For A Client, A Landscaper Purchases 3 Oak Trees And 4 Maple Trees For $\$ 380$. For His Own Home, The Landscaper Purchases 2 Oak Trees And 5 Maple Trees For $\$ 370$[/tex\]. Which Augmented Matrix Represents The Situation?A.

Feb 28, 2025 by ADMIN 230 views

**Solving a Real-World Problem with Augmented Matrices**

Introduction

In this article, we will explore how to use augmented matrices to solve a real-world problem involving a landscaper's purchases of oak and maple trees. We will first create an augmented matrix to represent the situation and then use it to find the cost of each type of tree.

The Problem

A landscaper purchases 3 oak trees and 4 maple trees for $380. For his own home, the landscaper purchases 2 oak trees and 5 maple trees for $370. We need to find the cost of each type of tree.

Step 1: Create an Augmented Matrix

To create an augmented matrix, we need to represent the situation as a system of linear equations. Let's denote the cost of an oak tree as x and the cost of a maple tree as y.

We can create the following system of linear equations:

3x + 4y = 380 ... (1) 2x + 5y = 370 ... (2)

Now, we can create an augmented matrix to represent this system of linear equations.

| 3  4 | 380 |
| ---  --- | --- |
| 2  5 | 370 |

Step 2: Write the Augmented Matrix in Standard Form

To write the augmented matrix in standard form, we need to make sure that the coefficients of the variables are in the same order as the variables themselves. In this case, we have x and y as the variables.

So, the augmented matrix in standard form is:

| 3  4 | 380 |
| ---  --- | --- |
| 2  5 | 370 |

Step 3: Solve the System of Linear Equations

To solve the system of linear equations, we can use the method of substitution or elimination. In this case, we will use the elimination method.

First, we will multiply equation (1) by 2 and equation (2) by 3 to make the coefficients of x the same.

| 6  8 | 760 |
| ---  --- | --- |
| 6  15 | 1110 |

Now, we can subtract equation (1) from equation (2) to eliminate the variable x.

| 0  7 | 350 |

Step 4: Find the Cost of Each Type of Tree

Now that we have eliminated the variable x, we can find the cost of each type of tree.

From equation (1), we have:

3x + 4y = 380

Substituting y = 50 into this equation, we get:

3x + 4(50) = 380

Simplifying this equation, we get:

3x + 200 = 380

Subtracting 200 from both sides, we get:

3x = 180

Dividing both sides by 3, we get:

x = 60

So, the cost of an oak tree is $60.

Conclusion

In this article, we used an augmented matrix to solve a real-world problem involving a landscaper's purchases of oak and maple trees. We created an augmented matrix to represent the situation, wrote it in standard form, and solved the system of linear equations using the elimination method. We found that the cost of an oak tree is $60 and the cost of a maple tree is $50.

References

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

Note

Introduction

In our previous article, we explored how to use augmented matrices to solve a real-world problem involving a landscaper's purchases of oak and maple trees. In this article, we will answer some frequently asked questions (FAQs) about augmented matrices.

Q: What is an augmented matrix?

A: An augmented matrix is a mathematical representation of a system of linear equations, where the coefficients of the variables are arranged in a matrix and the constants are placed on the right-hand side.

Q: How do I create an augmented matrix?

A: To create an augmented matrix, you need to represent the system of linear equations as a matrix, where the coefficients of the variables are in the same order as the variables themselves. You can then add the constants to the right-hand side of the matrix.

Q: What is the difference between a matrix and an augmented matrix?

A: A matrix is a mathematical representation of a set of numbers, where the numbers are arranged in rows and columns. An augmented matrix, on the other hand, is a matrix that includes the constants on the right-hand side, making it a system of linear equations.

Q: How do I solve a system of linear equations using an augmented matrix?

A: To solve a system of linear equations using an augmented matrix, you can use various methods such as substitution, elimination, or Gaussian elimination. You can also use row operations to transform the augmented matrix into a simpler form.

Q: What are some common row operations used in augmented matrices?

A: Some common row operations used in augmented matrices include:

Multiplying a row by a scalar
Adding a multiple of one row to another row
Interchanging two rows
Multiplying a row by -1

Q: How do I determine the number of solutions to a system of linear equations?

A: To determine the number of solutions to a system of linear equations, you can look at the augmented matrix. If the matrix has a unique solution, it will have a single solution. If the matrix has no solution, it will have no solution. If the matrix has infinitely many solutions, it will have infinitely many solutions.

Q: What is the significance of the rank of an augmented matrix?

A: The rank of an augmented matrix is the maximum number of linearly independent rows in the matrix. It is an important concept in linear algebra and is used to determine the number of solutions to a system of linear equations.

Q: Can I use augmented matrices to solve systems of linear equations with more than two variables?

A: Yes, you can use augmented matrices to solve systems of linear equations with more than two variables. The process is similar to solving a system of linear equations with two variables, but you will need to use more row operations to transform the augmented matrix into a simpler form.

Q: Are there any limitations to using augmented matrices?

A: Yes, there are some limitations to using augmented matrices. For example, if the system of linear equations has infinitely many solutions, the augmented matrix may not be able to capture all the solutions. Additionally, if the system of linear equations has no solution, the augmented matrix may not be able to determine this.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about augmented matrices. We hope that this article has provided you with a better understanding of augmented matrices and how they can be used to solve systems of linear equations.

References

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

Note

The information provided in this article is for educational purposes only and is not intended to be used as a substitute for professional advice or guidance.