Problem 19 - Algebra 1: Concepts and Skills

Question

Find the opposite of the number. $$ -10 $$

Step-by-Step Solution

Answer

The opposite of -10 is 10.

Step 1: Identify the original number

In this case, the given number is -10.

Step 2: Find the opposite of the number

To find the opposite of a number, it is multiplied by -1. So, the opposite of -10 will be \(-10 \times -1\).

Step 3: Calculate the result

Multiplication of -10 and -1 gives 10.

Key Concepts

Opposite Numbers

Opposite numbers are integers that have the same magnitude but different signs. To visualize this, imagine the number line. Positive numbers are to the right of zero, while negative numbers are to the left. The opposite of any positive number is its negative counterpart, and vice versa.

Here’s how you can find an opposite number:

• If the number is positive, its opposite is negative.
• If the number is negative, its opposite is positive.

In mathematical terms, finding an opposite implies reversing the sign. For example, if you are given -10, its opposite is 10. This involves multiplying by -1 to shift from negative to positive, or vice versa. It’s a simple yet crucial concept in algebra that helps us handle more complex calculations.

Multiplication

In algebra, multiplication is a fundamental operation involving two or more numbers (or variables). When you multiply numbers, you are essentially adding a number to itself a certain number of times. The process of multiplication is straightforward but can have more profound implications.

Multiplication of numbers follows specific rules:

• Sign Rules: When you multiply two numbers:
• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative
• Order of Operations: Typically, multiplication is performed before addition or subtraction unless parentheses indicate otherwise.

In the example given, multiplying -10 by -1 is straightforward using these rules. A negative multiplied by another negative results in a positive, leading to the result of 10. This is crucial for understanding how opposites are calculated.

Integers

Integers are a set of numbers that include all positive whole numbers, zero, and their negative counterparts. Understanding integers is vital in mathematics as they are used in simple operations like addition, subtraction, multiplication, and division.

Integer properties include:

• Whole Numbers: Integers are complete units without fractional or decimal parts.
• Continuity: Every integer is a step on the number line; there are no gaps between them.
• Additive Inverses: Every integer has an opposite, also known as its additive inverse, satisfying the equation: number + opposite = 0.

For example, in the problem -10 has an opposite, which is 10, demonstrating the property of additive inverses. This simple concept of integers is essential for solving various algebraic expressions and problems in mathematics.