Select The Correct Answer.This Week, Theo Walked $x$ Hours At A Constant Rate Of 4 Miles Per Hour And Jogged $y$ Hours At A Constant Rate Of 6 Miles Per Hour. The Total Distance He Walked And Jogged This Week Was 36 Miles. The

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Introduction

In the world of mathematics, problems often present themselves in the form of equations, requiring us to apply various mathematical concepts to arrive at a solution. This week, we are presented with a scenario involving distance, rate, and time, which we will use to explore the world of algebra and mathematical reasoning.

The Problem

Theo, a keen walker and jogger, has been keeping track of his activities this week. He walked for $x$ hours at a constant rate of 4 miles per hour and jogged $y$ hours at a constant rate of 6 miles per hour. The total distance he covered through walking and jogging is 36 miles. We are tasked with determining the values of $x$ and $y$ that satisfy this condition.

Mathematical Formulation

To approach this problem, we can use the formula for distance, which is given by:

Distance=Rate×Time\text{Distance} = \text{Rate} \times \text{Time}

In this scenario, we have two types of activities: walking and jogging. The distance covered through walking is given by:

Distance walked=4x\text{Distance walked} = 4x

Similarly, the distance covered through jogging is given by:

Distance jogged=6y\text{Distance jogged} = 6y

The total distance covered is the sum of the distances walked and jogged, which is given as 36 miles. Therefore, we can write the equation:

4x+6y=364x + 6y = 36

Solving the Equation

To solve for $x$ and $y$, we can use various algebraic techniques. One approach is to isolate one of the variables and then substitute it into the other equation. Let's isolate $x$ by subtracting $6y$ from both sides of the equation:

4x=36−6y4x = 36 - 6y

Dividing both sides by 4, we get:

x=36−6y4x = \frac{36 - 6y}{4}

Simplifying the expression, we get:

x=9−32yx = 9 - \frac{3}{2}y

Substitution Method

Another approach to solving the equation is to use the substitution method. Let's substitute the expression for $x$ into the original equation:

4(9−32y)+6y=364(9 - \frac{3}{2}y) + 6y = 36

Expanding the expression, we get:

36−6y+6y=3636 - 6y + 6y = 36

Simplifying the equation, we get:

36=3636 = 36

This equation is true for all values of $y$. Therefore, we can conclude that $y$ can take any value, and the corresponding value of $x$ will be given by:

x=9−32yx = 9 - \frac{3}{2}y

Graphical Representation

To visualize the solution, we can plot the equation on a coordinate plane. Let's plot the equation $x = 9 - \frac{3}{2}y$ on a coordinate plane.

import matplotlib.pyplot as plt
import numpy as np

y = np.linspace(-10, 10, 400)



x = 9 - 1.5 * y



plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y')
plt.title('Graph of the Equation')
plt.grid(True)
plt.show()

Conclusion

In this article, we explored a mathematical problem involving distance, rate, and time. We used algebraic techniques to solve for the values of $x$ and $y$ that satisfy the given condition. We also used the substitution method to arrive at the solution and visualized the result using a graphical representation. This problem serves as a reminder of the importance of mathematical reasoning and problem-solving skills in real-world applications.

Final Answer

The final answer is not a single value, but rather a range of values that satisfy the given condition. The values of $x$ and $y$ can be expressed as:

x=9−32yx = 9 - \frac{3}{2}y

y$ can take any value, and the corresponding value of $x$ will be given by the above equation.<br/> **The Distance Dilemma: A Mathematical Conundrum - Q&A** =====================================================

Introduction

In our previous article, we explored a mathematical problem involving distance, rate, and time. We used algebraic techniques to solve for the values of $x$ and $y$ that satisfy the given condition. In this article, we will address some of the frequently asked questions related to this problem.

Q: What is the significance of the equation $4x + 6y = 36$?

A: The equation $4x + 6y = 36$ represents the total distance covered by Theo through walking and jogging. The coefficients 4 and 6 represent the rates at which he walks and jogs, respectively. The constant term 36 represents the total distance covered.

Q: How do I solve for $x$ and $y$ using the substitution method?

A: To solve for $x$ and $y$ using the substitution method, you can substitute the expression for $x$ into the original equation. This will give you an equation in terms of $y$, which you can then solve for.

Q: Can I use the graphical representation to find the values of $x$ and $y$?

A: Yes, you can use the graphical representation to find the values of $x$ and $y$. The graph of the equation $x = 9 - \frac{3}{2}y$ is a straight line. You can use the graph to find the values of $x$ and $y$ that satisfy the given condition.

Q: What is the relationship between $x$ and $y$?

A: The relationship between $x$ and $y$ is given by the equation $x = 9 - \frac{3}{2}y$. This equation represents a linear relationship between $x$ and $y$.

Q: Can I use the equation $x = 9 - \frac{3}{2}y$ to find the values of $x$ and $y$ for any value of $y$?

A: Yes, you can use the equation $x = 9 - \frac{3}{2}y$ to find the values of $x$ and $y$ for any value of $y$. This equation represents a linear relationship between $x$ and $y$, and it is valid for all values of $y$.

Q: How do I interpret the graph of the equation $x = 9 - \frac{3}{2}y$?

A: The graph of the equation $x = 9 - \frac{3}{2}y$ is a straight line. The x-axis represents the values of $x$, and the y-axis represents the values of $y$. The graph shows the relationship between $x$ and $y$, and it can be used to find the values of $x$ and $y$ that satisfy the given condition.

Q: Can I use the equation $4x + 6y = 36$ to find the values of $x$ and $y$ for any value of $x$?

A: No, you cannot use the equation $4x + 6y = 36$ to find the values of $x$ and $y$ for any value of $x$. This equation represents a linear relationship between $x$ and $y$, and it is valid only for specific values of $x$ and $y$.

Conclusion

In this article, we addressed some of the frequently asked questions related to the mathematical problem involving distance, rate, and time. We provided explanations and examples to help clarify the concepts and techniques used to solve the problem. We hope that this article has been helpful in providing a deeper understanding of the subject matter.

Final Answer

The final answer is not a single value, but rather a range of values that satisfy the given condition. The values of $x$ and $y$ can be expressed as:

x = 9 - \frac{3}{2}y </span></p> <p class='katex-block'><span class="katex-error" title="ParseError: KaTeX parse error: Can&#x27;t use function &#x27;{{content}}amp;#x27; in math mode at position 2: y$̲$ can take any …" style="color:#cc0000">y$ can take any value, and the corresponding value of $x$ will be given by the above equation. </span></p>