PreCalculus - Matrices & Matrix Applications (10 of 33) Gaussian Elimination: Example of Solving 3x3

Michel van Biezen
13 Jun 201508:27
EducationalLearning
32 Likes 10 Comments

TLDRThe video script presents a step-by-step guide on solving a system of linear equations using the method of Gaussian elimination. The process involves creating an augmented matrix with coefficients of the variables and constants, and then systematically manipulating the matrix to achieve a diagonal of ones with zeros elsewhere. The final row operations yield the values for x, y, and z, which are the solution to the given equations. The method is detailed with clear explanations and is referred to as the 'giian elimination method' by the speaker.

Takeaways
  • 📌 The video discusses solving a system of linear equations with three variables (x, y, z) using the method of Gaussian elimination.
  • 🔢 The process starts by forming an augmented matrix with the coefficients of the variables and constants from the equations.
  • 🎯 The goal is to manipulate the matrix so that the diagonal elements are ones and all other elements in the same rows and columns are zeros.
  • 🔄 Row interchanges are performed to move the largest coefficient to the diagonal position to avoid fractions.
  • 📉 Rows are then scaled and added or subtracted from each other to create zeros in the columns below the diagonal.
  • ✅ The process is repeated for each column, ensuring that only one non-zero entry remains in each column.
  • 🔢 After completing the elimination process, the values of x, y, and z can be read directly from the matrix.
  • 🌟 The video emphasizes the importance of keeping track of the row operations to correctly solve for the variables.
  • 📚 The method of Gaussian elimination is a powerful tool for solving systems of linear equations and finding the coordinates where three planes meet in space.
  • 👍 The video demonstrates a step-by-step approach, making it easier for viewers to understand and follow the process.
  • 💡 The final result provides the values of the variables that satisfy all three equations simultaneously.
Q & A

  • What is the main topic of the video?

    -The main topic of the video is solving a system of linear equations using the method of Gaussian elimination.

  • What is an augmented matrix?

    -An augmented matrix is a matrix that consists of the coefficients of the variables in a system of linear equations, along with the constants from the right side of the equal sign, used to help visualize and solve the system.

  • What is the goal of Gaussian elimination?

    -The goal of Gaussian elimination is to manipulate the augmented matrix into a form where the diagonal elements are 1s and all other elements in the corresponding columns and rows are 0s, making it easier to read off the solution values for the variables.

  • How does the speaker decide which row to start with in the Gaussian elimination process?

    -The speaker decides to start with the row that does not contain fractions or decimals to avoid complications early in the process.

  • What operation did the speaker perform to get a 1 in the upper left corner of the matrix?

    -The speaker performed a row interchange, swapping row one with row three, to get a 1 in the upper left corner of the matrix.

  • How does the speaker eliminate variables from the second column?

    -The speaker eliminates variables from the second column by using back-substitution, where they replace rows with calculations involving the 1s in the diagonal and add them to other rows to get zeros above and below the 1s.

  • What does the final matrix look like after the Gaussian elimination process is complete?

    -The final matrix has 1s down the diagonal, with 0s in the positions directly above and below each 1, and the solution values for x, y, and z are found in the columns corresponding to these 1s.

  • How does the speaker interpret the final matrix to find the values of x, y, and z?

    -The speaker interprets the final matrix by looking at the columns with the 1s and reading the values in the last row, which correspond to the values of x, y, and z.

  • What is the solution to the given system of linear equations based on the video?

    -The solution to the given system of linear equations is x = -2, y = 1, and z = 3.

  • Why is it important to have a 1 in each diagonal position during the Gaussian elimination process?

    -Having a 1 in each diagonal position simplifies the process of eliminating variables and allows for easy back-substitution to find the solution values for each variable.

  • What is the method of elimination also referred to by the speaker?

    -The speaker also refers to the method of elimination as the 'giian elimination' method.

Outlines
00:00
📚 Introduction to Solving Systems of Linear Equations

This paragraph introduces the method of Gaussian elimination for solving systems of linear equations. It explains the process of finding values for variables x, y, and z that satisfy all three equations by arranging the coefficients of the variables in an augmented matrix. The goal is to manipulate the matrix so that the diagonal elements are ones and all other elements in the same row are zeros, which will then allow for the determination of the solution. The paragraph outlines the initial steps of this method, including the interchanging of rows and the use of operations to achieve the desired matrix form.

05:02
🔢 Elaboration on Gaussian Elimination Steps

This paragraph delves deeper into the specific steps of Gaussian elimination, detailing the row operations required to transform the augmented matrix. It describes how to achieve a matrix with ones along the diagonal and zeros elsewhere by using multiplication and addition of rows. The paragraph also explains how to deal with fractions and decimals that may arise during the process. It concludes by showing how the final form of the matrix allows for the extraction of the solution values for x, y, and z, thus solving the system of linear equations using this method.

