Problem 63 - Introductory and Intermediate Algebra for College Students 4th

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. The difference between 20 and a number is 5 .

Step-by-Step Solution

Answer

The algebraic equation derived from the given sentence is: \(20 - x = 5\)

Step 1: Identify the unknown and the constants

The unknown in this scenario is a number, which the question has suggested we denote by \(x\). The constants are 20 and 5, which are given in the problem statement.

Step 2: Interpret the word problem

The word 'difference' suggests a subtraction operation. In the given word problem, the difference between 20 and an unknown number is 5. Therefore, the situation can be written in mathematical terms as \(20 - x = 5\).

Step 3: Formulate the algebraic equation

The algebraic representation of the given sentence is thus \(20 - x = 5\). This equation means that if we subtract \(x\) from 20, we should obtain 5.

Key Concepts

Word Problems

Word problems can often seem tricky at first, but the key is to translate the words into a mathematical equation. Word problems describe a situation in everyday language that involves numbers and requires solving.
The crucial steps to tackle word problems are:

• Understand the problem: Read it carefully and ensure you know what it is asking.
• Identify important information: Look for numbers and keywords that imply mathematical operations.
• Translate into an equation: Turn the words into a math statement using variables and operations.

In our example, the problem states, "The difference between 20 and a number is 5." The key term here is "difference," which suggests subtraction. The phrase "a number" is what you'll need to find, and often suggests using a variable like \(x\). Understanding this process is crucial for forming equations that you can solve.

Variable Representation

With word problems, effectively using variables is essential. Variables are symbols used to represent unknown values in an equation, and they allow us to work with mathematical concepts abstractly.
In our exercise, we used the variable \(x\) to represent the unknown number. This choice is not arbitrary; we use \(x\) or other letters like \(y\) or \(z\) according to preference or convention.
The steps in representing variables are:

• Choose a letter to represent the unknown quantity.
• Replace descriptive phrases like "a number" with your chosen variable.
• Use this variable consistently to set up and solve the equation.

By using \(x\), the word problem "The difference between 20 and a number is 5" becomes "The difference between 20 and \(x\) is 5" and leads us towards setting up the subtraction equation \(20 - x = 5\). This variable-based approach simplifies solving and helps us confidently find the solution.

Subtraction in Algebra

Subtraction is a fundamental operation in algebra, indicated by terms like "difference" in word problems. In algebraic settings, subtraction helps us form equations that show how quantities relate to each other.
In our example, the "difference between 20 and a number is 5" translates to the equation \(20 - x = 5\). Here’s how subtraction is applied:

• Identify the minuend (the number from which another number is subtracted), which is 20 in this instance.
• Spot the subtrahend, which is the unknown \(x\).
• The result of the subtraction is 5, known as the difference.

Solving this equation involves finding the value of \(x\) that makes it true. The operation \(20 - x = 5\) is straightforward once you grasp that "difference" means subtraction, and it helps depict the relationship between the numbers. Understanding subtraction in this context allows you to solve similar algebraic equations with confidence.