Problem 58 - Algebra 2

Question

Name the property illustrated by each equation. \((5+9)+13=13+(5+9)\)

Step-by-Step Solution

Answer

Commutative Property of Addition

Step 1: Identify the Structure

Look closely at the structure of the equation \((5+9)+13=13+(5+9)\). Notice that it involves the rearrangement of terms without changing any values.

Step 2: Recall the Properties

Think about the different properties of addition. In algebra, the property that deals with the rearrangement of terms is very specific.

Step 3: Select the Appropriate Property

Recognize that the equation shows the sides being swapped, indicating a rearrangement. This is a key characteristic of the commutative property.

Step 4: Review the Commutative Property

The commutative property states that the order of numbers can be changed within an operation, such as addition or multiplication, without affecting the result. In this case, \(a+b=b+a\).

Key Concepts

Properties of Addition

Addition has several useful properties that help simplify and solve equations. These include:

• Commutative Property: This property teaches us that the order of numbers in an addition operation can be changed without affecting the sum. For example, if you have two numbers, 3 and 5, then according to the commutative property, 3+5 is the same as 5+3. This is exactly what happens in the original exercise equation, where the numbers are rearranged.
• Associative Property: This property indicates that how numbers are grouped in an addition does not change the outcome. For example, (1+2)+3 equals 1+(2+3). In this scenario, the grouping of numbers alters, but the sum stays the same.

Understanding these properties allows you to experiment with equations and get the same results consistently, no matter the order or grouping of numbers.

Equation Structure

To understand the property illustrated by an equation, it is crucial to examine its structure. The structure reveals how the components of the equation are arranged and what rules govern those arrangements. In the exercise, the equation \[(5+9)+13=13+(5+9)\] demonstrates a specific structure where the terms are shuffled. Observing this rearrangement is the first step towards identifying which algebraic property is being utilized. Take note:

• Both sides of the equation contain the same numbers, just organized differently.
• No numbers are altered in the process, only repositioned.

Recognizing these structural components is essential for selecting the appropriate property being illustrated.

Number Rearrangement

Number rearrangement is central to the commutative property of addition. This concept implies changing the position of numbers within an equation without changing the result. In the given exercise, you see that `(5+9)+13` is transformed into `13+(5+9)`, while both expressions equal each other.

This kind of rearrangement under the commutative property includes swapping any numbers within an addition operation as needed. Here’s how it benefits you:

• Allows flexibility in solving and simplifying equations.
• Makes mental calculations easier by adjusting the order of numbers.

Practicing number rearrangement sharpens your skills in recognizing patterns and properties in equations.

Algebraic Properties

Algebra is equipped with several properties that define how numbers behave, especially during operations such as addition. The main algebraic properties include:

• Commutative Property (focus of this exercise): As discussed, it involves rearranging numbers in both addition and multiplication without affecting the outcome. In symbolic form, it’s represented as \(a+b = b+a\).
• Associative Property: It involves grouping. For addition, \((a+b)+c = a+(b+c)\) illustrates that how you group numbers in addition does not change the sum.
• Identity Property: Adding zero to any number leaves it unchanged, such as \(a+0 = a\).

These properties offer a robust framework to manipulate and solve algebraic equations effectively. Understanding each helps in mastering complex equations and identifying patterns in mathematical operations.