Problem 32 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. three subtracted from a number

Step-by-Step Solution

Answer

The algebraic expression is \(x - 3\).

Step 1: Identify the Operations

‘Three subtracted from a number’. The operation involved here is subtraction, indicated by the verb 'subtracted.'

Step 2: Identify the Variables

It is mentioned in the problem to let \(x\) represent 'a number'.

Step 3: Formulate the Expression

In this case, 'three subtracted from a number' means that we subtract three from the variable \(x\). It will be represented as \(x - 3\) in algebraic form.

Key Concepts

Understanding Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to express mathematical relationships. These symbols are often represented as letters, known as variables. Algebra allows us to formulate equations and expressions which can model real-world scenarios or solve practical problems.

One of the key skills in algebra is the translation of verbal statements or sentences into mathematical expressions. For example, phrases like 'three times a number' or 'five less than a number' can be transformed into expressions using variables and operations like multiplication or subtraction.

To effectively use algebra, understanding how different operations affect expressions is critical. Each operation, whether it's addition, subtraction, multiplication, or division, follows specific rules that we must apply consistently to solve problems. This organized approach helps simplify and solve equations accurately.

Exploring Variables

Variables are fundamental in algebra, representing unknown or changeable quantities. Typically, variables are denoted by letters such as \(x\), \(y\), or \(z\). In an expression or equation, they stand in for numbers that might be unknown or that we want to vary.

For instance, when we say 'three subtracted from a number,' we use a variable, say \(x\), to represent 'a number'. This allows us to write the expression \(x - 3\). The variable \(x\) in this expression signifies any number we might choose, providing a comprehensive way to represent different possibilities.

Understanding variables and how to use them effectively is a cornerstone of algebra. It enables you to create versatile mathematical models that can adapt to different values and provide solutions to numerous types of problems. This flexibility is what makes algebra so powerful.

The Art of Subtraction

Subtraction is one of the basic arithmetic operations we use to solve algebraic expressions. It involves finding the difference between numbers or expressions. When given a phrase such as 'three subtracted from a number,' subtraction is the operative technique needed to form the mathematical expression.

In algebraic terms, subtraction can be expressed using a minus sign (−). The phrase 'three subtracted from a number' translates to \(x - 3\). This structure is critical: the number 3 is taken away from the variable \(x\).

• This process requires you to be attentive to the order of terms. It's crucial to subtract the right way round (three from the number, not the number from three).
• Understanding subtraction helps you interpret various algebraic problems more accurately.

Subtraction isn't just about removing quantity; it's also about understanding relationships between numbers and variables, which is integral to solving algebraic equations.