Problem 6 - A Graphical Approach to Precalculus with Limits

Question

Work Exercises \(1-6\) without pencil and paper. Do not use a calculator. Consider the following problem: The difference between six times a number and 9 is equal to five times the sum of the number and \(2 .\) Find the number. If \(x\) represents the number, which equation is correct for solving this problem? A. \(6 x-9=5(x+2)\) B. \(9-6 x=5(x+2)\) C. \(6 x-9=5 x+2\) D. \(9-6 x=5 x+2\)

Step-by-Step Solution

Answer

A. \(6x - 9 = 5(x + 2)\)

Step 1: Understand the Problem

The problem involves finding a number based on a verbal description. Understanding the word problem is key. We need to express the difference between six times a number and 9 as equal to five times the sum of that number and 2.

Step 2: Express in Mathematical Terms

Let's denote the number as \(x\). The phrase 'six times a number' translates to \(6x\), and 'the difference between six times a number and 9' becomes \(6x - 9\). Similarly, 'the sum of the number and 2' is \(x + 2\). Therefore, 'five times the sum of the number and 2' translates to \(5(x + 2)\).

Step 3: Formulate the Equation

According to the problem, the difference \(6x - 9\) is equal to five times the sum \(5(x + 2)\). Thus, the equation that represents this condition is \(6x - 9 = 5(x + 2)\).

Step 4: Match the Equation to the Choices

We compare the formulated equation \(6x - 9 = 5(x + 2)\) with the provided options. Option A matches this equation exactly.

Key Concepts

Word Problems

Word problems are a common feature in math exercises where real-life situations are described using everyday language. The challenge is to translate this into mathematical expressions and equations.
Understanding word problems involves identifying key phrases and numbers that relate to mathematical operations. In this particular problem, **'the difference between six times a number and 9'** and **'five times the sum of the number and 2'** are critical phrases. Here is a simple approach to tackle word problems:

• Read the problem several times to comprehend the scenario.
• Highlight or note key numbers and phrases.
• Decide what mathematical operation each phrase is indicating (addition, subtraction, multiplication, etc.).

By practicing these steps, students can turn complex language into manageable equations.

Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents higher than one. These are written in the form of **ax + b = c**, where the variables are raised only to the first power.
In word problems, we often translate descriptions into linear equations to find unknown numbers. For the exercise at hand, we identify the number as **x** and translate the phrases into mathematical terms to create a linear equation:
* **'Six times a number'** becomes **6x**.
* **'The difference between six times a number and 9'** transforms into **6x - 9**.
* **'The sum of the number and 2'** becomes **x + 2**.
* Finally, **'five times the sum of the number and 2'** is represented as **5(x + 2)**.
These transformations allow us to set up and solve the linear equation, comparing it to the multiple-choice options to find the correct match.

Problem Solving Steps

Approaching a word problem requires a systematic method to decode the information and find a solution. In this exercise, the key solving steps are:

Understanding the Problem: Clearly determine what is being asked by pinpointing the relationships between quantities.
Express in Mathematical Terms: Use algebraic expressions to represent the relationships. For instance, descriptions like "six times a number" or "the sum of the number and 2" are converted into expressions using a variable such as **x**.
Formulate the Equation: Based on the information, set up a linear equation. Here, it translates to **6x - 9 = 5(x + 2)**.
Match the Equation to the Choices: Compare your equation to given options to decide which one is correct. The exercise's options provide various forms, and our result matches option **A**, thereby confirming the solution.
Following these steps ensures clarity and consistency in solving similar algebraic word problems.