Problem 77 - Algebra for College Students

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Ten times the difference of a number and 6

Step-by-Step Solution

Answer

The algebraic expression is \(10(n - 6)\).

Step 1: Identify Keywords

The phrase "ten times the difference of a number and 6" contains key terms. "Difference" suggests subtraction, and "ten times" suggests multiplication by 10.

Step 2: Define the Expression Inside

The phrase "the difference of a number and 6" indicates subtraction between the unknown number and 6, which can be written as \(n - 6\).

Step 3: Apply the Multiplication

The phrase "ten times" means multiplying the expression \(n - 6\) by 10, leading to the algebraic expression \(10(n - 6)\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a fundamental part of algebra. They allow us to represent real-world situations using mathematical symbols and operations. An algebraic expression consists of numbers, variables, and arithmetic operations such as addition, subtraction, multiplication, and division. In our exercise, we are asked to translate a phrase into an algebraic expression. The key elements of the phrase are recognized and then mathematically formulated. This process involves understanding the language of math and identifying keywords that guide the formation of the expression. Recognizing terms like "ten times" and "the difference" helps to know the operations—multiplication and subtraction—we need to use. Understanding how to create algebraic expressions from word problems is crucial for solving real-world problems. It essentially bridges the gap between words and mathematical equations.

Unknown Variables

In algebra, unknown variables are symbols, often letters, used to represent values that are not known yet. In most problems, the unknown variable is what we are trying to solve for or simplify an expression around. In our example, we use the letter \(n\) to represent an unknown number. Understanding unknown variables is key because they allow us to manipulate expressions and solve equations to find particular values. Using \(n\) helps maintain the general form of a problem until we have enough information to find out what number \(n\) actually represents. This technique helps in setting up equations and working through complex problems by providing a placeholder that can take any value. It gives us the flexibility to handle different scenarios by substituting the variable with specific numbers when additional information is available.

Multiplication and Subtraction

Multiplication and subtraction are two of the basic arithmetic operations used to form algebraic expressions. In our problem, we need to identify the operations indicated by the language of the phrase.

• Multiplication: The phrase "ten times" indicates we will be multiplying a number by 10. In algebra, multiplication compresses repeated addition into a more straightforward operation. Applying this to algebraic expressions, it allows us to expand or simplify expressions efficiently.
• Subtraction: The term "difference" suggests a subtraction operation. Here, "the difference of a number and 6" means we take a number \(n\) and subtract 6. Subtraction is used for calculating decreases or finding how much more one value is than another.

By understanding and applying these operations, we can transform a verbal phrase into a mathematical statement: \(10(n - 6)\). This encompasses both multiplication (ten times) and subtraction (difference). Properly interpreting and applying these operations is crucial in solving algebraic problems efficiently.