Problem 2 - Beginning and Intermediate Algebra

Question

Complete each of the following. The sum of a number and its opposite will always be

Step-by-Step Solution

Answer

The sum of a number and its opposite will always be zero.

Step 1: Understanding Opposite Numbers

Opposite numbers, also known as additive inverses, are pairs of numbers that, when added together, equal zero. For example, the opposite of 5 is -5.

Step 2: Identify the Number and Its Opposite

Let the number be represented by a variable, say \( x \). Its opposite would then be \( -x \).

Step 3: Formulate the Sum

Write the equation representing the sum of the number and its opposite: \( x + (-x) \).

Step 4: Calculate the Sum

The sum can be simplified as follows: \( x + (-x) = 0 \).

Step 5: Conclude the Result

Therefore, the sum of a number and its opposite will always be zero.

Key Concepts

Additive Inverses

Additive inverses are pairs of numbers that, when added together, result in zero. These numbers are also called opposite numbers. For instance, the additive inverse of 7 is -7. When you add them, the sum is 0:

• 7 + (-7) = 0

Understanding additive inverses is important because it helps simplify equations and solve problems faster. Whenever you see a number and its opposite being added together, you can immediately know the result will be zero.

Variables in Algebra

A variable in algebra is a symbol, often represented by letters like x or y, that stands in for an unknown value. Variables are used because they make it easier to write and solve equations. For example, instead of saying 'a number plus its opposite,' you can write x + (-x).Using variables helps you generalize mathematical rules. In our example, let x be any number. Then, its opposite is -x. When you add x and -x, the result is zero. This means that the rule applies to any number you choose to represent by x.

Basic Algebraic Operations

Basic algebraic operations include addition, subtraction, multiplication, and division. These are foundational skills for solving more complex math problems. Let's look at addition with a focus on our example of additive inverses.If you have a number x and its opposite -x, adding these together simplifies to zero:

• x + (-x) = 0

This is because -x cancels out x, leaving you with nothing. This principle of opposites canceling each other is a key part of understanding more advanced algebraic concepts. It also shows how useful basic operations are in breaking down and solving problems.