Problem 74 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. nine times the sum of 3 and a number

Step-by-Step Solution

Answer

The English phrase 'nine times the sum of 3 and a number' translates to the algebraic expression '9 * (3 + x)', which simplifies to '27 + 9x'.

Step 1: Rewrite the English phrase as an algebraic expression

The key words 'nine times' implies multiplication, and 'the sum of 3 and a number' implies addition. Therefore, the English phrase can be written as an algebraic expression as ‘9 * (3 + x)’.

Step 2: Simplify the Algebraic Expression

To simplify the expression, we distribute '9' to each term inside the parentheses, using the distributive property of multiplication over addition. The simplified algebraic expression is \( 9*3 + 9*x \), which simplifies further to \( 27 + 9x \).

Key Concepts

Distributive Property

The distributive property is a useful tool in algebra that helps us to simplify expressions and make calculations easier. It essentially involves multiplying a single term by all terms within a set of parentheses. This property allows us to eliminate the parentheses and rewrite the expression in a more manageable form.
In mathematical terms, the distributive property can be stated as:

• If you have an expression in the form of \( a(b+c) \), it can be expanded to \( ab + ac \).

Let's take a closer look at how this property was applied in the original exercise:

• We had the expression \( 9(3 + x) \).
• By applying the distributive property, we multiplied 9 by each of the terms inside the parentheses: \( 9 imes 3 \) and \( 9 imes x \).
• This gave us the expanded expression \( 27 + 9x \).

Simplification of Expressions

Simplifying an expression means rewriting it in the simplest or most compact form possible. This often involves combining like terms, applying the distributive property, and performing any straightforward arithmetic operations.
In the example we worked with:

• Initially, we started with \( 9(3 + x) \).
• Using the distributive property, we expanded this to \( 27 + 9x \).

Each part of the expression can then be examined to see if further simplification is possible. In this case, \( 27 + 9x \) is already in its simplest form:

• There are no like terms to combine, as 27 is a constant and \( 9x \) is a linear term,
• Thus, we've reached the end of the simplification process.

Translating Phrases into Expressions

One foundational skill in algebra is translating verbal phrases into algebraic expressions. This is crucial because it allows us to model real-world situations mathematically. To do this, we need to understand the language of algebra and the operations it describes.
In our given exercise:

• The phrase was "nine times the sum of 3 and a number."
• "Nine times" indicates a multiplication operation.
• "The sum of 3 and a number" indicates adding 3 and an unknown number, which is represented as \( x \).

So, the phrase translates into the algebraic expression \( 9(3 + x) \).
A careful interpretation of words and using a placeholder like \( x \) for an unknown or variable makes this process effective and accurate. Understanding the keywords and their mathematical operations helps bridge speech and numbers seamlessly, forming the backbone of algebraic expressions.