

Understanding addition in algebra involves knowing how to group and regroup terms to simplify expressions or solve equations.

Mathematical Properties

Mathematical properties are rules that apply to numbers and operations. They help us perform calculations more efficiently and are fundamental to understanding algebra.

One such property is the associative law of addition, which states: \( (a + b) + c = a + (b + c) \). This property tells us that no matter how we group the numbers when adding, the result is the same.

In the exercise, we applied this property to transform \( (5 + m) + r \) into \( 5 + (m + r) \). By recognizing and using mathematical properties like this, we can simplify complex expressions and solve problems systematically.