Problem 64 - Elementary Algebra Within Reach

Question

In Exercises 63-68, translate the verbal phrase into an algebraic expression. Simplify the expression. $$ n \text { times the difference of } 6 \text { and } n $$

Step-by-Step Solution

Answer

The simplified expression of 'n times the difference of 6 and n' is \(6n - n^2\).

Step 1: Translate the verbal phrase into an algebraic expression

Translate 'n times the difference of 6 and n' into \(n * (6 - n)\).

Step 2: Distribute multiplication over subtraction

Use the distributive property to multiply \(n\) with all terms inside the brackets. This results in the new expression \(6n - n^2\).

Step 3: Final Simplification

There is no further simplification possible for the expression \(6n - n^2\). It is already in its simplest form.

Key Concepts

Distributive Property

The Distributive Property is a fundamental concept in algebra that helps in expanding expressions and simplifying calculations. This principle allows multiplication to spread out over addition or subtraction inside parentheses. For example, when we encounter an expression like \( n \times (6 - n) \), the distributive property tells us we can perform the following steps:

• Multiply \( n \) by each term inside the parentheses separately.
• This provides the expression \( 6n - n^2 \).

By distributing the multiplication across each term, you ensure each element inside the parentheses is given equal treatment in regards to multiplication. This results in a tidy algebraic expression that is often easier to simplify or analyze further. It is a crucial technique that comes into play in many algebraic scenarios.

Simplification

Simplification in algebra involves making an expression easier to understand and work with, while still maintaining its original value. After applying the distributive property, you might end up with something like \( 6n - n^2 \). Here, we check if any like terms can be combined. Like terms are terms that have the same variable raised to the same power. In this example, there are no like terms to combine, which means the expression is already as simple as possible.

Sometimes, simplification can include factoring, canceling common factors, or reducing fractions. However, in this particular case involving \( 6n - n^2 \), there is no further simplification needed. It's important to recognize when an expression is simplified, as this prevents unnecessary and sometimes confusing extra steps. Keeping expressions simplified makes solving equations or applying further algebraic methods easier and more efficient.

Verbal to Algebraic Translation

Translating verbal phrases into algebraic expressions is a crucial skill in algebra that allows you to convert real-world problems into mathematical formulas. This process requires understanding mathematical operations and recognizing how they are described in words. For example, the phrase "\( n \) times the difference of 6 and \( n \)" was translated into the algebraic expression \( n \times (6 - n) \).

• "Times" signals multiplication.
• "Difference" indicates subtraction.

First, identify the order of operations as expressed in the verbal phrase. Recognize the keywords, such as 'times' and 'difference', which indicate the operations required. Then, accurately represent the structure of operations using appropriate mathematical symbols and grouping symbols like parentheses to keep the operations in the correct order.

This translation process is not just a foundational skill in algebra, but it also enhances problem-solving abilities in real-life situations. By practicing converting from words to algebraic expressions, you improve in interpreting problems into a mathematical context, facilitating easier calculation and analysis.