Solve The System Of Equations Using The Gaussian Elimination Method To Obtain The Matrix In Row Echelon Form. Choose The Correct Answer Below.Given System Of Equations:$\[ \left\{ \begin{array}{l} x + 4y = 0 \\ x + 5y + Z = 1 \\ 5x - Y - Z =
Introduction
The Gaussian elimination method is a widely used technique for solving systems of linear equations. It involves transforming the augmented matrix into row echelon form, which makes it easier to find the solution. In this article, we will use the Gaussian elimination method to solve the given system of equations and obtain the matrix in row echelon form.
The Given System of Equations
The given system of equations is:
Step 1: Write the Augmented Matrix
To apply the Gaussian elimination method, we need to write the augmented matrix of the system of equations. The augmented matrix is a matrix that combines the coefficient matrix and the constant matrix.
Step 2: Perform Row Operations
The next step is to perform row operations to transform the augmented matrix into row echelon form. We will use the following row operations:
- Swap two rows
- Multiply a row by a non-zero constant
- Add a multiple of one row to another row
Step 2.1: Swap Rows 1 and 2
To get a 1 in the top-left position, we will swap rows 1 and 2.
Step 2.2: Multiply Row 1 by -1
To get a 0 in the second column of row 1, we will multiply row 1 by -1.
Step 2.3: Add 5 Times Row 1 to Row 3
To get a 0 in the first column of row 3, we will add 5 times row 1 to row 3.
Step 2.4: Multiply Row 2 by -1
To get a 1 in the second row, we will multiply row 2 by -1.
Step 2.5: Add Row 2 to Row 1
To get a 0 in the second column of row 1, we will add row 2 to row 1.
Step 2.6: Multiply Row 3 by -1/26
To get a 1 in the third row, we will multiply row 3 by -1/26.
Step 2.7: Add 9 Times Row 3 to Row 1
To get a 0 in the third column of row 1, we will add 9 times row 3 to row 1.
Step 2.8: Add 4 Times Row 3 to Row 2
To get a 0 in the third column of row 2, we will add 4 times row 3 to row 2.
Step 2.9: Multiply Row 2 by -1
To get a 1 in the second row, we will multiply row 2 by -1.
Step 2.10: Add 90/13 Times Row 2 to Row 1
To get a 0 in the second column of row 1, we will add 90/13 times row 2 to row 1.
Step 2.11: Add -1 Times Row 2 to Row 3
To get a 0 in the second column of row 3, we will add -1 times row 2 to row 3.
Step 2.12: Multiply Row 3 by 13/51
To get a 1 in the third row, we will multiply row 3 by 13/51.
Step 2.13: Add 26/13 Times Row 3 to Row 1
To get a 0 in the third column of row 1, we will add 26/13 times row 3 to row 1.
Step 2.14: Multiply Row 1 by -1
To get a 1 in the first row, we will multiply row 1 by -1.
Step 2.15: Add 26/13 Times Row 3 to Row 1
To get a 0 in the third column of row 1, we will add 26/13 times row 3 to row 1.
Step 2.16: Multiply Row 1 by 13/26
To get a 1 in the first row, we will multiply row 1 by 13/26.
\left[ \begin{array}{<br/> **Frequently Asked Questions (FAQs)** ===================================== **Q: What is the Gaussian elimination method?** -------------------------------------------- A: The Gaussian elimination method is a widely used technique for solving systems of linear equations. It involves transforming the augmented matrix into row echelon form, which makes it easier to find the solution. **Q: What is the row echelon form?** ------------------------------ A: The row echelon form is a matrix that has the following properties: * All the rows consisting entirely of zeros are grouped at the bottom of the matrix. * Each row that is not entirely zeros has a 1 as its first nonzero entry (this entry is called a **pivot**). * The column in which a pivot is found has all zeros elsewhere, so a column containing a pivot will have zeros everywhere except for one place. **Q: How do I apply the Gaussian elimination method?** ------------------------------------------------ A: To apply the Gaussian elimination method, you need to follow these steps: 1. Write the augmented matrix of the system of equations. 2. Perform row operations to transform the augmented matrix into row echelon form. 3. Use the row echelon form to find the solution of the system of equations. **Q: What are the common row operations?** ----------------------------------------- A: The common row operations are: * Swap two rows * Multiply a row by a non-zero constant * Add a multiple of one row to another row **Q: How do I choose the pivot?** ------------------------------ A: To choose the pivot, you need to select the first nonzero entry in each row. The pivot is the entry that will be used to eliminate the other entries in the same column. **Q: How do I eliminate the other entries in the same column?** --------------------------------------------------------- A: To eliminate the other entries in the same column, you need to add a multiple of the row containing the pivot to the other rows. **Q: What is the advantage of using the Gaussian elimination method?** ---------------------------------------------------------------- A: The advantage of using the Gaussian elimination method is that it allows you to solve systems of linear equations in a systematic and efficient way. It also helps to identify the dependent and independent variables in the system. **Q: What are the limitations of the Gaussian elimination method?** ---------------------------------------------------------------- A: The limitations of the Gaussian elimination method are: * It can be time-consuming for large systems of equations. * It may not be suitable for systems of equations with complex coefficients. * It may not be suitable for systems of equations with a large number of variables. **Q: Can I use the Gaussian elimination method for non-linear systems of equations?** -------------------------------------------------------------------------------- A: No, the Gaussian elimination method is only suitable for linear systems of equations. For non-linear systems of equations, you need to use other methods such as the Newton-Raphson method or the Lagrange multiplier method. **Q: Can I use the Gaussian elimination method for systems of equations with complex coefficients?** ----------------------------------------------------------------------------------------- A: No, the Gaussian elimination method is not suitable for systems of equations with complex coefficients. For systems of equations with complex coefficients, you need to use other methods such as the Gauss-Jordan elimination method or the LU decomposition method. **Q: Can I use the Gaussian elimination method for systems of equations with a large number of variables?** ----------------------------------------------------------------------------------------- A: No, the Gaussian elimination method is not suitable for systems of equations with a large number of variables. For systems of equations with a large number of variables, you need to use other methods such as the QR decomposition method or the singular value decomposition method. **Conclusion** ---------- The Gaussian elimination method is a powerful tool for solving systems of linear equations. It involves transforming the augmented matrix into row echelon form, which makes it easier to find the solution. However, it has some limitations, such as being time-consuming for large systems of equations and not being suitable for systems of equations with complex coefficients or a large number of variables.