Problem 84 - Algebra 1: Concepts and Skills

Question

Name the property shown by the statement. $$ -10+(-25)=-25+(-10) $$

Step-by-Step Solution

Answer

The mathematical property shown is the commutative property of addition.

Step 1: Identify the elements in the equation

The equation given is \(-10+(-25)=-25+(-10)\). The numbers on both sides are the same but their order has been changed. Moreover, both sides of the equation have numbers added up.

Step 2: Recognize the mathematical property

The fact that changing the order of the numbers does not change the result leads to the conclusion that the addition showcased in the problem statement is an example of the commutative property.

Key Concepts

Mathematical Properties

Understanding the basic mathematical properties is essential for solving a variety of problems in mathematics. Properties, such as commutative, associative, and distributive, serve as rules that explain how numbers behave under certain operations like addition and multiplication.

In the given exercise, we delve into one such property by looking at the equation \( -10 + (-25) = -25 + (-10) \). Here, despite rearranging the numbers, the equality holds true. This characteristic where the order does not affect the outcome is specifically called the commutative property of addition. It is important to remember these fundamental properties as they apply to different areas of mathematics and help streamline the process of calculation.

Associative Property

While the associative property might seem similar to the commutative property, they are distinct. The associative property deals with how numbers are grouped within parentheses. Specifically, in addition and multiplication, this property tells us that no matter how the numbers are associated (grouped), the result will be the same.

Example of Associative Property

Consider the expression \( (2 + 3) + 4 = 2 + (3 + 4) \). Despite the different groupings of numbers, the sum remains unchanged. This idea can be tremendously helpful when dealing with complex algebraic expressions where strategic grouping can simplify the problem-solving process.

Order of Operations

Grasping the order of operations is crucial for correctly evaluating mathematical expressions. The order of operations, often remembered by the acronym PEMDAS, stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Understanding this order prevents confusion and ensures consistency in the way problems are solved. For instance, in the equation \( 3 + 4 \times 5 \), first we do the multiplication \( 4 \times 5 \) and then add the result to 3. Failing to follow the correct order can lead to wrong answers, thus it's an important principle to follow in algebra.

Algebraic Expressions

An algebraic expression is a mathematical statement that includes numbers, variables, and operation symbols. A simple example could be \( 6x + 2 \) where \( x \) is a variable representing a number we do not yet know. Such expressions can be manipulated using mathematical properties and the order of operations to simplify or solve them.

Using properties like the commutative property in algebra can be especially helpful. It allows for flexibility in rearranging expressions for simplification or solving equations. This can make more complex problems tractable and easier to understand.