Problem 87 - Introductory Algebra for College Students

Question

Let x represent the number and write the phrase as an algebraic expression. The quotient of 13 and a number, decreased by 7 times the number

Step-by-Step Solution

Answer

The algebraic expression for the given phrase is \(\frac{13}{x} - 7x\).

Step 1: Translating Phrases

The first part of the sentence, 'The quotient of 13 and a number', could be written as \(\frac{13}{x}\) where x represents the number. The phrase 'a quotient' typically refers to the result of a division.

Step 2: Identifying Mathematical Operations

'Decreased by' usually means subtraction. '7 times the number' could be written as \(7x\). The entire phrase 'decreased by 7 times the number' translates to '- 7x'

Step 3: Combining the Translations and Writing the Final Expression

Finally, putting both parts together, the phrase 'The quotient of 13 and a number, decreased by 7 times the number' can be written as an algebraic expression: \(\frac{13}{x} - 7x\).

Key Concepts

Understanding Quotient

A quotient is the answer you get when you divide one number by another. In algebra, it's helpful to think of it as a division operation between two numbers.
When you see the phrase "the quotient of\( a \) and\( b \)," it means \( \frac{a}{b} \). This is a common way to express division in an algebraic expression.

• For example, the quotient of 13 and\( x \) is written as\( \frac{13}{x} \). This expression shows how 13 is divided by\( x \), where\( x \) represents a yet-to-be-known number or value.

Understanding quotients helps us translate phrases into mathematical expressions, enabling us to solve problems more effectively.

Working with Variables

In algebra, variables are symbols that represent numbers. They are like placeholders.
The most common variable is \( x \), and we often see phrases asking us to express relationships in terms of \( x \).

• Using variables, we can describe complex relationships simply and clearly. If you are given, \( 7x \), this means 7 multiplied by the variable \( x \).
  This type of expression is useful when we do not know the exact number\( x \) represents yet, but understand that it stands for an unknown value.

Variables allow us to write equations and expressions that can be manipulated to find unknown values.

Basics of Subtraction

Subtraction is a basic mathematical operation that indicates removing a quantity from another. In phrases like "decreased by," subtraction is the operation at play.
When spoken about in an algebraic context, subtraction helps adjust calculations or measurements, showing how much less one value is compared to another.

• For the expression \( 7x \), if we have a context where something is decreased by this, it signifies \(-7x\), meaning we subtract 7 times the unknown \( x \) from another value.

Subtraction is not just about counting down; it's also a fundamental operation that helps us form and understand algebraic expressions.

Translating Phrases into Expressions

Translating phrases into algebraic expressions is about turning words into mathematical symbols.
This translation is crucial in solving problems and understanding conditions that are described verbally.

• For example, the phrase "the quotient of 13 and\( x \), decreased by 7 times the number" translates into the expression \( \frac{13}{x} - 7x \). Here, each part of the phrase corresponds to a specific mathematical operation or term.

The process involves identifying key words that indicate mathematical operations. Understanding this translation process is essential for tackling algebraic problems efficiently.