Mindmap
Keywords
💡Gaussian Elimination
Gaussian elimination is a systematic method used to solve systems of linear equations by transforming the augmented matrix associated with the system into an upper triangular form. In the video, this method is used to find the values of x, y, and z that satisfy all three given equations. The process involves manipulating the matrix to create a diagonal of ones and zeros elsewhere, from which the solution can be read directly.
💡Augmented Matrix
An augmented matrix is a mathematical tool used in linear algebra to simultaneously represent the system of linear equations and their constants. It consists of the coefficients of the variables and the constants from the right-hand side of the equations. In the context of the video, the augmented matrix is used to visualize and solve the system of equations for x, y, and z through Gaussian elimination.
💡Coefficients
Coefficients are numerical factors that are multiplied by variables in a linear equation. In the context of the video, the coefficients refer to the numbers in front of the variables x, y, and z in the given equations. These coefficients are crucial for solving the system of equations, as they are used to form the augmented matrix and guide the steps of Gaussian elimination.
💡Variables
In the context of the video, variables refer to the unknown quantities represented by x, y, and z in the system of linear equations. The goal of the video is to find the specific values of these variables that satisfy all three equations simultaneously. Variables are essential in algebra and are used to represent quantities that can change or are unknown.
💡Linear Equations
Linear equations are mathematical equations in which the highest power of the variables is one. They are the simplest form of algebraic equations and are used to model relationships between quantities that are proportional to each other. In the video, the linear equations are represented by the three given equations involving x, y, and z that need to be solved simultaneously.
💡Solutions
In the context of the video, a solution refers to the specific values of the variables x, y, and z that satisfy all three linear equations simultaneously. Finding the solution is the ultimate goal of the Gaussian elimination process demonstrated in the video. A solution to a system of linear equations can be unique, non-existent, or infinite, depending on the nature of the system.
💡Row Operations
Row operations are a set of basic operations performed on rows of a matrix during Gaussian elimination or other linear algebra methods. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting one row from another. The purpose of row operations is to simplify the matrix and make it easier to find the solutions to the system of linear equations.
💡Diagonal
In the context of matrices, the diagonal refers to the elements that run from the top left to the bottom right of the matrix, forming a line from the first element to the last. In Gaussian elimination, the goal is often to create a diagonal of ones (also known as an identity matrix), with zeros elsewhere in the diagonal, to facilitate the solution of the system of linear equations.
💡Intersection
In the context of the video, intersection refers to the point or points in space where the three planes represented by the linear equations meet. The solution to the system of equations corresponds to the coordinates of this intersection point. Finding the intersection is crucial in many areas of mathematics and its applications, such as geometry and physics.
💡Constants
Constants in the context of the video are the numerical values that appear on the right-hand side of the linear equations. They represent known quantities that, along with the coefficients and variables, determine the solution to the system of equations. Constants are essential in forming the augmented matrix and are part of the process to find the values of the variables.
💡Planes
In the context of the video, planes refer to the two-dimensional surfaces defined by the linear equations. Each equation represents a plane in three-dimensional space. The video's objective is to find the point at which all three planes intersect, which is equivalent to solving the system of linear equations that define these planes.
💡Interchange
Interchange, in the context of the video, refers to the process of swapping the positions of two rows within the augmented matrix. This is one of the row operations used in Gaussian elimination to manipulate the matrix in a way that facilitates the solution of the system of linear equations.
Highlights

Introduction to the method of Gaussian elimination for solving systems of linear equations.

Explanation of how to use the coefficients of variables to form an augmented matrix.

The process of transforming the matrix to have ones on the diagonal and zeros elsewhere.

Interchanging rows to achieve a one in the upper left corner.

Using the first row to turn other elements in the same column to zero.

Transforming the second row to have a one in the middle column.

Multiplying entire rows by negative one to change signs and achieve a one in the middle column.

Using the second row to turn other elements in the middle column to zero.

Transforming the third row to have a one in the last column by dividing by the coefficient.

Using the third row to turn other elements in the last column to zero.

Final step of obtaining the values for x, y, and z by back substitution.

Solution of the set of linear equations using Gaussian elimination.

The practical application of finding the coordinates where three planes meet in space.

The importance of maintaining a systematic approach to solving complex mathematical problems.

The innovative method of visual separation of variables and constants in the augmented matrix.

The detailed explanation of each step in the Gaussian elimination process.

The use of the augmented matrix as a visual tool for solving systems of linear equations.

Transcripts

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