Problem 35 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(6+(7+8)=(6+7)+8\)

Step-by-Step Solution

Answer

The algebra rule illustrated by the statement \(6+(7+8)=(6+7)+8\) is the Associative Property of Addition.

Step 1: Understand the equation structure

The equation can be viewed as \(a+(b+c)=(a+b)+c\), where \(a=6\), \(b=7\), and \(c=8\). Then, look at the similarity of this structure to the rules of algebra.

Step 2: Identify the Algebraic Rule

The structure of the equation correlates with the 'Associative Property of Addition'. This property states that when three or more numbers are added, the sum is the same regardless of the way in which the numbers are grouped or in other words, how the parentheses are placed. Therefore, \(a+(b+c)=(a+b)+c\) demonstrates the Associative Property of Addition.

Key Concepts

Algebraic Rules

In algebra, a vast array of rules governs how we manipulate numbers and equations. These rules are pivotal because they ensure consistency and reliability when solving math problems. One of the fundamental rules is the *Associative Property of Addition*. This property simply states that when three or more numbers are added, the sum remains the same regardless of how the numbers are grouped. The specific grouping of numbers is altered using parentheses without changing the actual outcome. The general form is \( a + (b + c) = (a + b) + c \).

• It focuses on the grouping of numbers.
• Does not alter the sum.
• Applies only to addition (and multiplication).

By understanding and applying such rules, we can simplify complex equations and ensure the accuracy of our results.

Equation Structure

Understanding the structure of an equation is critical in identifying algebraic properties. In the example given, the equation \( 6 + (7 + 8) = (6 + 7) + 8 \) showcases how the numbers are grouped differently yet equal the same result. Here, recognizing the pattern is essential. We break the equation down:

• First group: \( 6 + (7 + 8) \)
• Second group: \( (6 + 7) + 8 \)

Each expression is a valid representation of the same sum, highlighting the *Associative Property of Addition*. Grasping this concept helps solve problems more fluidly because it lets us rearrange terms for simpler calculations while preserving equality.

Addition Properties

Addition has particular properties that make calculations straightforward and flexible. These include *Commutativity* and *Associativity*. Focusing on the *Associative Property* in our exercise, it's vital to remember:

• Order of grouping does not influence the sum.
• Only applicable in addition and multiplication.

For example, regardless of whether you add \(6\), \(7\), and \(8\) as \(6 + (7 + 8)\) or \((6 + 7) + 8\), the outcome \(21\) remains consistent. This property allows flexibility in solving equations and is a critical piece of algebra that assists in simplifying expressions effectively.