Problem 68 - Algebra A Combined Function

Question

Write each phrase as an algebraic expression. Use \(x\) for the unknown number. Three times a number

Step-by-Step Solution

Answer

The algebraic expression is \(3x\).

Step 1: Understanding the Phrase

The phrase we are given is 'Three times a number'. We need to express this phrase as an algebraic expression using a variable, in this case, \(x\), for the unknown number.

Step 2: Identifying the Operation

The word 'times' in the phrase indicates multiplication. Therefore, we will be multiplying by three to represent the 'three times' part of the phrase.

Step 3: Constructing the Expression

To construct the algebraic expression, multiply the variable \(x\) (the unknown number) by 3. This gives us the expression: \(3x\).

Key Concepts

Multiplication

When we encounter the word "times" in mathematical language, it immediately signals the operation of multiplication. This operation is essentially a shortcut for repeated addition. Instead of adding the same number several times, we can multiply it by how many times it is being added.
For example, saying "3 times 4" means you want to add 4 together 3 times, which is the same as multiplying 4 by 3 to get 12.
In algebra, multiplication is used extensively to simplify expressions and equations. When you see a phrase like "three times a number," we're expressing a relationship between two numbers where one number (a constant) is multiplied by an unknown number (a variable). In this instance, we quickly translate the word "times" into a multiplication problem using the symbol implicitly or explicitly. Hence, the expression "three times a number" becomes the algebraic expression: \(3x\).
Using multiplication in such expressions allows us to represent and solve problems efficiently, making it a fundamental tool in algebra and beyond.

Variable Representation

In algebra, the term "variable" refers to a symbol, often a letter, that stands for an unknown value. In many problems, particularly those involving word phrases, we use variables to translate words into mathematical language. In the problem we focus on, "use \(x\) for the unknown number," \(x\) represents that unknown number.
Variables serve several purposes in mathematics:

• They can represent numbers from a specific set.
• They provide a way to create general statements or rules.
• They allow for the representation of unknowns in an easily understandable manner.

In the example of the phrase "three times a number," \(x\) is the placeholder that lets us form an algebraic expression without knowing the actual number's value. This ability to use a variable makes it easier to explore, calculate, and solve equations and is foundational to algebraic thinking.

Algebra Basics

Algebra is a branch of mathematics that uses symbols to represent numbers and operations. It allows us to formulate equations and expressions that can solve problems involving unknown quantities. Understanding its basics is crucial for grasping more complex mathematical concepts later on.
One of the core ideas in algebra is converting phrases or real-life situations into mathematical expressions. This process involves:

• Identifying the unknowns and representing them with variables.
• Recognizing the operations described by the words (like "times" indicating multiplication).
• Constructing a mathematical statement or equation based on the given information.

In our exercise about "three times a number," we demonstrate these basics by turning the verbal phrase into a workable algebraic expression, \(3x\). This expression represents multiplying the unknown variable \(x\) by three. Such an approach simplifies complex problems into manageable steps, which can then be further analyzed or solved.