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Chapter 4: Problem 98

What is the sum of two numbers that are opposites?

Short Answer

Expert verified
The sum is 0.

Step by step solution

01

Define Opposite Numbers

Opposite numbers are pairs of numbers that are the same distance from zero on a number line but in opposite directions. For example, +3 and -3 are opposite numbers.
02

Identify the Opposite Numbers

Let the two opposite numbers be represented by a number and its negation. For example, if one number is represented by 'a', the opposite number is '-a'.
03

Set Up the Addition Equation

Add the two numbers together: \(a + (-a)\)
04

Simplify the Equation

Combine the numbers: \(a - a = 0\)
05

State the Sum

The sum of any number and its opposite is 0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Opposite Numbers
Opposite numbers are pretty straightforward once you visualize them. Think of opposite numbers as cousins on a number line. They are the same distance from zero but are on different sides. For example, +4 and -4 are opposites because both are four steps away from zero. You might find pairs like +5 and -5 or +2 and -2. These opposite pairs balance each other out perfectly.
Number Line
A number line is a great tool for understanding opposite numbers. Imagine a horizontal line with zero in the middle, positive numbers on the right, and negative numbers on the left. When you place a number like +3 on the right of zero and its opposite, -3, on the left side, you can see they are equally spaced from zero. This visual can help you easily identify pairs of opposite numbers.
Addition of Integers
When dealing with integers, it's essential to grasp addition. Adding two positive numbers moves you right on the number line. Adding two negative numbers moves you left. But what happens when you add a positive number to its negative opposite? Let's say you have +5 and add its opposite, -5. Using the number line, you start at +5 and move five steps left to reach zero. Therefore, the sum of +5 and -5 is zero.
Negation
Negation is simply flipping the sign of a number. The negation of a positive number is its negative counterpart, and vice versa. For instance, the negation of +7 is -7. When you add a number to its negation, \(a + (-a)\), you end up with zero. This is because the positive and negative cancel each other out. So, whenever you add one number and its negation, you always get zero. This concept is a cornerstone in understanding that opposite numbers balance each other.

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Most popular questions from this chapter

Solve by either elimination or substitution. $$ \begin{aligned} x+y &=6 \\ 2 x-y &=-3 \end{aligned} $$

For exercises \(41-48\), solve by substitution. $$ \begin{aligned} &a=2 b-15 \\ &3 a+4 b=55 \end{aligned} $$

For exercises \(85-88\), the completed problem has one mistake. (a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Write a system of equations that represents the relationship of the variables. Each pound of Cereal A contains \(0.5 \mathrm{oz}\) of almonds and \(2 \mathrm{oz}\) of dried cranberries. Each pound of Cereal B contains 2 oz of almonds and 1 oz of dried cranberries. Find the amount of each cereal needed to make 400 lb of a new cereal that contains \(0.75 \mathrm{oz}\) of almonds per pound of cereal. (1 \(\mathrm{lb}=16 \mathrm{oz}\).) $$ \begin{aligned} &x=\text { amount of Cereal A } \\ &y=\text { amount of Cereal B } \end{aligned} $$ Incorrect Answer: \(x+y=400 \mathrm{lb}\) $$ \left(\frac{0.5 \mathrm{oz}}{1 \mathrm{lb}}\right) x+\left(\frac{2 \mathrm{oz}}{1 \mathrm{lb}}\right) y=0.75 \mathrm{oz} $$

The perimeter of a rectangle is \(60 \mathrm{in}\). The length is 14 in. longer than the width. Find the length and the width.

For exercises \(3-22\), graph. Label the solution region.a $$ \begin{aligned} &x+y \leq 8 \\ &x \geq 2 \end{aligned} $$
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