Indira Has 18 Coins Valued At { $ 4.20$} . S H E H A S O N L Y Q U A R T E R S A N D D I M E S . W H I C H A U G M E N T E D M A T R I X C A N B E U S E D T O D E T E R M I N E T H E N U M B E R O F Q U A R T E R S A N D D I M E S I N T H E C O L L E C T I O N ? A . \[ . She Has Only Quarters And Dimes. Which Augmented Matrix Can Be Used To Determine The Number Of Quarters And Dimes In The Collection?A. \[ . S H E Ha So N L Y Q U A R T Ers An Dd Im Es . Whi C Ha Ug M E N T E D Ma T R I X C Anb E U Se D T O D E T Er Min E T H E N U Mb Ero F Q U A R T Ers An Dd Im Es In T H Eco Ll Ec T I O N ? A . \[ \left[\begin{array}{cc|c}1 & 1 & 18 \ 0.25 & 0.10 &
Introduction
In this article, we will explore how to use an augmented matrix to solve a problem involving coins. Indira has a collection of coins valued at $4.20, consisting only of quarters and dimes. We will use an augmented matrix to determine the number of quarters and dimes in her collection.
Understanding the Problem
Let's break down the problem:
- Indira has 18 coins in total.
- Each quarter is worth $0.25, and each dime is worth $0.10.
- The total value of the coins is $4.20.
We can represent the number of quarters and dimes as variables x and y, respectively. We can then write two equations based on the given information:
- The total number of coins is 18: x + y = 18
- The total value of the coins is $4.20: 0.25x + 0.10y = 4.20
Creating the Augmented Matrix
An augmented matrix is a matrix that combines the coefficients of the variables with the constant terms. In this case, we have two equations with two variables, so our augmented matrix will have three columns: two for the coefficients and one for the constant terms.
The augmented matrix for this problem is:
| 1 | 1 | 18 |
|---|---|---|
| 0.25 | 0.10 | 4.20 |
Solving the System of Equations
To solve the system of equations, we can use various methods, such as substitution or elimination. However, in this case, we will use the augmented matrix to perform row operations and solve the system.
Step 1: Multiply the First Row by 0.25
To eliminate the 0.25 term in the second row, we can multiply the first row by 0.25. This will give us:
| 0.25 | 0.25 | 4.50 |
|---|---|---|
| 0.25 | 0.10 | 4.20 |
Step 2: Subtract the First Row from the Second Row
Now, we can subtract the first row from the second row to eliminate the 0.25 term:
| 0.25 | 0.25 | 4.50 |
|---|---|---|
| 0 | -0.15 | -0.30 |
Step 3: Divide the Second Row by -0.15
To solve for y, we can divide the second row by -0.15:
| 0.25 | 0.25 | 4.50 |
|---|---|---|
| 0 | 1 | 2 |
Step 4: Substitute y into the First Equation
Now that we have the value of y, we can substitute it into the first equation to solve for x:
x + 2 = 18
Step 5: Solve for x
Subtracting 2 from both sides gives us:
x = 16
Conclusion
In this article, we used an augmented matrix to solve a problem involving coins. We represented the number of quarters and dimes as variables x and y, respectively, and wrote two equations based on the given information. We then used row operations to solve the system of equations and found that Indira has 16 quarters and 2 dimes in her collection.
Discussion
This problem is a great example of how augmented matrices can be used to solve systems of linear equations. By representing the coefficients and constant terms in a matrix, we can perform row operations to solve the system. This method is particularly useful when dealing with systems of equations with multiple variables.
References
- [1] "Linear Algebra and Its Applications" by Gilbert Strang
- [2] "Introduction to Linear Algebra" by Gilbert Strang
Additional Resources
- Khan Academy: Linear Algebra
- MIT OpenCourseWare: Linear Algebra
- Wolfram Alpha: Linear Algebra
Frequently Asked Questions: Augmented Matrices and Coin Problems ====================================================================
Q: What is an augmented matrix?
A: An augmented matrix is a matrix that combines the coefficients of the variables with the constant terms. It is used to represent a system of linear equations in a compact and efficient way.
Q: How do I create an augmented matrix?
A: To create an augmented matrix, you need to write the coefficients of the variables in the first two columns and the constant terms in the third column. For example, if you have a system of equations like:
x + y = 18 0.25x + 0.10y = 4.20
The augmented matrix would be:
| 1 | 1 | 18 |
|---|---|---|
| 0.25 | 0.10 | 4.20 |
Q: What are the steps to solve a system of equations using an augmented matrix?
A: The steps to solve a system of equations using an augmented matrix are:
- Multiply the first row by a constant to eliminate a term in the second row.
- Subtract the first row from the second row to eliminate a term.
- Divide the second row by a constant to solve for a variable.
- Substitute the value of the variable into the first equation to solve for another variable.
- Solve for the remaining variable.
Q: Can I use an augmented matrix to solve a system of equations with more than two variables?
A: Yes, you can use an augmented matrix to solve a system of equations with more than two variables. However, the process becomes more complex and may require more row operations.
Q: What are some common mistakes to avoid when using an augmented matrix?
A: Some common mistakes to avoid when using an augmented matrix include:
- Not multiplying the correct row by the correct constant.
- Not subtracting the correct row from the correct row.
- Not dividing the correct row by the correct constant.
- Not substituting the correct value of the variable into the correct equation.
Q: Can I use an augmented matrix to solve a system of equations with non-integer coefficients?
A: Yes, you can use an augmented matrix to solve a system of equations with non-integer coefficients. However, you may need to use decimal arithmetic and round your answers to the correct number of decimal places.
Q: Are there any real-world applications of augmented matrices?
A: Yes, there are many real-world applications of augmented matrices, including:
- Solving systems of linear equations in physics and engineering.
- Finding the minimum or maximum of a function in optimization problems.
- Solving systems of linear equations in computer graphics and game development.
- Solving systems of linear equations in data analysis and machine learning.
Q: Can I use an augmented matrix to solve a system of equations with a large number of variables?
A: Yes, you can use an augmented matrix to solve a system of equations with a large number of variables. However, the process may become computationally intensive and may require the use of specialized software or algorithms.
Q: Are there any online resources available to help me learn about augmented matrices?
A: Yes, there are many online resources available to help you learn about augmented matrices, including:
- Khan Academy: Linear Algebra
- MIT OpenCourseWare: Linear Algebra
- Wolfram Alpha: Linear Algebra
- Mathway: Augmented Matrices
Conclusion
In this article, we have answered some frequently asked questions about augmented matrices and coin problems. We have discussed the steps to create an augmented matrix, the steps to solve a system of equations using an augmented matrix, and some common mistakes to avoid. We have also discussed some real-world applications of augmented matrices and provided some online resources to help you learn more about this topic.