Problem 48 - Algebra A Combined Function

Question

Rewrite each sentence using mathematical symbols. Do not solve the equations. If the quotient of twice a number and 7 is subtracted from the reciprocal of the number, the result is 2 .

Step-by-Step Solution

Answer

\( \frac{1}{x} - \frac{2x}{7} = 2 \)

Step 1: Identify the Unknown

First, let's identify the unknown number in the sentence. We'll use the variable \( x \) to represent this number.

Step 2: Translate "Twice a Number"

Twice a number can be expressed mathematically as \( 2x \).

Step 3: Translate "Quotient of Twice a Number and 7"

The quotient of twice a number and 7 is given by \( \frac{2x}{7} \).

Step 4: Translate "Reciprocal of the Number"

The reciprocal of a number \( x \) is given by \( \frac{1}{x} \).

Step 5: Formulate the Overall Expression

According to the sentence, the quotient of twice a number and 7 is subtracted from the reciprocal of the number. This can be expressed as \( \frac{1}{x} - \frac{2x}{7} \).

Step 6: Set the Equation

The result of the expression is equal to 2. Thus, we set the equation: \( \frac{1}{x} - \frac{2x}{7} = 2 \).

Key Concepts

Mathematical Symbols

Mathematical symbols serve as the language of mathematics. They allow us to represent complex concepts in a simple and efficient manner. When we encounter a text problem, identifying the right symbols is crucial to translating it accurately into a mathematical expression.

Here are a few common mathematical symbols and their meanings:

• Plus (+): Used for addition.
• Minus (-): Represents subtraction.
• Times (× or *): Indicates multiplication.
• Division (/): Used to express division or a quotient.
• Equals (=): Shows that two expressions are equivalent.

In algebra, symbols also include more abstract representations like variables and formulas. For example, the expression for twice a number might look like this: \( 2x \).

Understanding these symbols enables us to decode problems and set the foundation for further algebraic manipulation.

Translating into Algebra

Translating verbal statements into algebraic expressions involves recognizing patterns and relationships in a sentence. Start by identifying keywords that suggest mathematical operations or relationships.

Consider the phrase: "the quotient of twice a number and 7." Here, "twice a number" becomes \( 2x \), and "quotient" suggests division. Therefore, it turns into \( \frac{2x}{7} \).

Another key term, "reciprocal," translates a number \( x \) into \( \frac{1}{x} \). Using these translations, we convert complex sentences into manageable algebraic forms.

This process of translation is essential for forming equations that model real-world situations. Effective translation relies on understanding both the mathematics and the language used in the problem.

Setting Up Equations

To tackle problems involving unknown values, setting up equations is essential. An equation shows the relationship between different quantities. It is formed by setting expressions equal to each other and often looks like a balanced scale.

When creating an equation from a sentence, ensure that the expression reflects the relationships described in the text. For the example given: "If the quotient of twice a number... the result is 2," first translate the verbal description to \( \frac{1}{x} - \frac{2x}{7} = 2 \).

This equation tells us that the difference between the reciprocal of a number and the quotient of twice the number by 7 equals 2. Equations allow us to understand the balance in a mathematical statement and serve as a tool for finding unknown values.

Variables in Mathematics

Variables are symbols used to represent unknown numbers or values in mathematical statements. They are typically denoted by letters like \( x \), \( y \), or \( z \). In our exercise, the variable \( x \) is used to denote the unknown number.

Why use variables? They provide flexibility and generality, allowing equations to be more universally applicable. With variables, you can solve not just for a specific number but for any number that fits the equation's conditions.

Understanding variables is key in algebra as they transition abstract concepts into tangible equations. This knowledge empowers you to explore a range of possible solutions, making math not just about numbers but about discovering relationships between them.