Problem 5 - College Algebra with Corequisite Support

Question

If Bill was traveling \(v \mathrm{mi} / \mathrm{h}\), how would you represent Daemon's speed if he was traveling \(10 \mathrm{mi} / \mathrm{h}\) faster?

Step-by-Step Solution

Answer

Daemon's speed is \(v + 10\) mi/h.

Step 1: Understanding the Problem

We need to express Daemon's speed in terms of Bill's speed. Bill travels at a speed of \(v\) miles per hour, and Daemon travels 10 miles per hour faster than Bill.

Step 2: Expressing Daemon's Speed in Words

Daemon's speed is Bill's speed plus an additional 10 miles per hour.

Step 3: Formulating the Equation

Based on the previous step, we can formulate the equation for Daemon's speed as: \[ ext{Daemon's speed} = v + 10 \]

Step 4: Final Representation

The final expression for Daemon's speed, in terms of \(v\), is \(v + 10\).

Key Concepts

Variable Representation in Algebra

Variables are symbols used to represent numbers in equations or expressions. In algebra, they serve as placeholders for values that can change or vary. They are often denoted by letters like \(x\), \(y\), or \(v\).

Understanding how to accurately represent variables is essential for solving algebraic problems. In the given exercise, Bill’s speed is represented as \(v\) miles per hour. This means that \(v\) is a variable standing for any potential value representing his speed.

When we talk about Daemon's speed, we use the idea of variable representation to state that Daemon's speed is \(v + 10\) miles per hour. Here, \(v + 10\) is an expression using the initial variable \(v\), demonstrating that Daemon travels 10 miles per hour faster than Bill. This clear understanding of variable representation allows us to set up equations effectively.

Understanding Linear Equations

Linear equations are foundational to algebra and are used to describe relationships between variables using constants and variables.

They are written in the form \(ax + b = c\), where:

• \(a\), \(b\), and \(c\) are constants
• \(x\) is the variable

The equation \(v + 10\) from the original exercise is a simplified example of a linear equation. This translates Daemon's speed relative to Bill’s speed.

Converting real-world word problems, like comparing two speeds, into linear equations is a vital step in algebra. It allows us to work with the equation to find unknown values easily. Additionally, understanding how to set up and manipulate these linear equations is crucial for problem-solving and further mathematics study.

Enhancing Problem-Solving Skills

Problem-solving skills in algebra involve using logical steps to find solutions to mathematical problems. This exercise develops those skills by guiding you through a process of representing unknown quantities with variables and turning verbal information into a mathematical equation.

To tackle similar problems effectively, consider these strategies:

• Understand and define what each variable represents in the context of the problem.
• Translate words into mathematical expressions or equations.
• Perform operations step-by-step, simplifying as you go to maintain clarity.

By breaking down the problem into smaller manageable parts, like determining what Daemon's speed would be relative to Bill's, you can solve even complex algebraic problems more comfortably. Remember, practice is key to improving problem-solving skills. The more you engage with these steps, the more intuitive they will become.