Problem 29 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

\(x + 4\)

Step 1: Identify the Operation

The phrase 'the sum of' indicates an addition operation in mathematics.

Step 2: Identify the Numbers

The given phrase also mentions 'a number' and '4'. In this context, 'a number' refers to the variable \(x\) and '4' is a constant.

Step 3: Formulate the Algebraic Expression

Since 'sum' indicates an addition operation, the English phrase 'the sum of a number and 4' can be written as the algebraic expression: \(x + 4\). This expression represents the sum of the variable \(x\) and the constant 4.

Key Concepts

Translating English to Algebra

Understanding how to convert natural language phrases into algebraic expressions is a foundational skill in algebra. This process involves interpreting words and phrases to determine which mathematical operations they represent and then constructing an expression using variables and constants.

For example, the phrase 'the sum of a number and 4' requires identifying the operation and the elements involved. Here, 'sum' translates to addition, 'a number' implies a variable (commonly represented as \(x\)), and '4' is a constant. The algebraic expression formulates as \(x + 4\).

• 'The sum of' translates to '+' (addition).
• 'A number' is a placeholder for an unknown and is represented by a variable.
• Specific numbers, like '4' in this case, are constants in the expression.

Practice becomes essential, as phrases can indicate different operations such as 'the product of' for multiplication or 'the quotient of' for division. Recognizing these keywords and their corresponding operations is crucial for accurate translation.

Variables in Algebra

In algebra, a variable is a symbol, often a letter, that represents a numeric value that can change or vary. It is the unknown element in an expression or an equation.

Using a variable like \(x\) in our example, allows for the representation of a range of numbers rather than a single fixed number. Variables are essential because they give us a way to articulate general rules about how numbers behave.

• A variable can take any value: \(x\) could be 1, 2, 100, or any other number.
• In an algebraic expression, a variable interplays with constants and other variables through arithmetic operations.
• Variables can represent quantities in real-world scenarios, making algebra a powerful tool for solving practical problems.

Understanding how to work with variables is important for progressing in algebra and for later application in real-world problem solving and higher mathematics.

Arithmetic Operations in Algebra

Arithmetic operations are fundamental processes in algebra that involve numbers and variables. The basic operations are addition (+), subtraction (-), multiplication (×), division (÷), and exponentiation (^).

When constructing or simplifying algebraic expressions, these operations have a defined order of execution known as the order of operations, which can be remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

• Addition is represented by '+', indicating the combination of numbers or variables.
• Subtraction is represented by '-', expressing the difference between numbers or variables.
• Multiplication can be denoted by '×', a dot '\(\cdot\)', or even parentheses '\(()\)' when variables are involved.
• Division is commonly symbolized by '÷', a slash '/', or a fraction bar.
• Exponentiation indicates a number or variable raised to a power, like \(x^2\), which means \(x\) multiplied by itself.

Recognizing how these operations are expressed in algebraic terms is vital for solving equations and understanding more complex mathematical concepts.