Gaussian Elimination With 4 Variables Using Elementary Row Operations With Matrices

The Organic Chemistry Tutor

18 Feb 201818:04

EducationalLearning

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TLDRThis transcript outlines a step-by-step process of solving a system of four-variable linear equations using Gaussian elimination. The method involves forming an augmented matrix, performing row operations to achieve zeros in specific locations, and ultimately simplifying the system into a row echelon form. The detailed explanation guides the user through each step, resulting in a clear solution for the variables w, x, y, and z.

Takeaways

📚 The lesson focuses on solving a system of equations with four variables using Gaussian Elimination.
🔢 The initial step is to convert the system of equations into an augmented matrix, separating coefficients and constants with a vertical bar.
✍️ The goal of the process is to manipulate the matrix into a form where each column has a leading zero, except for the pivot columns.
🌟 Row operations, including adding and multiplying rows, are used to create zeros in specific locations and move the matrix towards row echelon form.
🔄 The first row operation combines row 2 and row 4 to eliminate the first variable from the second and third rows.
📈 Subsequent row operations target the third row, using multiples of row 2 to create zeros in the first and second columns.
🎯 Careful attention must be paid to maintain the correct operations to avoid errors that could ruin the solution process.
📌 Once the matrix is in row echelon form, the system of equations can be solved by back substitution, starting with the simplest equation.
📝 The final solution is found by substituting the values of y and z into the other equations to solve for x and w.
📋 The solution to the system of equations is w = 1, x = 2, y = 3, and z = 4.
🔍 Gaussian Elimination is a powerful tool for solving systems of linear equations, demonstrated effectively in this lesson.

Q & A

What is the method used in the lesson to solve the system of equations?
-The method used in the lesson to solve the system of equations is Gaussian Elimination.
How many variables are in the system of equations discussed in the lesson?
-There are four variables in the system of equations: w, x, y, and z.
What is the augmented matrix in the context of the given system of equations?
-The augmented matrix is a matrix that combines the coefficients of the variables and the constants from the right-hand side of the equations, separated by a vertical bar.
What is the goal of the row operations performed in Gaussian Elimination?
-The goal of the row operations in Gaussian Elimination is to transform the matrix into a form where the coefficients of the variables are zero, except for the leading ones, which helps in solving the system of equations.
How does the讲师 (instructor) transform the first column to have zeros below the leading entry?
-The讲师 (instructor) adds row 2 and row 4 to transform the first column and get zeros below the leading entry.
What row operation is used to make the second column's leading entry zero?
-The讲师 (instructor) multiplies row 2 by 2 and adds it to row 3 to make the second column's leading entry zero.
How does the讲师 (instructor) ensure that the zeros in the matrix remain unchanged while performing row operations?
-The讲师 (instructor) carefully selects and applies row operations, such as adding or subtracting rows, to ensure that the existing zeros do not change.
What is the final system of equations obtained after performing Gaussian Elimination?
-The final system of equations is: w + 2x - y + z = 6, 3x + y = 9, -17y = -51, and 17z = 68.
What is the row echelon form of the matrix after the Gaussian Elimination process?
-The row echelon form of the matrix is: [1 2 -1 1 6; 0 1 0 0 3; 0 0 1 -51/17 3; 0 0 0 1 4].
What are the solutions for w, x, y, and z after converting the system into row echelon form?
-The solutions are w = 1, x = 2, y = 3, and z = 4.
How does the讲师 (instructor) correct the mistake made in the column operations?
-The讲师 (instructor) corrects the mistake by identifying the incorrect entry, making necessary calculations, and adjusting the row operations to ensure the accuracy of the solution.
What is the significance of reducing the system to a regular row echelon form?
-Reducing the system to a regular row echelon form simplifies the process of solving the system of equations and makes it easier to identify the values of the variables.

Outlines

00:00

📚 Introduction to Gaussian Elimination

The paragraph introduces the concept of using Gaussian elimination to solve a system of equations with four variables. The system consists of four equations: w + 2x - y + z = 6, -w + x + 2y - z = 3, 2w - x + 2y + 2z = 14, and w + x - y + 2z = 8. The first step is to convert this system into an augmented matrix, separating the coefficients from the constants with a vertical bar. The goal is to manipulate the matrix into a form where each row represents an equation, and each column corresponds to a variable, ultimately leading to a solution for the variables w, x, y, and z.

05:04

🔄 Row Operations to Achieve Zeros

This paragraph delves into the process of performing row operations to achieve zeros in specific positions within the matrix. The speaker describes adding row 2 and row 4 to create the first zero and then focuses on obtaining zeros in the third row by using row 2 and 3. The operations are carefully explained, emphasizing the importance of accuracy to avoid errors that could compromise the entire problem-solving process. The goal is to transform the matrix into a form where zeros are in the desired locations to simplify the system of equations and bring it closer to a solvable state.

10:07

📉 Solving the System of Equations

The speaker continues the process of solving the system of equations by making further adjustments to the matrix. After performing a series of row operations, the matrix is transformed into a form where the equations can be easily interpreted. The speaker then proceeds to solve for the variables w, x, y, and z by using the simplified matrix. The process involves back substitution and careful arithmetic to arrive at the values of the variables. The final solution is presented as w = 1, x = 2, y = 3, and z = 4, demonstrating the effectiveness of Gaussian elimination in solving complex systems of equations.

