Creating the System of Linear Equations

Oct 13, 2025 by recoveryshake.com 101 views

Introduction

In this article, we will explore a classic problem in mathematics that involves solving a system of linear equations. The problem is as follows: Yolanda paid for her movie ticket using 28 coins, all nickels and quarters. The ticket cost $4. We need to find the number of nickels, n, and the number of quarters, q, that Yolanda used to pay for the ticket.

Understanding the Problem

Let's break down the problem and understand what we are trying to solve. We know that Yolanda used 28 coins to pay for the ticket, and these coins are a combination of nickels and quarters. We also know that the total cost of the ticket is $4.

Defining the Variables

Let's define the variables that we will use to solve the problem. We will use n to represent the number of nickels and q to represent the number of quarters.

Creating the System of Linear Equations

We can create a system of linear equations to represent the problem. The first equation represents the total number of coins used to pay for the ticket:

n + q = 28

This equation states that the total number of nickels and quarters used to pay for the ticket is equal to 28.

The second equation represents the total cost of the ticket:

0.05n + 0.25q = 4

This equation states that the total cost of the ticket is equal to $4, and that this cost is made up of the value of the nickels and quarters used to pay for the ticket.

Solving the System of Linear Equations

To solve the system of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution.

First, we can solve the first equation for n:

n = 28 - q

Substituting this expression for n into the second equation, we get:

0.05(28 - q) + 0.25q = 4

Expanding and simplifying the equation, we get:

1.4 - 0.05q + 0.25q = 4

Combine like terms:

0.2q = 2.6

Divide both sides by 0.2:

q = 13

Now that we have found the value of q, we can substitute this value back into the first equation to find the value of n:

n + 13 = 28

Subtract 13 from both sides:

n = 15

Conclusion

In this article, we have solved a system of linear equations to find the number of nickels and quarters that Yolanda used to pay for her movie ticket. We have used the method of substitution to solve the system of linear equations, and we have found that Yolanda used 15 nickels and 13 quarters to pay for the ticket.

Final Answer

The final answer is that Yolanda used 15 nickels and 13 quarters to pay for her movie ticket.

Mathematical Representation

The system of linear equations that we used to solve the problem is:

n + q = 28

0.05n + 0.25q = 4

This system of linear equations can be represented graphically as a pair of lines that intersect at the point (15, 13).

Graphical Representation

The graphical representation of the system of linear equations is a pair of lines that intersect at the point (15, 13). The first line represents the equation n + q = 28, and the second line represents the equation 0.05n + 0.25q = 4.

Real-World Applications

This problem has real-world applications in finance and economics. For example, a business may need to determine the number of coins that a customer used to pay for a product or service. This problem can be solved using a system of linear equations, and the solution can be used to make informed business decisions.

Conclusion

In conclusion, this article has demonstrated how to solve a system of linear equations to find the number of nickels and quarters that Yolanda used to pay for her movie ticket. We have used the method of substitution to solve the system of linear equations, and we have found that Yolanda used 15 nickels and 13 quarters to pay for the ticket. This problem has real-world applications in finance and economics, and it can be solved using a system of linear equations.

Introduction

In our previous article, we solved a system of linear equations to find the number of nickels and quarters that Yolanda used to pay for her movie ticket. In this article, we will answer some frequently asked questions related to the problem.

Q: What is the total value of the coins used to pay for the ticket?

A: The total value of the coins used to pay for the ticket is $4.

Q: How many coins did Yolanda use to pay for the ticket?

A: Yolanda used 28 coins to pay for the ticket.

Q: What is the value of a nickel?

A: The value of a nickel is $0.05.

Q: What is the value of a quarter?

A: The value of a quarter is $0.25.

Q: How many nickels did Yolanda use to pay for the ticket?

A: Yolanda used 15 nickels to pay for the ticket.

Q: How many quarters did Yolanda use to pay for the ticket?

A: Yolanda used 13 quarters to pay for the ticket.

Q: What is the total number of nickels and quarters used to pay for the ticket?

A: The total number of nickels and quarters used to pay for the ticket is 28.

Q: How can I solve a system of linear equations like this one?

A: You can solve a system of linear equations like this one by using the method of substitution or elimination. In this case, we used the method of substitution to solve the system of linear equations.

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

A: Some real-world applications of solving a system of linear equations include finance, economics, and business. For example, a business may need to determine the number of coins that a customer used to pay for a product or service.

Q: Can I use a calculator to solve a system of linear equations?

A: Yes, you can use a calculator to solve a system of linear equations. However, it's also important to understand the underlying math and be able to solve the system of linear equations by hand.

Q: What are some common mistakes to avoid when solving a system of linear equations?

A: Some common mistakes to avoid when solving a system of linear equations include:

Not reading the problem carefully
Not defining the variables correctly
Not using the correct method to solve the system of linear equations
Not checking the solution for errors

Conclusion

In this article, we have answered some frequently asked questions related to solving a system of linear equations. We have also provided some tips and advice for solving a system of linear equations, including how to avoid common mistakes.

Final Answer

The final answer is that Yolanda used 15 nickels and 13 quarters to pay for her movie ticket.

Mathematical Representation

The system of linear equations that we used to solve the problem is:

n + q = 28

0.05n + 0.25q = 4

This system of linear equations can be represented graphically as a pair of lines that intersect at the point (15, 13).

Graphical Representation

Real-World Applications

Conclusion

In conclusion, this article has provided some frequently asked questions and answers related to solving a system of linear equations. We have also provided some tips and advice for solving a system of linear equations, including how to avoid common mistakes.