Problem 60 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Five times a number is 35

Step-by-Step Solution

Answer

The equation is \(5x = 35\).

Step 1: Identify the unknown

The problem mentions a specific 'number'. Let's denote this unknown number as \(x\).

Step 2: Translate the words into mathematical terms

'Five times a number' can be written mathematically as \(5x\). 'Is' usually means '=', and '35' remains as 35.

Step 3: Formulate the equation

Putting the pieces together, 'Five times a number is 35' translates to the equation \(5x = 35\).

Key Concepts

Algebraic Expressions

Understanding algebraic expressions is like learning a new language that describes patterns and relationships in mathematics. In the exercise, we encounter the phrase 'five times a number', which needs to be translated into a mathematical expression. An algebraic expression is a concise way of representing such verbal statements using variables, numbers, and arithmetic operations.

In our case, the 'number' is the unknown we want to represent, assigned the variable name 'x'. The term 'five times' indicates multiplication. Thus, 'five times a number' becomes the algebraic expression '5x'. These expressions are crucial building blocks that lead us towards forming equations.

It's important for students to recognize key phrases and their corresponding mathematical operations: 'times' for multiplication, 'sum' for addition, 'difference' for subtraction, and 'quotient' for division. Practice with various examples helps reinforce these concepts and improves one's ability to form correct algebraic expressions.

Equation Formulation

When it comes to equation formulation, the goal is to create a mathematical sentence that accurately represents a given situation or condition. An equation is made up of two algebraic expressions connected by an equals sign \(=\), indicating that the two expressions have the same value.

In our textbook exercise, we convert the statement 'Five times a number is 35' into the equation \(5x = 35\). The word 'is' translates to the equals sign, showing that the product of 'five times a number' is equal to '35'. Formulating an equation often involves this process of identifying connections between language and mathematical symbols.

Students should practice translating verbal statements into equations step by step, starting by identifying the unknowns, translating words into mathematical operations, and then writing the complete equation. This step-by-step approach simplifies complex problems and lays a strong foundation for solving equations.

Variable Representation

Variable representation plays a critical role in mathematics, especially in algebra. It allows us to symbolize unknown or changing quantities, which we can then work with in equations and expressions. In the given exercise, the variable \(x\) is used to represent the unknown number we're looking to find.

Choosing an appropriate variable is like giving a name to an unknown quantity, which we can reference throughout our calculations. It's important to note that while 'x' is commonly used, any letter or symbol can serve as a variable. What's essential is consistency; once we define a variable, we use the same symbol to represent that quantity throughout the problem.

Variables help generalize problems, making it easier to see patterns and solve more complex problems. By mastering variable representation, students gain the ability to tackle a wide range of mathematical challenges, as it is the foundation of working with equations and functions in algebra.