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Chapter 1: Problem 25

Name the property illustrated by each equation. $$ (2+14)+3=3+(2+14) $$

Short Answer

Expert verified
Commutative Property of Addition

Step by step solution

01

Identify the Property Types

In algebra, there are several properties related to addition and multiplication. These include the commutative property, which states that a + b = b + a, and the associative property, which states that (a + b) + c = a + (b + c). In this particular equation, we will determine which property is illustrated.
02

Examine the Equation

The given equation is \((2+14)+3=3+(2+14)\). Notice how the terms \((2+14)\) and \(3\) have been rearranged, with all terms being on the same side of the equation in different order, which helps identify the property.
03

Identify the Property

Based on the rearrangement of the operands in the equation, \((2+14)+3=3+(2+14)\) matches the definition of the commutative property of addition. This property states that changing the order of numbers in an addition operation doesn't affect the sum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Addition
Addition is one of the basic operations in mathematics, symbolized by the plus sign \(+\). In its simplest form, addition involves combining two or more numbers to get a total sum. For example, if you add 2 and 3, the result is 5. It is important to understand that
  • addition is both associative and commutative, meaning numbers can be grouped differently without changing the results, and their order can be switched to the same effect, respectively.
  • The concept is not only applied to numbers but also to variables and algebraic expressions, making it fundamental in algebra.
This operation is a cornerstone of more complex math topics, helping to simplify and solve equations in algebra and beyond.
Associative Property
The associative property is a concept that makes dealing with addition much simpler, especially in algebraic expressions. Simply put, it tells us that the way we group numbers in an addition operation does not change their result.
For instance, with numbers \(a\), \(b\), and \(c\), the associative property means \[(a + b) + c = a + (b + c)\]
  • This tells us that if you are adding three or more numbers, it doesn't matter how the numbers are grouped.
  • The use of parentheses is purely for grouping and does not affect the actual sum.
Remember that the associative property applies to addition and multiplication, but **not** subtraction or division. In algebra, it's especially useful for simplifying calculations and rearranging terms.
Algebra Properties
In algebra, there are various properties or rules that can be applied to numbers and equations to simplify calculations and solve problems effectively. Among these properties are the associative and commutative properties, which often get used hand in hand.
  • The **commutative property** of addition states that the order of numbers does not affect their sum \( (a + b = b + a) \).
  • On the other hand, the **associative property** deals with the grouping of numbers, showing that different groupings still produce the same total.
These properties form a fundamental part of algebra, providing a foundation for more complex problem-solving methods. Mastering them is important as they make manipulating algebraic expressions easier, aiding in the simplification of equations and the discovery of solutions.

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