Problem 24 - Prealgebra

Question

Name the property shown by each statement. $$(5+x)+6=5+(x+6)$$

Step-by-Step Solution

Answer

Associative Property of Addition.

Step 1: Analyze the Expression

Start by examining the given expression: \ \((5 + x) + 6 = 5 + (x + 6)\). \ Notice the sequence of operations, specifically the grouping of the numbers.

Step 2: Identify the Property

In this expression, the numbers are regrouped without altering the order, which suggests that the Associative Property of Addition is being used. This property states that the way in which numbers are grouped in an addition operation does not change their sum.

Step 3: Reaffirm the Property Application

Verify that both sides of the equation, \((5 + x) + 6\) and \(5 + (x + 6)\), indeed only differ in their grouping, which confirms the application of the Associative Property.

Key Concepts

Properties of Addition

Addition, a fundamental mathematical operation, comes with properties that make calculations easier and conceptually intuitive. One of these properties is the **Associative Property of Addition**. This property reveals that how we group numbers in addition doesn't change their total sum. For instance, consider the expression \((5+x)+6\) and its rearrangement to \(5+(x+6)\). Both combinations result in the same sum, which showcases the associative characteristic. This property is beneficial as it allows us to simplify problems and break down complex operations into smaller, more manageable parts.### Key Properties of Addition:- **Commutative Property**: Changing the order of the addends does not affect the sum.- **Associative Property**: Changing the grouping of the addends does not change the sum.- **Identity Property**: Adding zero to any number does not change the value of that number.Understanding these properties allows students to manipulate expressions more effectively and with greater confidence, especially in algebra.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. Variables, often represented by letters like \(x\) or \(y\), stand in place for numbers we don't yet know. This flexibility allows us to solve many kinds of problems using algebra.Consider the expression \(5+x\). Here, the number 5 is a constant, and \(x\) is a variable. Together, in addition with another number or variable, they form an algebraic expression. ### Characteristics of Algebraic Expressions:- **Variables**: Symbols that represent unknown quantities.- **Constants**: Fixed numbers.- **Coefficients**: Numbers that multiply variables.- **Operations**: Addition, subtraction, multiplication, and division.By knowing how to manipulate these components, students can solve equations, simplify expressions, and better understand the behavior of numbers in equations.

Mathematical Operations

Mathematical operations form the backbone of all calculations in mathematics. These operations include addition, subtraction, multiplication, and division. Each operation has its own set of rules and properties that govern how we use them in equations and expressions.### Important Operations in Focus:- **Addition**: Combining numbers to get a total. Governed by properties like associative and commutative.- **Subtraction**: Finding the difference between numbers. Not inherently associative or commutative.- **Multiplication**: Repeated addition of a number. Similar properties to addition.- **Division**: Splitting a number into equal parts. Unlike multiplication, division does not have an associative property.When solving algebraic equations and expressions, understanding these operations and their properties is crucial for simplifying and finding solutions efficiently. For instance, swapping the grouping in addition due to the associative property makes complex problems simpler, as we've seen in our example: \((5+x)+6=5+(x+6)\). Recognizing these properties can significantly help students in tackling numerous mathematical challenges.