15:08

📈 Converting to Row Echelon Form

In the final paragraph, the speaker discusses the conversion of the matrix into row echelon form, which is a further step beyond the standard form achieved earlier. The process involves scaling the rows to have leading ones and using row operations to create a diagonal of ones. The speaker then rewrites the equations based on the new matrix form, providing a clear and simplified version of the original system. The solution is reiterated with the values of w, x, y, and z, and the speaker emphasizes the educational value of understanding how to achieve the row echelon form, which is a fundamental technique in linear algebra and systems of equations.

Mindmap

Keywords

💡Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It involves transforming the system's matrix into an upper triangular form through a series of row operations. In the video, Gaussian Elimination is the primary technique used to simplify the system of equations with four variables, allowing for the eventual solution of the system.

💡Augmented Matrix

An Augmented Matrix is a matrix that combines the coefficients of a system of linear equations with the constants from the right-hand side of the equations. This matrix is used as a visual tool in Gaussian Elimination to organize the system and facilitate the row operations needed to solve for the variables.

💡Row Operations

Row Operations are a set of mathematical maneuvers performed on rows of a matrix during Gaussian Elimination. These operations include swapping rows, multiplying a row by a scalar, and adding or subtracting one row from another. The goal is to manipulate the matrix into a more simplified form where solving the system of equations becomes possible.

💡Pivot Element

The Pivot Element is a term used in Gaussian Elimination to describe the diagonal element in a matrix that is currently being used to create zeros in the columns below it. The process of making the largest possible number in a pivot position is known as pivoting and is crucial for the efficiency of the elimination process.

💡Row Echelon Form

Row Echelon Form is a particular way of arranging the rows of a matrix after applying Gaussian Elimination. In this form, the leading coefficient (the first non-zero number from the left in a row) is always 1, and the matrix is structured so that each leading coefficient is to the right of the one above it. This form of the matrix simplifies the process of back-substitution to solve for the variables.

💡Back-Substitution

Back-Substitution is a technique used after a system of linear equations has been simplified using Gaussian Elimination and put into Row Echelon Form. It involves solving the system from the bottom up, starting with the equation at the bottom and using the solutions of the lower equations to find the values of the variables in the higher equations.

💡Variables

In the context of the video, Variables refer to the unknowns in the system of linear equations being solved. The process of Gaussian Elimination aims to find the values of these variables that satisfy all the equations simultaneously.

💡Linear Equations

Linear Equations are mathematical equations in which the highest power of the variables is 1. They represent straight lines in two-dimensional space and planes in three-dimensional space. The video is about solving a system of such equations with multiple variables.

💡Systems of Equations

A System of Equations is a set of simultaneous equations that need to be solved together. Each equation in the system represents a condition or a relationship between variables. The video is focused on using Gaussian Elimination to find the values of variables that satisfy all equations in the system simultaneously.

💡Zeroing Out

Zeroing Out is a term used in the context of Gaussian Elimination to describe the process of creating zeros in specific positions in the matrix. This is done through row operations to simplify the system and make it easier to solve for the variables.

💡Pivot Position

Pivot Position refers to the location in a row where the main focus is during the Gaussian Elimination process. The goal is to have a non-zero number, known as the pivot element, in this position to effectively use it for creating zeros in the columns below.

Highlights

Introduction to Gaussian Elimination for solving a system of equations with four variables. (Start time: 0s)

Formation of an augmented matrix from the given system of equations. (Start time: 10s)

First row operation to achieve a zero in the first variable's column. (Start time: 30s)

Second row operation to eliminate variables from the second and third columns. (Start time: 50s)

Third row operation to make the second variable's column zeros. (Start time: 1m 10s)

Fourth row operation to adjust the third and fourth rows for further elimination. (Start time: 2m 30s)

Explanation of the necessity to maintain zeros in certain columns during row operations. (Start time: 3m 10s)

Conversion of the system into a row echelon form. (Start time: 4m 20s)

Solving for the variable z using the simplified matrix. (Start time: 4m 50s)

Solving for the variable y using the updated row echelon form. (Start time: 5m 30s)

Solving for the variable x using the back-substitution method. (Start time: 6m 10s)

Solving for the variable w using the completed row echelon form. (Start time: 7m 20s)

Final answer presentation in the form of w, x, y, z. (Start time: 8m 0s)

Summary of the Gaussian Elimination process and its application in solving complex systems of equations. (Start time: 9m 0s)

Emphasis on the importance of careful calculation to avoid errors in the Gaussian Elimination process. (Start time: 10m 30s)

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Takeaways

Q & A

What is the method used in the lesson to solve the system of equations?

How many variables are in the system of equations discussed in the lesson?

What is the augmented matrix in the context of the given system of equations?

What is the goal of the row operations performed in Gaussian Elimination?

How does the讲师 (instructor) transform the first column to have zeros below the leading entry?

What row operation is used to make the second column's leading entry zero?

How does the讲师 (instructor) ensure that the zeros in the matrix remain unchanged while performing row operations?

What is the final system of equations obtained after performing Gaussian Elimination?

What is the row echelon form of the matrix after the Gaussian Elimination process?

What are the solutions for w, x, y, and z after converting the system into row echelon form?

How does the讲师 (instructor) correct the mistake made in the column operations?

What is the significance of reducing the system to a regular row echelon form?