The Soccer Team Collected $\$800$ At A Car Wash Fundraiser. They Charged $\$5.00$ For Small Vehicles And $\$10.00$ For Larger Vehicles. The Amount Collected Can Be Modeled By The Equation $5x + 10y = 800$, Where

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Introduction

In this article, we will explore a real-world scenario where a soccer team collected $800\$800 at a car wash fundraiser. The team charged $5.00\$5.00 for small vehicles and $10.00\$10.00 for larger vehicles. We will use the concept of linear equations to model the amount collected and solve for the number of small and large vehicles that were washed.

The Mathematical Model

The amount collected can be modeled by the equation 5x+10y=8005x + 10y = 800, where xx represents the number of small vehicles and yy represents the number of large vehicles. This equation is a linear equation in two variables, and it can be solved using various methods.

Understanding the Equation

The equation 5x+10y=8005x + 10y = 800 can be broken down into two parts:

  • The first part, 5x5x, represents the total amount collected from small vehicles.
  • The second part, 10y10y, represents the total amount collected from large vehicles.
  • The constant term, 800800, represents the total amount collected.

Solving the Equation

To solve the equation 5x+10y=8005x + 10y = 800, we can use the method of substitution or elimination. In this case, we will use the substitution method.

Substitution Method

Let's start by isolating one of the variables. We can isolate xx by subtracting 10y10y from both sides of the equation:

5x=800−10y5x = 800 - 10y

Next, we can divide both sides of the equation by 55 to solve for xx:

x=800−10y5x = \frac{800 - 10y}{5}

Finding the Number of Small Vehicles

Now that we have the expression for xx, we can substitute it back into the original equation to find the number of small vehicles. We can substitute x=800−10y5x = \frac{800 - 10y}{5} into the equation 5x+10y=8005x + 10y = 800:

5(800−10y5)+10y=8005\left(\frac{800 - 10y}{5}\right) + 10y = 800

Simplifying the equation, we get:

800−10y+10y=800800 - 10y + 10y = 800

This equation is true for all values of yy, which means that the number of small vehicles is not dependent on the number of large vehicles.

Finding the Number of Large Vehicles

To find the number of large vehicles, we can substitute x=800−10y5x = \frac{800 - 10y}{5} into the equation x=800−10y5x = \frac{800 - 10y}{5}:

x=800−10y5x = \frac{800 - 10y}{5}

We can solve for yy by multiplying both sides of the equation by 55:

5x=800−10y5x = 800 - 10y

Next, we can add 10y10y to both sides of the equation to get:

5x+10y=8005x + 10y = 800

This equation is the same as the original equation, which means that the number of large vehicles is also not dependent on the number of small vehicles.

Conclusion

In this article, we used the concept of linear equations to model the amount collected by a soccer team at a car wash fundraiser. We solved the equation 5x+10y=8005x + 10y = 800 using the substitution method and found that the number of small vehicles and the number of large vehicles are not dependent on each other.

Real-World Applications

The concept of linear equations has many real-world applications, including:

  • Finance: Linear equations can be used to model the growth of an investment over time.
  • Science: Linear equations can be used to model the relationship between two variables in a scientific experiment.
  • Engineering: Linear equations can be used to model the behavior of a system in engineering applications.

Tips and Tricks

Here are some tips and tricks for solving linear equations:

  • Use the substitution method: The substitution method is a powerful tool for solving linear equations.
  • Use the elimination method: The elimination method is another powerful tool for solving linear equations.
  • Check your work: Always check your work to make sure that the solution is correct.

Practice Problems

Here are some practice problems to help you practice solving linear equations:

  • Problem 1: Solve the equation 2x+3y=122x + 3y = 12 using the substitution method.
  • Problem 2: Solve the equation x+2y=8x + 2y = 8 using the elimination method.
  • Problem 3: Solve the equation 3x−2y=103x - 2y = 10 using the substitution method.

Conclusion

Introduction

In our previous article, we explored a real-world scenario where a soccer team collected $800\$800 at a car wash fundraiser. We used the concept of linear equations to model the amount collected and solved for the number of small and large vehicles that were washed. In this article, we will answer some frequently asked questions about the soccer team's car wash fundraiser.

Q&A

Q: What is the total amount collected by the soccer team?

A: The total amount collected by the soccer team is $800\$800.

Q: How much did the team charge for small vehicles?

A: The team charged $5.00\$5.00 for small vehicles.

Q: How much did the team charge for large vehicles?

A: The team charged $10.00\$10.00 for large vehicles.

Q: What is the equation that models the amount collected?

A: The equation that models the amount collected is 5x+10y=8005x + 10y = 800, where xx represents the number of small vehicles and yy represents the number of large vehicles.

Q: How did you solve the equation?

A: We used the substitution method to solve the equation.

Q: What did you find out about the number of small vehicles?

A: We found out that the number of small vehicles is not dependent on the number of large vehicles.

Q: What did you find out about the number of large vehicles?

A: We found out that the number of large vehicles is also not dependent on the number of small vehicles.

Q: What are some real-world applications of linear equations?

A: Some real-world applications of linear equations include finance, science, and engineering.

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by using online resources or by working on practice problems.

Q: What are some tips and tricks for solving linear equations?

A: Some tips and tricks for solving linear equations include using the substitution method, using the elimination method, and checking your work.

Real-World Applications of Linear Equations

Linear equations have many real-world applications, including:

  • Finance: Linear equations can be used to model the growth of an investment over time.
  • Science: Linear equations can be used to model the relationship between two variables in a scientific experiment.
  • Engineering: Linear equations can be used to model the behavior of a system in engineering applications.

Practice Problems

Here are some practice problems to help you practice solving linear equations:

  • Problem 1: Solve the equation 2x+3y=122x + 3y = 12 using the substitution method.
  • Problem 2: Solve the equation x+2y=8x + 2y = 8 using the elimination method.
  • Problem 3: Solve the equation 3x−2y=103x - 2y = 10 using the substitution method.

Conclusion

In this article, we answered some frequently asked questions about the soccer team's car wash fundraiser. We discussed the total amount collected, the charges for small and large vehicles, and the equation that models the amount collected. We also discussed some real-world applications of linear equations and provided some practice problems to help you practice solving linear equations. Finally, we provided some tips and tricks for solving linear equations.

Additional Resources

If you want to learn more about linear equations, here are some additional resources:

  • Online Resources: There are many online resources available that can help you learn about linear equations, including Khan Academy, Mathway, and Wolfram Alpha.
  • Textbooks: There are many textbooks available that can help you learn about linear equations, including "Linear Algebra and Its Applications" by Gilbert Strang and "Linear Equations and Inequalities" by Michael Sullivan.
  • Practice Problems: There are many practice problems available that can help you practice solving linear equations, including the ones listed above.

Conclusion

In conclusion, linear equations are a powerful tool for modeling real-world scenarios. By understanding how to solve linear equations, you can apply this knowledge to a wide range of fields, including finance, science, and engineering. We hope that this article has been helpful in answering some of your questions about linear equations and providing you with some additional resources to help you learn more.