Problem 25 - College Algebra and Calculus: An Applied Approach

Question

Identify the rule(s) of algebra illustrated by the statement.\(-15+15=0\)

Step-by-Step Solution

Answer

The algebraic rule illustrated by the equation \(-15+15=0\) is the Additive inverse rule.

Step 1: Recognize the Type of Numbers in the Equation

In the equation, both the numbers are integers and they are additive inverses of each other. An additive inverse is defined as the number that when added to the original number results in zero.

Step 2: Apply the Algebraic Rule

The given equation proves the Additive inverse rule. The additive inverse rule states that any number plus its opposite equals zero. In this case, \(-15\) is the opposite of \(15\), and when they are added together, they equal zero.

Step 3: State the Rule Illustrated

So the algebra rule illustrated by the given statement \(-15+15=0\) is the Additive Inverse Rule.

Key Concepts

Algebraic Rules

Algebraic rules form the foundation of algebra, which is a branch of mathematics that deals with symbols and the rules for manipulating these symbols. These symbols represent numbers and quantities in formulae and equations. One fundamental rule that we often encounter is the Additive Inverse Rule.

This rule is a simple yet powerful concept that helps to simplify complex equations by eliminating terms to isolate variables. In essence, it allows us to balance equations and solve for unknowns. To understand this rule in the context of algebra, it's crucial to comprehend how numbers interact within equations, which brings us to the concept of integers and their properties.

Integers

Integers are a set of numbers that include all whole numbers and their negatives. It does not include fractions or decimals. Integers are crucial in everyday mathematics and fall on a number line infinitely in both directions. They can be positive, negative, or zero.

Understanding integers is essential in algebra because they are the backbone of many algebraic structures. When working with equations, it's important to note how these whole numbers behave under different operations like addition, subtraction, multiplication, and division. Their predictable behavior under these operations makes solving algebraic equations more systematic.

Additive Inverse

An additive inverse is a number that when added to a given number yields zero. This is a key concept in algebra that is used to solve equations. For any integer 'a', the additive inverse is '-a', and vice versa. Thus, the formula defining this relationship is: \( a + (-a) = 0 \).

In the context of our exercise \( -15+15=0 \) both -15 and 15 are additive inverses of each other. When dealing with equations, the additive inverse is used to eliminate terms, allowing us to isolate and solve for the desired variable. It is this principle that often simplifies the process of balancing equations.

Equations

Equations are statements that assert the equality of two expressions. They are composed of two sides separated by an equal sign, such as \( a + b = c \). In algebra, equations are solved to find the value(s) of the unknown variable(s).

The solution process involves applying algebraic rules, like the additive inverse, to simplify the equation and isolate the variable. It's akin to a balance scale where both sides must remain equal as we perform operations. As seen in the exercise \( -15+15=0 \), equations can be simple, yet they illustrate fundamental principles crucial for understanding more complex algebraic concepts. By mastering basic equations and their underlying rules, students build a strong foundation in algebra.