Anatoly Has A Combination Of 104 Nickels And Quarters Totaling \$22. Which System Of Linear Equations Can Be Used To Find The Number Of Nickels, \[$ N \$\], And The Number Of Quarters, \[$ Q \$\], Anatoly Has?A.

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Introduction

In this article, we will explore a problem involving a combination of nickels and quarters, and how to use a system of linear equations to find the number of each type of coin. Anatoly has a combination of 104 nickels and quarters totaling $22. We will use this information to set up a system of linear equations and solve for the number of nickels and quarters.

Understanding the Problem

Let's break down the problem and understand what we are given. Anatoly has a combination of 104 nickels and quarters, and the total value of these coins is $22. We know that the value of a nickel is $0.05 and the value of a quarter is $0.25. We can use this information to set up a system of linear equations.

Setting Up the System of Linear Equations

Let's use the variables n to represent the number of nickels and q to represent the number of quarters. We can set up two equations based on the given information:

  • The total number of coins is 104, so we can write the equation: n + q = 104
  • The total value of the coins is $22, so we can write the equation: 0.05n + 0.25q = 22

Solving the System of Linear Equations

We can solve this system of linear equations using substitution or elimination. Let's use the substitution method.

First, we can solve the first equation for n: n = 104 - q

Now, we can substitute this expression for n into the second equation: 0.05(104 - q) + 0.25q = 22

Expanding and simplifying the equation, we get: 5.2 - 0.05q + 0.25q = 22

Combine like terms: 0.2q = 16.8

Divide both sides by 0.2: q = 84

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 + 84 = 104

Subtract 84 from both sides: n = 20

Conclusion

In this article, we have set up and solved a system of linear equations to find the number of nickels and quarters that Anatoly has. We have used the variables n and q to represent the number of nickels and quarters, and we have set up two equations based on the given information. We have solved the system of linear equations using the substitution method and have found that Anatoly has 20 nickels and 84 quarters.

Key Takeaways

  • A system of linear equations can be used to solve problems involving multiple variables.
  • The substitution method can be used to solve a system of linear equations.
  • The variables n and q can be used to represent the number of nickels and quarters.

Real-World Applications

This problem can be applied to real-world situations where we need to solve problems involving multiple variables. For example, a store owner may need to determine the number of nickels and quarters in a cash register to calculate the total value of the coins.

Future Directions

In the future, we can explore more complex problems involving systems of linear equations. We can also explore other methods for solving systems of linear equations, such as the elimination method.

References

Glossary

  • System of linear equations: A set of two or more linear equations that are solved simultaneously.
  • Substitution method: A method for solving a system of linear equations by substituting one equation into another.
  • Elimination method: A method for solving a system of linear equations by eliminating one variable.
  • Nickel: A coin worth $0.05.
  • Quarter: A coin worth $0.25.
    Frequently Asked Questions: Solving the Coin Problem =====================================================

Q: What is the main goal of the problem?

A: The main goal of the problem is to find the number of nickels and quarters that Anatoly has, given that he has a combination of 104 nickels and quarters totaling $22.

Q: What are the two equations that we need to solve?

A: The two equations that we need to solve are:

  • n + q = 104 (the total number of coins is 104)
  • 0.05n + 0.25q = 22 (the total value of the coins is $22)

Q: How do we solve the system of linear equations?

A: We can solve the system of linear equations using the substitution method. First, we can solve the first equation for n: n = 104 - q. Then, we can substitute this expression for n into the second equation and solve for q.

Q: What is the value of q (the number of quarters)?

A: The value of q (the number of quarters) is 84.

Q: What is the value of n (the number of nickels)?

A: The value of n (the number of nickels) is 20.

Q: Can we use the elimination method to solve the system of linear equations?

A: Yes, we can use the elimination method to solve the system of linear equations. To do this, we can multiply the first equation by 0.25 and the second equation by 1, and then subtract the two equations to eliminate the variable q.

Q: What is the advantage of using the substitution method over the elimination method?

A: The advantage of using the substitution method over the elimination method is that it is often easier to solve for one variable and substitute it into the other equation, rather than multiplying and subtracting equations.

Q: Can we apply this problem to real-world situations?

A: Yes, we can apply this problem to real-world situations where we need to solve problems involving multiple variables. For example, a store owner may need to determine the number of nickels and quarters in a cash register to calculate the total value of the coins.

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

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

  • Not checking for extraneous solutions
  • Not using the correct method (substitution or elimination)
  • Not simplifying the equations before solving
  • Not checking the solution for consistency with the original problem

Q: How can we extend this problem to more complex situations?

A: We can extend this problem to more complex situations by adding more variables or equations, or by using different methods for solving the system of linear equations. For example, we could add a third variable to represent the number of dimes, or we could use the elimination method to solve the system of linear equations.

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

A: Some real-world applications of systems of linear equations include:

  • Finance: solving systems of linear equations to determine the value of investments or the amount of money in a bank account
  • Science: solving systems of linear equations to model the behavior of physical systems or to analyze data
  • Engineering: solving systems of linear equations to design and optimize systems or to analyze the behavior of complex systems

Q: How can we use technology to solve systems of linear equations?

A: We can use technology, such as graphing calculators or computer software, to solve systems of linear equations. This can be especially helpful for complex systems or for systems with many variables.