Problem 51 - College Algebra

Question

state the name of the property illustrated. $$ 6+(2+7)=(6+2)+7 $$

Step-by-Step Solution

Answer

The property illustrated is the associative property of addition.

Step 1: Examine the Equation

We are given:
state the name of the property illustrated.
$$
6+(2+7)=(6+2)+7
$$

Step 2: Identify the Pattern

We compare both sides to identify the algebraic property (commutative, associative, distributive, identity, inverse).

Step 3: Name the Property

The property illustrated is the The property illustrated is the associative property of addition..

Key Concepts

Properties of Addition

Understanding the properties of addition is essential for solving algebraic problems efficiently. One fundamental property is the associative property, which illustrates that when adding three or more numbers, the way in which the numbers are grouped does not affect the sum. For example, the expression
\( 6 + (2 + 7) \) can be regrouped without altering the result as \( (6 + 2) + 7 \).
This principle allows for flexibility in computation and simplifies complex operations by rearranging the numbers for easier addition.
Other properties of addition include the commutative property, where the order of numbers does not affect the sum (e.g., \( 3 + 5 \) equals \( 5 + 3 \)), and the identity property, where adding zero to any number does not change its value (e.g., \( 7 + 0 = 7 \)). Understanding these properties aids in developing algebraic fluency and problem-solving strategies.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operation symbols. For instance, \( 3x + 4y - 5 \) is an algebraic expression with variables \( x \) and \( y \). Such expressions become the foundational language through which algebraic relationships are communicated.
To manipulate these expressions effectively, one must understand properties of addition, as these properties enable us to reorder and group terms strategically. For instance, according to the associative property, \( x + (y + z) \) is equivalent to \( (x + y) + z \), allowing for a more manageable form to work with.
Algebraic expressions are the building blocks of equations and functions, roles that are integral in the field of algebra. By mastering the manipulation of these expressions using addition properties, students prepare themselves to tackle more complex operations in algebra.

College Algebra

Moving into college algebra, students are expected to grasp more abstract concepts and apply mathematical operations to a broader range of problems. Here, the importance of understanding the properties of addition and manipulation of algebraic expressions can't be overstated.
College algebra typically covers a wide array of topics including functions, systems of equations, matrices, and polynomial operations. Each of these areas involves addition in some form, where properties like the associative property come into play to simplify expressions and solve equations.
For example, when dealing with polynomials, one might need to group terms to combine like terms effectively. The associative property ensures that these rearrangements are valid. A strong foundation in the basic properties of arithmetic operations facilitates success in college-level mathematics courses and beyond, ensuring students are well-equipped for the analytical challenges they will encounter.