Creating the Augmented Matrix

Oct 13, 2025 by windailys.com 116 views

Introduction

Indira has a collection of 18 coins valued at $\$4.20$ . The coins in her collection are only quarters and dimes. In this article, we will explore how to determine the number of quarters and dimes in her collection using an augmented matrix.

Understanding the Problem

Let's assume that the number of quarters in Indira's collection is represented by the variable $q$ and the number of dimes is represented by the variable $d$ . We know that the total number of coins is 18, so we can write an equation:

q + d = 18

We also know that the total value of the coins is $\$4.20$ . Since quarters are worth $\$0.25$ each and dimes are worth $\$0.10$ each, we can write another equation:

0.25q + 0.10d = 4.20

Creating the Augmented Matrix

To solve this system of equations, we can create an augmented matrix. An augmented matrix is a matrix that combines the coefficients of the variables with the constant terms. In this case, our augmented matrix will have two rows and three columns.

	$q$	$d$	Constant
1	1	1	18
2	0.25	0.10	4.20

Simplifying the Augmented Matrix

To simplify the augmented matrix, we can multiply the second row by 10 to eliminate the decimal points.

	$q$	$d$	Constant
1	1	1	18
2	2.5	1	42

Using Gaussian Elimination

To solve the system of equations, we can use Gaussian elimination. We will start by eliminating the variable $d$ from the second row.

	$q$	$d$	Constant
1	1	1	18
2	2.5	1	42

We can eliminate the variable $d$ by subtracting the first row from the second row.

	$q$	$d$	Constant
1	1	1	18
2	1.5	0	24

Finding the Solution

Now that we have eliminated the variable $d$ , we can find the value of $q$ . We can do this by dividing the constant term in the second row by the coefficient of $q$ .

q = \frac{24}{1.5} = 16

Now that we have found the value of $q$ , we can find the value of $d$ by substituting $q$ into one of the original equations.

d = 18 - q = 18 - 16 = 2

Conclusion

In this article, we used an augmented matrix to determine the number of quarters and dimes in Indira's collection. We created an augmented matrix, simplified it, and used Gaussian elimination to solve the system of equations. We found that Indira has 16 quarters and 2 dimes in her collection.

Discussion

The use of augmented matrices and Gaussian elimination is a powerful tool for solving systems of linear equations. This approach can be used to solve a wide range of problems in mathematics, science, and engineering.

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon

Glossary

Augmented Matrix: A matrix that combines the coefficients of the variables with the constant terms.
Gaussian Elimination: A method for solving systems of linear equations by eliminating variables.
Linear Equation: An equation in which the variables are raised to the power of 1.
System of Linear Equations: A set of linear equations that are solved simultaneously.
Indira's Coin Collection: A Mathematical Approach - Q&A =====================================================

Introduction

In our previous article, we used an augmented matrix to determine the number of quarters and dimes in Indira's collection. We created an augmented matrix, simplified it, and used Gaussian elimination to solve the system of equations. In this article, we will answer some frequently asked questions about the problem and the solution.

Q: What is the total value of the coins in Indira's collection?

A: The total value of the coins in Indira's collection is $\$4.20$ .

Q: How many quarters and dimes are in Indira's collection?

A: Indira has 16 quarters and 2 dimes in her collection.

Q: Why did we multiply the second row by 10 to eliminate the decimal points?

A: We multiplied the second row by 10 to eliminate the decimal points because it makes the calculations easier and faster.

Q: What is the purpose of the augmented matrix in this problem?

A: The augmented matrix is used to represent the system of linear equations and to solve for the variables.

Q: How did we eliminate the variable $d$ from the second row?

A: We eliminated the variable $d$ by subtracting the first row from the second row.

Q: What is the value of $q$ after we eliminated the variable $d$ ?

A: The value of $q$ is 16.

Q: How did we find the value of $d$ after we found the value of $q$ ?

A: We found the value of $d$ by substituting $q$ into one of the original equations.

Q: What is the relationship between the number of quarters and the number of dimes in Indira's collection?

A: The number of quarters and the number of dimes in Indira's collection are related by the equation $q + d = 18$ .

Q: What is the significance of the system of linear equations in this problem?

A: The system of linear equations represents the relationship between the number of quarters and the number of dimes in Indira's collection.

Q: How can we use this problem to teach students about linear equations and systems of linear equations?

A: We can use this problem to teach students about linear equations and systems of linear equations by having them work through the solution and explaining the steps involved.

Conclusion

In this article, we answered some frequently asked questions about the problem and the solution. We hope that this Q&A article has been helpful in clarifying any confusion and providing a better understanding of the problem and the solution.

Discussion

References

[1] "Linear Algebra and Its Applications" by Gilbert Strang
[2] "Introduction to Linear Algebra" by Jim Hefferon

Glossary

Augmented Matrix: A matrix that combines the coefficients of the variables with the constant terms.
Gaussian Elimination: A method for solving systems of linear equations by eliminating variables.
Linear Equation: An equation in which the variables are raised to the power of 1.
System of Linear Equations: A set of linear equations that are solved simultaneously.