Problem 10 - Fundamentals of Mathematics

Question

Translate each phrase or sentence into a mathematical expression or equation. Nine from some number is four.

Step-by-Step Solution

Answer

The mathematical expression for the phrase is \( x - 9 = 4 \), and the number is 13.

Step 1: Understand the Phrase

The phrase "Nine from some number is four" means that when you take nine away from an unknown number, the result is four.

Step 2: Define the Variables

Identify the unknown in the problem. Let the unknown number be represented by the variable \( x \).

Step 3: Set Up the Equation

Translate the phrase into an equation. "Nine from some number" can be written as \( x - 9 \), so the equation becomes \( x - 9 = 4 \).

Step 4: Solve for the Unknown Number

To find \( x \), rearrange the equation to isolate \( x \) on one side: 1. Add 9 to both sides of the equation: \( x - 9 + 9 = 4 + 9 \) 2. Simplify: \( x = 13 \).

Step 5: Verify the Solution

Substitute \( x = 13 \) back into the original equation to check: \( 13 - 9 = 4 \), which is true.

Key Concepts

Variable Definition

When tackling problems that involve translating words into mathematical expressions or equations, the first critical step is understanding the role of variables. Variables are symbols, often letters like \( x \) or \( y \), that represent unknown values we aim to find. Using a variable allows us to transform a verbal description into a manipulatable equation. It's like adding a placeholder that will eventually be filled with a specific number after solving the equation.

In this particular exercise, the phrase "nine from some number is four" entails an unknown number. Here, we use \( x \) to denote this unknown number. Keeping your variable definitions clear and consistent is essential because they serve as the backbone of your mathematical expression. Variable definition is usually straightforward but needs to be done correctly to avoid errors in subsequent steps.

Equation Setup

Once you have defined a variable, the next step is setting up the equation based on the problem statement. This involves converting the entire phrase into a mathematical sentence using your defined variable. The goal is to write an equation that precisely matches the description given.

For our phrase, "Nine from some number is four," the term "nine from some number" translates mathematically into \( x - 9 \). The phrase "is four" indicates the result of the operation, thus forming the equation \( x - 9 = 4 \).

The setup is crucial as it forms the foundation for solving the problem. It is like constructing an architectural blueprint. A detailed and accurate setup will ensure your problem-solving process is smooth and effective.

Solving Equations

Solving the equation is where the magic happens. This involves steps to isolate the variable and find its value. For the equation \( x - 9 = 4 \), our task is to solve for \( x \). You can think of this as a balance, where you maintain equality on both sides of the equation as you perform operations.

Here are the straightforward steps to solve this equation:

• First, perform the opposite operation to eliminate \( -9 \) from the left side. Add 9 to both sides of the equation: \( x - 9 + 9 = 4 + 9 \).
• This simplification will yield \( x = 13 \).

Solving equations involves systematically simplifying until you isolate the variable. Always check your work at the end by substituting the found variable back into the original equation to verify your solution, ensuring every step maintains the balance and correctness of your mathematical statement.