Problem 88 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Seven subtracted from a number is \(0 .\)

Step-by-Step Solution

Answer

The equation is \(x - 7 = 0\) and the solution is \(x = 7\).

Step 1: Identify the Unknown Number

We will represent the unknown number using the variable \(x\).

Step 2: Translate the Sentence to Mathematical Operation

The phrase 'Seven subtracted from a number' translates to \(x - 7\).

Step 3: Set Up the Equation

The sentence says that the result is \(0\). Therefore, we set the equation as \(x - 7 = 0\).

Step 4: Solve the Equation

Add \(7\) to both sides of the equation to solve for \(x\): \(x - 7 + 7 = 0 + 7\). Hence, \(x = 7\).

Key Concepts

Variable Representation

When faced with a problem involving unknown numbers, we typically use a variable to represent these unknowns, with "\(x\)" commonly serving as the standard representation. This approach allows us to translate real-world problems into algebraic expressions or equations.

• A variable acts as a placeholder for the unknown value until we solve the problem.
• Choosing \(x\) is mostly a matter of convention; any letter could technically be used.
• Variables help simplify and solve problems by providing a clear way to recognize and manipulate unknown quantities.

In our exercise, we represent the phrase 'a number' with the variable \(x\). This sets the stage for translating and solving the equation.

Mathematical Translation

Translating words into mathematical language involves converting phrases into mathematical expressions or equations. This process is crucial because it forms a bridge from a problem stated in words to one that can be solved mathematically.

• The phrase 'Seven subtracted from a number' can be converted into \(x - 7\).
• Key terms such as 'subtracted from,' 'plus,' 'times,' and 'equals' often indicate specific mathematical operations.
• Understanding the language of math enables you to set equations correctly based on word problems.

For example, in this exercise, recognizing that 'seven subtracted from a number' suggests taking \(7\) away from \(x\) is a critical step in forming the equation.

Solving Equations

Solving equations is about finding the value of the variable that makes the equation true. Once we have translated the words into an equation, we can apply algebraic techniques to solve it.

• We start with an equation like \(x - 7 = 0\).
• The goal is to isolate the variable on one side of the equation.
• To do this, add \(7\) to both sides to reverse the subtraction: \(x - 7 + 7 = 0 + 7\).

This operation simplifies the equation to \(x = 7\), thereby revealing that the unknown number is \(7\). The process involves maintaining the balance of the equation and performing valid operations to simplify it in a logical way.