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

Mar 11, 2025 by ADMIN 278 views

**The Soccer Team's Car Wash Fundraiser: A Mathematical Model**

Introduction

In this article, we will explore a real-world scenario where a soccer team collected $\$800$ at a car wash fundraiser. The team charged $\$5.00$ for small vehicles and $\$10.00$ 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.

The Mathematical Model

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

Understanding the Equation

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

The first part, $5x$ , represents the amount collected from small vehicles. Since each small vehicle costs $\$5.00$ , the total amount collected from small vehicles is $5x$ .
The second part, $10y$ , represents the amount collected from large vehicles. Since each large vehicle costs $\$10.00$ , the total amount collected from large vehicles is $10y$ .

Solving the Equation

To solve the equation $5x + 10y = 800$ , we can use the method of substitution or elimination. Let's use the substitution method.

Step 1: Isolate One Variable

We can isolate one variable by subtracting $5x$ from both sides of the equation:

10y = 800 - 5x

Step 2: Solve for the Other Variable

Now, we can solve for the other variable by dividing both sides of the equation by $10$ :

y = \frac{800 - 5x}{10}

Step 3: Substitute the Expression

We can substitute the expression for $y$ into the original equation:

5x + 10\left(\frac{800 - 5x}{10}\right) = 800

Step 4: Simplify the Equation

Now, we can simplify the equation by multiplying both sides by $10$ :

50x + 800 - 50x = 8000

Step 5: Solve for the Variable

Finally, we can solve for the variable $x$ by subtracting $800$ from both sides of the equation:

50x = 0

Step 6: Find the Value of x

Now, we can find the value of $x$ by dividing both sides of the equation by $50$ :

x = 0

Step 7: Find the Value of y

Now that we have the value of $x$ , we can find the value of $y$ by substituting $x$ into the expression for $y$ :

y = \frac{800 - 5(0)}{10}

y = 80

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 $5x + 10y = 800$ using the substitution method and found that the team collected $\$800$ from $80$ large vehicles and $0$ small vehicles.

Real-World Applications

This problem has real-world applications in various fields, such as:

Business: A company can use this model to determine the number of small and large vehicles that need to be washed to collect a certain amount of money.
Economics: A government can use this model to determine the number of small and large vehicles that need to be taxed to collect a certain amount of revenue.
Mathematics: This problem can be used to teach students about linear equations and how to solve them using various methods.

Future Research Directions

This problem can be extended in various ways, such as:

Adding More Variables: We can add more variables to the equation to represent other factors that affect the amount collected.
Using Different Methods: We can use different methods, such as the elimination method, to solve the equation.
Using Real-World Data: We can use real-world data to test the model and see how well it fits the actual data.

References

[1] "Linear Equations". Khan Academy.
[2] "Systems of Linear Equations". Math Is Fun.
[3] "Car Wash Fundraiser". Wikipedia.

Appendix

This appendix provides additional information and resources that may be helpful for readers who want to learn more about the topic.

Additional Resources

[1] "Linear Equations". Wolfram MathWorld.
[2] "Systems of Linear Equations". MIT OpenCourseWare.
[3] "Car Wash Fundraiser". National Association of School Resource Officers.

Glossary

Linear Equation: An equation in which the highest power of the variable(s) is 1.
System of Linear Equations: A set of two or more linear equations that are solved simultaneously.
Car Wash Fundraiser: A fundraising event in which a group of people wash cars to collect money for a cause.
The Soccer Team's Car Wash Fundraiser: A Q&A Article =====================================================

Introduction

In our previous article, we explored a real-world scenario where a soccer team collected $\$800$ 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. In this article, we will answer some frequently asked questions (FAQs) about the problem.

Q&A

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

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

Q: How do we solve the equation?

A: We can solve the equation using the substitution method. We can isolate one variable by subtracting $5x$ from both sides of the equation, and then solve for the other variable by dividing both sides of the equation by $10$ .

Q: What is the value of x?

A: The value of $x$ is $0$ , which means that the team did not collect any money from small vehicles.

Q: What is the value of y?

A: The value of $y$ is $80$ , which means that the team collected $\$800$ from $80$ large vehicles.

Q: Can we use this model to determine the number of small and large vehicles that need to be washed to collect a certain amount of money?

A: Yes, we can use this model to determine the number of small and large vehicles that need to be washed to collect a certain amount of money. We can simply substitute the desired amount of money into the equation and solve for the number of small and large vehicles.

Q: Can we add more variables to the equation to represent other factors that affect the amount collected?

A: Yes, we can add more variables to the equation to represent other factors that affect the amount collected. For example, we can add a variable to represent the number of people washing cars, or a variable to represent the amount of time spent washing cars.

Q: Can we use real-world data to test the model and see how well it fits the actual data?

A: Yes, we can use real-world data to test the model and see how well it fits the actual data. We can collect data on the number of small and large vehicles washed, the amount of money collected, and other relevant factors, and then use the model to predict the amount of money that should be collected.