Problem 66 - Algebra for College Students

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

The problem statement mentions "a number," which we need to represent with a variable. We will use \( n \) to represent this unknown number.

Step 2: Interpret the Phrase

The phrase "decreased by 7" indicates that from the unknown number, 7 is subtracted.

Step 3: Write the Algebraic Expression

Combine the information from the previous steps to write the algebraic expression. Since we're subtracting 7 from \( n \), the expression is \( n - 7 \).

Key Concepts

Translating Phrases into Expressions

When faced with real-world situations or word problems, one key skill in algebra is translating English phrases into algebraic expressions. This allows us to work mathematically with statements that initially appear as words. You need to identify key terms in the sentence that relate to mathematical operations. For instance, words like "sum," "difference," "increased by," and "decreased by" all suggest specific operations.

• "Sum" means addition.
• "Difference" refers to subtraction.
• "Product" indicates multiplication.
• "Quotient" signifies division.

Once you spot these terms, translate them into an equation using symbols and numbers. For example, if the phrase is "a number decreased by 7," recognize "a number" as a variable we often denote as \( n \), and "decreased by 7" as subtracting 7. Thus, the expression becomes \( n - 7 \). Understanding this mapping between language and mathematics is crucial for problem solving.

Unknown Variables

In algebra, an unknown variable represents a number that we don't yet know. It acts like a placeholder in equations and expressions, allowing us to manipulate and solve the mathematical statement. Variables are typically denoted by letters such as \( x \) or \( n \), but you can use any letter.

When you encounter phrases like "a number," "an amount," or "the price," these indicate variables. Since we don't have specific values, these variables allow flexibility and are essential in forming algebraic expressions. In our example, "a number" is represented by the variable \( n \). This permits addressing a range of possible values, making algebra a powerful tool for generalization and abstraction in mathematics.

Subtraction in Algebra

Subtraction in algebra is not much different from basic subtraction, but it involves working with variables and expressions. The operation of subtraction is indicated by terms like "minus," "less than," "decreased by," and "subtract." These keywords alert you to the task of taking one term away from another.Using subtraction in algebraic expressions requires attention to the order of terms, as it directly affects the outcome. For instance, in an expression like \( n - 7 \), we subtract 7 from \( n \), correctly reflecting the word problem: "a number decreased by 7."

• Be mindful of subtraction involving negative numbers, which can add complexity.
• The associative and commutative properties do not apply to subtraction, so the order matters.

Subtraction helps in solving equations and modeling situations where you're dealing with decreases or deficits, making it crucial for real-world applications and mathematical reasoning.