Problem 20 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number plus the opposite of six.

Step-by-Step Solution

Answer

The expression is \( x - 6 \).

Step 1: Define the Variable

First, let's define the unknown number mentioned in the sentence. We will denote this number as \( x \).

Step 2: Identify the Phrase 'Opposite of Six'

The phrase 'opposite of six' refers to the number that is opposite to 6 on the number line, which is \(-6\). So, the opposite of six is \(-6\).

Step 3: Translate 'A Number Plus the Opposite of Six'

Now, we translate the phrase 'a number plus the opposite of six' into a mathematical expression. The number is \( x \) and the opposite of six is \(-6\), so the expression is \( x + (-6) \).

Step 4: Simplify the Expression

We can simplify the expression \( x + (-6) \) to \( x - 6 \). This is the final mathematical expression that represents the given phrase.

Key Concepts

Understanding Variables

Variables are symbols used in algebra to represent unknown numbers or values. In this context, we use variables like letters (e.g., \( x \)) to denote numbers we don’t know yet.
This allows us to form expressions that can model real-world problems.

• Think of variables as placeholders. They hold the place for any number until we know its value.
• For example, when we say "a number plus the opposite of six," we don't know the exact number. So, we use \( x \) to represent it.

Variables are fundamental in algebra because they help us format equations and expressions, ultimately leading us to solutions.

Opposite Numbers in Algebra

Opposite numbers are numbers that, when added together, result in zero. They are located on opposite sides of zero on the number line.

• For instance, the opposite of 6 is \(-6\) and vice versa.
• Opposites are also called additive inverses. When you add a number to its opposite, like 6 and \(-6\), the sum is always 0.

Understanding opposite numbers is crucial in algebra because it helps us simplify expressions and solve equations effectively. They are especially helpful when dealing with subtraction and negative values.

Simplification of Algebraic Expressions

Simplification involves reducing expressions to their simplest form.
In the expression \( x + (-6) \), simplification allows us to write it as \( x - 6 \), making it easier to handle.

• To simplify, combine like terms or use basic arithmetic operations.
• For our example, because adding a negative number is the same as subtracting, \( x + (-6) \) becomes \( x - 6 \).

Simplifying expressions is important because it makes equations easier to manipulate and understand, which can streamline problem-solving processes significantly.