Problem 63 - Introductory Algebra for College Students

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 equation representing the sentence is: \(20 - x = 5\).

Step 1: Define Variables

Let's define \(x\) as the unknown number mentioned in the problem.

Step 2: Translate Sentence into Mathematical Terms

`The difference between 20 and a number` translates to `20 - x` in mathematical terms.

Step 3: Formulate the Equation

We know that this difference is equal to 5. So, we can write the equation as `20 - x = 5`.

Key Concepts

Variable Definition

In algebra, variables are symbols that stand in for unknown values and allow us to work with them in equations. Defining a variable is the first step in solving any algebra problem. This is crucial because it sets the ground for formulating equations and finding solutions.

In our given exercise, we define the variable as follows:

• Let \( x \) represent the unknown number.

By choosing a simple letter like \( x \), we can easily identify it as the placeholder for the number we need to find.

It is important to define your variables clearly in the beginning to avoid any confusion later.

Translating Sentences to Equations

Translating sentences into equations is a key skill in algebra. It involves interpreting the given information and expressing it as mathematical expressions.

Let's break down the sentence from the exercise:

• "The difference between 20 and a number" implies a subtraction operation.
• The phrase can be translated into the expression \( 20 - x \) where \( x \) represents the unknown number.

Understanding phrases like "difference between" as subtraction or "sum of" as addition is crucial.

This skill helps bridge the gap between verbal problems and mathematical expressions, enabling you to work through more complex algebraic solutions systematically.

Problem-Solving Steps

Problem-solving in algebra often involves a series of steps that guide you to the solution. Following these steps systematically ensures you correctly interpret and solve the problem.

In the given exercise, the steps include:

• Define the variable involved (\( x \)).
• Translate the sentence into a mathematical equation \( 20 - x = 5 \).

The final step is solving the equation to find the value of \( x \). To solve \( 20 - x = 5 \), you'll need to isolate \( x \).

Subtract 20 from both sides to move constants to one side:
\( 20 - 20 - x = 5 - 20 \), simplifying to \(-x = -15\).
Divide by -1 to find \( x \):
\( x = 15 \).

This map of steps helps organize your thoughts and ensure each part of the problem is handled methodically. This structure is vital for mastering algebra and dealing with more challenging problems over time.