Problem 30 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. the sum of a number and 6

Step-by-Step Solution

Answer

The algebraic expression for the phrase 'the sum of a number and 6' is \(x + 6\)

Step 1: Identify the variable

To interpret this phrase mathematically, identify 'a number' as the variable. As instructed, we will use the variable \(x\) as 'a number'.

Step 2: Recognize the operation

The phrase uses the term 'sum', which signifies an operation of addition in mathematics.

Step 3: Translate the phrase into a mathematical operation

The sum of 'a number' and 6 would therefore translate into \(x + 6\), where \(x\) represents 'the number'.

Key Concepts

Variables in Algebra

In algebra, variables are symbols, often letters, used to represent unknown quantities. They play a fundamental role as placeholders for numbers in equations and expressions. In our exercise, the phrase "a number" is key. Here, the variable chosen to represent "a number" is denoted by the letter \( x \).

This assignment allows us to work with the value without knowing exactly what it is.

• Variables can stand for different numbers in different problems or for the same unknown in a single problem.
• They serve as a way to generalize problems and solutions.
• Choosing an appropriate variable is crucial for correctly setting up an algebraic expression.

By using a variable, complex relationships involving numbers become simpler to model and solve.

Mathematical Operations

Mathematical operations are basic actions that we use to manipulate numbers. The four main operations in mathematics are addition, subtraction, multiplication, and division. In word problems or algebraic expressions, these operations are often indicated by specific keywords.

For the exercise in question, the key word "sum" is used in the phrase "the sum of a number and 6". "Sum" signals that the operation of addition is required.

• Addition (indicated by the symbol \(+\)) combines numbers to find their total.
• Subtraction (indicated by \(-\)) finds the difference between numbers.
• Multiplication (indicated by \(\times\) or \(\cdot\)) calculates the total when numbers are repeated.
• Division (indicated by \(\div\) or fractions) distributes one number into a specified number of parts.

Understanding which operation to use based on keywords helps transform verbal instructions into algebraic expressions that can be solved or simplified.

Phrase Translation in Math

Phrase translation in mathematics involves converting English words into mathematical language. This skill is essential as many word problems and instructions are given in verbal form. To correctly translate a phrase, one must identify both the variable and the mathematical operation involved.

In this exercise, the phrase "the sum of a number and 6" needs translation. Here’s how it works:

• Identify the variable, which in this case is "a number" (\(x\)).
• Recognize the mathematical operation associated with "sum", which means addition.
• Combine the variable and the operation with other numbers mentioned to form the expression. Thus, "the sum of a number and 6" converts to \(x + 6\).

Effective translation allows you to transform real-world situations into mathematical problems that can be assessed and solved using algebra.