Problem 27 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ -6+(12+7)=(-6+12)+7 $$

Step-by-Step Solution

Answer

Associative property

Step 1 - Identify the operation

Look at the given equation: -6+(12+7)=(-6+12)+7The operation used here is addition.

Step 2 - Recognize the grouping of terms

Notice how the grouping of the numbers changes: The left side groups (12+7) together, while the right side groups (-6+12) together.

Step 3 - Recall the associative property

The associative property states that the way in which numbers are grouped does not change their sum or product. It applies to both addition and multiplication. For addition, it can be written as: (a + b) + c = a + (b + c) In this case, we see: -6 + (12 + 7) = (-6 + 12) + 7

Step 4 - Conclusion

Since the given equation changes the grouping of terms without altering the result, it exemplifies the associative property of addition.

Key Concepts

Algebra Concepts

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in equations and expressions. Understanding algebra involves recognizing and applying various mathematical properties and rules. One critical concept in algebra is using properties like the associative property, which helps simplify and solve equations. By familiarizing yourself with these properties, you can make the problem-solving process more efficient and straightforward. For example, in the exercise given, the associative property helps us understand that changing the grouping of numbers doesn't affect the overall sum.

Addition Properties

Addition properties are essential for simplifying mathematical expressions and equations. These properties provide rules that can make complex calculations more manageable. Here are some key addition properties:

• Commutative Property: This property states that changing the order of addends does not affect the sum. Mathematically, it is written as \(a + b = b + a\).
• Associative Property: This property suggests that the way in which numbers are grouped does not change their sum. For addition, it is represented as \((a + b) + c = a + (b + c)\).
• Identity Property: This property states that adding zero to any number will not change the value of that number. It can be written as \(a + 0 = a\).

In the exercise provided, the associative property is showcased because the grouping of terms changes from \(-6 + (12 + 7)\) to \((-6 + 12) + 7\), but the overall sum remains the same.

Grouping of Terms

Grouping of terms refers to how numbers or variables are combined in a mathematical expression. Proper understanding and application of grouping can simplify problem-solving.

For example, in the mathematical expression \(-6 + (12 + 7)\), the terms \(12\) and \(7\) are grouped together inside parentheses. According to the associative property, we can regroup these terms as \((-6 + 12) + 7\) without changing the sum.

The concept of grouping is crucial because it allows you to rearrange terms in expressions to simplify them or to better see patterns and relationships. This rearrangement can be done effectively using the associative property, particularly in complex problems involving multiple operations.

In conclusion, recognizing and applying the appropriate properties of algebra, such as the associative property for grouping terms, leads to more straightforward and accurate computation.