Problem 73 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Then simplify the expression. Let \(x\) represent the number. six times the sum of 4 and a number

Step-by-Step Solution

Answer

The English phrase 'six times the sum of 4 and a number' as an algebraic expression is \(6 * (4 + x)\) and when simplified it becomes \(24 + 6x\).

Step 1: Interpret the English Phrase

We are given the phrase 'six times the sum of 4 and a number'. Here, 'six times' indicates multiplication, 'the sum of 4 and a number' means we have to add 4 to a certain number, which is represented by \(x\).

Step 2: Translate the Phrase into an Algebraic Expression

Now that we know 'six times the sum of 4 and a number' means we multiply 6 with the result of adding 4 to a number \(x\) we write it as an algebraic expression: \(6 * (4 + x)\).

Step 3: Simplify the Algebraic Expression

To simplify this expression, we use the distributive property of multiplication over addition. So multiply 6 with both 4 and \(x\). The simplified expression would be \(24 + 6x\).

Key Concepts

basic algebra

In basic algebra, we translate real-world scenarios into mathematical expressions. This involves recognizing keywords such as "sum," "times," and "difference," which correspond to mathematical operations – addition, multiplication, and subtraction, respectively.
Understanding these keywords is crucial for forming correct algebraic expressions. For example, in the phrase "six times the sum of 4 and a number," "six times" signals that multiplication is involved, while "sum of 4 and a number" indicates addition.
Converting phrases into algebraic expressions requires recognizing that any unknown amount can be represented by a variable. Variables such as \(x\) serve as placeholders, making it easier to manipulate the expression algebraically.

simplifying expressions

Simplifying expressions is a basic skill in algebra, and it involves reducing the expression to its simplest form while maintaining its original value. This often includes combining like terms or performing arithmetic operations.
When faced with a phrase such as 'six times the sum of 4 and a number,' after translating it into the algebraic expression \(6 * (4 + x)\), the next aim is simplification.

• First, ensure all arithmetic operations inside the parentheses have been completed. In this example, ensure the sum within the parentheses, \(4 + x\), is clear.
• Next, use mathematical properties to simplify, such as the distributive property, to break down any complex expressions.

Remember, simplifying makes the expression easier to understand and work with in further calculations or applications.

distributive property

The distributive property is a fundamental concept in algebra that facilitates simplifying expressions involving multiplication over addition or subtraction.
Mathematically, it states that for any three numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\). This property allows us to expand expressions and combine terms efficiently.
Consider the expression \(6 * (4 + x)\). Here, you distribute \(6\) across each term inside the parentheses, resulting in \(6 * 4 + 6 * x\).

• The product \(6 * 4\) equals \(24\).
• The product \(6 * x\) translates into \(6x\).

Thus, the expression simplifies to \(24 + 6x\). Understanding and applying the distributive property is essential as it sets the stage for solving more complex algebraic equations.