Problem 74 - Introductory Algebra for College Students

Question

Write each sentence as an equation. Let the variable \(x\) represent the number. Four times a number is equal to 25 decreased by the number.

Step-by-Step Solution

Answer

The equation is \(4x = 25 - x\).

Step 1: Identify the Variable

The sentence specifies that \(x\) is the number referenced in the statement.

Step 2: Translate the Sentence Piece by Piece

The sentence 'Four times a number is equal to 25 decreased by the number' can be broken down piece by piece. 'Four times a number' translates to \(4x\), 'is equal to' translates to \(=\), and '25 decreased by the number' translates to \(25 - x\).

Step 3: Combine the Translated Parts

Combining the translated pieces of the sentence gives the equation \(4x = 25 - x\).

Key Concepts

Variables in Algebra

Understanding variables is fundamental in algebra. A variable is simply a symbol or letter used to represent an unknown value or number. It allows us to write equations that can model real-world situations or abstract problems. In our exercise, the variable is represented by the letter \(x\). The variable can take on different values depending on the context of the problem or the constraints provided.

Here’s why variables are so useful:

• Flexibility: Use them to represent any unknown quantity.
• Simplicity: Replace complex expressions or repetitive values with a single letter.
• Solving Problems: Create equations to find unknown values by manipulating the variable.

In our example, the variable \(x\) stands for the number that satisfies the given relationship in the sentence. The ability to define and manipulate variables is a powerful tool in math that helps in everything from solving simple algebraic equations to understanding complex calculus problems.

Translating Sentences into Equations

The key to translating sentences into equations lies in understanding the language of mathematics. Every part of a sentence can correspond to a mathematical expression. Here’s how to translate the sentence from the exercise: ‘Four times a number is equal to 25 decreased by the number’.

• Four times a number: This tells us to multiply the variable by 4. Mathematically, it means \(4x\).
• Is equal to: Represents equality in mathematics, so we use the equal sign \(=\).
• 25 decreased by the number: Indicates subtraction, which can be written as \(25 - x\).

After translating each part, these pieces are combined into the equation \(4x = 25 - x\). This process can transform a word problem into a mathematical model, making it easier to solve and understand.

Solving Equations

Once we have our algebraic equation, the next step is solving it. Solving means finding the value of the variable that makes the equation true.Let’s look at how to solve the equation from the exercise: \(4x = 25 - x\).

• Step 1: Move all terms involving \(x\) to one side: Add \(x\) to both sides to get \(4x + x = 25\).
• Step 2: Simplify the equation: Combine like terms to get \(5x = 25\).
• Step 3: Solve for \(x\): Divide both sides by 5 to find \(x = 5\).

This step-by-step approach helps in isolating the variable and determining its value. Solving equations is like peeling an onion layer-by-layer until only the core (the variable's value) remains uncovered. Through practice, solving equations becomes a straightforward process that enhances problem-solving skills.