Problem 72 - Intermediate Algebra

Question

Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. Seven more than three times a number

Step-by-Step Solution

Answer

\(3n + 7\)

Step 1: Identify the Unknown

The unknown number in this problem is represented by the variable \(n\). So, let's start by identifying \(n\) as our variable.

Step 2: Translate 'Three Times a Number'

The phrase 'three times a number' means we need to multiply \(n\) by 3. This can be written algebraically as \(3n\).

Step 3: Add Seven to the Expression

Now, we need to incorporate 'seven more than' into our expression. This means we add 7 to \(3n\). Thus, the expression becomes \(3n + 7\).

Step 4: Final Expression

Combining all the steps, translating 'seven more than three times a number' to an algebraic expression gives us \(3n + 7\).

Key Concepts

Understanding Unknown Variables

When dealing with algebraic expressions, unknown variables are often used to represent quantities that are not specified. In the example from the exercise, we are tasked with determining the expression for a phrase. Here, the unknown variable is represented by the letter \(n\).
This variable \(n\) stands for the unknown number mentioned in the phrase. This allows us to manipulate and transform the expression algebraically, even though we don't know the exact value of \(n\) yet.
Using a variable is helpful because it lets us express the relationship between numbers abstractly. This is a crucial concept in algebra, where solving for unknowns is often the main goal. When using unknown variables, always assign them clearly to avoid confusion.

Translation of Phrases into Expressions

Translating phrases into algebraic expressions involves interpreting the language of the problem and converting it to mathematical symbols. Let's break down the exercise phrase: 'seven more than three times a number.'

• First, identify "three times a number." It means we multiply the variable \(n\) by 3, resulting in the expression \(3n\).
• Next, the phrase "seven more than" suggests an addition. It implies you take what you have (\(3n\)) and add 7 to it.

This process of translation requires understanding the meaning behind words like "more than," "times," or "add," among others, as these direct the mathematical operations we'll use. Such linguistic translation is a foundational skill for crafting the correct algebraic expressions from word problems.

Performing Algebraic Operations

Once we've translated phrases into parts of an expression, it's time to perform the necessary algebraic operations to find the final expression. With our example, we start with \(3n\), which is "three times a number".
To fit the phrase "seven more than," we add 7 to this term, resulting in \(3n + 7\). The operations involved here include:

• Multiplication: Represented by the product of 3 and \(n\), creating \(3n\).
• Addition: Adding 7 to \(3n\) translates the expression into \(3n + 7\).

Understanding and performing these operations correctly is crucial. The order matters: first, calculate the multiplicative part of the expression (\(3n\)), then proceed to addition (adding 7). This approach ensures you translate and perform operations in a logical and intended sequence.