Problem 35 - Introductory Algebra for College Students

Question

Write each English phrase as an algebraic expression. Let the variable \(x\) represent the number. three times a number, decreased by 5

Step-by-Step Solution

Answer

The algebraic expression for 'three times a number, decreased by 5' where \(x\) represents the number is \(3x - 5\).

Step 1: Identify the main operations

The main operations in this exercise are multiplication (represented by 'times') and subtraction (represented by 'decreased by'). Recognizing that these mathematical operations correspond to the words 'times' and 'decreased by' is the first step.

Step 2: Translate the operations and numbers into algebra

Next, substitute 'a number' with the variable \(x\). 'Three times a number' means we multiply \(x\) by 3, giving us \(3x\). We then subtract 5 from \(3x\) according to the 'decreased by 5' part of the phrase. So, the completed algebraic expression is \(3x - 5\).

Key Concepts

Mathematical Operations

Mathematics is a language of its own, using symbols and operations to express complex relationships. In this exercise, we look at two fundamental math operations: multiplication and subtraction. These operations help us translate words into algebraic expressions. When you hear "times", it signifies multiplication, where you multiply two numbers or variables. Similarly, when you encounter "decreased by", it indicates subtraction, reducing the value by a specific number or expression. Understanding these operations is the key to turning phrases into equations.

• Multiplication: Usually represented by variables or numbers with a multiplication sign, like '3 times a number' becomes \(3x\).
• Subtraction: Reduction by a value, expressed with a minus sign. 'Decreased by 5' translates to '-5'.

Recognizing these keywords simplifies the process of writing mathematical expressions from text.

Variable Representation

In algebra, variables play a crucial role. They are symbols that represent unknown values, often denoted by letters like \(x\), \(y\), or \(z\). These variables stand in for numbers we do not know yet in a problem or that can take on multiple values. In the given exercise, the phrase "a number" refers to our variable, \(x\). By using \(x\), we create a representation that can be used and manipulated in equations.

• Variables are placeholders for unknown values or numbers in equations.
• Commonly represented by letters, they allow us to generalize problems and find solutions.
• Choosing variables wisely simplifies the translation of verbal expressions into mathematical ones.

Grasping the concept of variables is fundamental; it acts as the foundation for constructing algebraic expressions from phrases.

Translation of Phrases into Algebra

Transforming English phrases into algebraic expressions is a critical skill in algebra. It requires you to pick out mathematical operations and apply them to variables. When translating, you must maintain the order and structure implied by the words. Let's break it down:- **Identify Operations**: Determine which operations are present. In our example, "three times" and "decreased by" point to multiplication and subtraction.- **Choose Variables**: Decide what the variables will stand for. The exercise states "a number", which we represent with the variable \(x\).- **Combine Elements**: Follow the instructional words to combine numbers and operations with the variable. "Three times a number, decreased by 5" translates to \(3x - 5\).Being adept at this translation process enhances problem-solving skills and boosts mathematical reasoning. It's all about dissecting the language and converting it into the precise language of math.