Problem 66 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. A number decreased by 7

Step-by-Step Solution

Answer

The expression is \( n - 7 \).

Step 1: Identify the Unknown

In the problem, the unknown number is mentioned as 'a number'. We are instructed to represent this unknown number using the variable \( n \).

Step 2: Understand the Operation

The phrase "decreased by 7" means that we need to subtract 7 from the unknown number. The word "decreased" translates to a subtraction operation.

Step 3: Formulate the Expression

Combine the identified variable \( n \) with the operation from step 2. Since we need to subtract 7 from \( n \), the algebraic expression becomes \( n - 7 \).

Key Concepts

Variable Representation

In algebra, a **variable** is a symbol used to represent an unknown or unspecified number. It acts as a placeholder that can hold different values. In our exercise, we represent the unknown number by using the variable \( n \). This choice is common because \( n \) is frequently used to denote numbers, though in general, any letter or symbol can serve as a variable.
Using variables makes it easier to write and manipulate expressions. They allow us to communicate complex mathematical ideas simply. In our problem, \( n \) stands for the number that we do not know yet, letting us easily construct algebraic expressions around it.

Subtraction Operation

Subtraction is one of the basic arithmetic operations. It represents the action of removing or taking away a certain quantity from another. In language, words like "decrease," "minus," "less," or "subtract" are commonly used to indicate this operation.
When we talk about a number decreased by 7, we mean that 7 is subtracted from that number. In mathematical terms, if we have a number represented by \( n \), subtracting 7 from \( n \) gives us \( n - 7 \).

• The minuend is \( n \) (the number we start with).
• The subtrahend is 7 (the amount we take away).

This concept is crucial in forming equations and expressions, helping us solve various mathematical problems.

Translating Phrases into Math

Translating words into mathematical expressions is a fundamental skill in algebra. It involves converting verbal phrases into algebraic expressions that accurately reflect their meaning.
This process often includes recognizing keywords that signify mathematical operations. For example, phrases like "increased by," "times," "divided by," and "decreased by" correspond to the operations of addition, multiplication, division, and subtraction, respectively.
In our exercise, the phrase "a number decreased by 7" translates into the algebraic expression \( n - 7 \). We:

• Identify the variable representing the unknown number (\( n \)).
• Recognize that "decreased by" implies subtraction.
• Construct the expression by subtracting 7 from \( n \).

This translation is an essential step in solving algebra problems, allowing us to create and solve equations based on word problems.