Solve The System Of Equations Using Matrices. Use The Gaussian Elimination Method With Back-substitution.$\[ \left\{ \begin{array}{r} x + Y - Z = -2 \\ 3x - Y + Z = 6 \\ -x + 4y - 3z = 1 \end{array} \right. \\]Use The Gaussian Elimination

Feb 28, 2025 by ADMIN 239 views

Solve the System of Equations Using Matrices: A Step-by-Step Guide to Gaussian Elimination with Back-Substitution

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Introduction

In this article, we will explore the concept of solving a system of linear equations using matrices and the Gaussian elimination method with back-substitution. This technique is a powerful tool for solving systems of linear equations and is widely used in various fields such as physics, engineering, and computer science.

What is Gaussian Elimination?

Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into upper triangular form using elementary row operations. This method is named after the German mathematician Carl Friedrich Gauss, who first developed it in the early 19th century.

The Augmented Matrix

The augmented matrix is a matrix that combines the coefficients of the variables and the constant terms of the system of equations. It is denoted by the symbol [A | b], where A is the coefficient matrix and b is the constant matrix.

For the given system of equations:

{ \left\{ \begin{array}{r} x + y - z = -2 \\ 3x - y + z = 6 \\ -x + 4y - 3z = 1 \end{array} \right. \}

The augmented matrix is:

{ \left[ \begin{array}{ccc|c} 1 & 1 & -1 & -2 \\ 3 & -1 & 1 & 6 \\ -1 & 4 & -3 & 1 \end{array} \right] \}

Gaussian Elimination

The first step in Gaussian elimination is to transform the augmented matrix into upper triangular form using elementary row operations. We will perform the following operations:

Multiply row 1 by -3 and add it to row 2.
Multiply row 1 by 1 and add it to row 3.

Performing these operations, we get:

{ \left[ \begin{array}{ccc|c} 1 & 1 & -1 & -2 \\ 0 & -4 & 4 & 18 \\ 0 & 5 & -2 & 3 \end{array} \right] \}

Back-Substitution

Now that we have transformed the augmented matrix into upper triangular form, we can use back-substitution to find the values of the variables. We will start with the last equation and work our way up to the first equation.

From the last equation, we have:

5y - 2z = 3

We can solve for y in terms of z:

y = (2z + 3) / 5

Now, we can substitute this expression for y into the second equation:

-4y + 4z = 18

Substituting the expression for y, we get:

-4((2z + 3) / 5) + 4z = 18

Simplifying, we get:

-8z - 12 + 20z = 90

Combine like terms:

12z = 102

Divide by 12:

z = 17/2

Now that we have found the value of z, we can substitute it into the expression for y:

y = (2(17/2) + 3) / 5

Simplifying, we get:

y = (17 + 3) / 5

y = 20/5

y = 4

Now that we have found the values of y and z, we can substitute them into the first equation to find the value of x:

x + y - z = -2

Substituting the values of y and z, we get:

x + 4 - 17/2 = -2

Simplifying, we get:

x - 17/2 + 8/2 = -2

Combine like terms:

x - 9/2 = -2

Add 9/2 to both sides:

x = -2 + 9/2

Simplifying, we get:

x = -4/2 + 9/2

x = 5/2

Conclusion

In this article, we have used the Gaussian elimination method with back-substitution to solve a system of linear equations. We have transformed the augmented matrix into upper triangular form using elementary row operations and then used back-substitution to find the values of the variables. This technique is a powerful tool for solving systems of linear equations and is widely used in various fields.

Example Use Cases

Gaussian elimination with back-substitution has many practical applications in various fields such as:

Physics: To solve systems of linear equations that describe the motion of objects in physics.
Engineering: To solve systems of linear equations that describe the behavior of electrical circuits, mechanical systems, and other engineering systems.
Computer Science: To solve systems of linear equations that arise in computer graphics, machine learning, and other areas of computer science.

Code Implementation

Here is an example implementation of Gaussian elimination with back-substitution in Python:

import numpy as np

def gaussian_elimination(A, b):
    n = len(A)
    for i in range(n):
        # Search for maximum in this column
        max_el = abs(A[i][i])
        max_row = i
        for k in range(i+1, n):
            if abs(A[k][i]) > max_el:
                max_el = abs(A[k][i])
                max_row = k

        # Swap maximum row with current row
        A[i], A[max_row] = A[max_row], A[i]
        b[i], b[max_row] = b[max_row], b[i]

        # Make all rows below this one 0 in current column
        for k in range(i+1, n):
            c = -A[k][i]/A[i][i]
            for j in range(i, n+1):
                if i == j:
                    A[k][j] = 0
                else:
                    A[k][j] += c * A[i][j]
            b[k] += c * b[i]

    # Solve equation Ax=b for an upper triangular matrix A
    x = [0 for i in range(n)]
    for i in range(n-1, -1, -1):
        x[i] = b[i]/A[i][i]
        for k in range(i-1, -1, -1):
            b[k] -= A[k][i] * x[i]
    return x

# Define the augmented matrix
A = [[1, 1, -1], [3, -1, 1], [-1, 4, -3]]
b = [-2, 6, 1]

# Solve the system of equations
x = gaussian_elimination(A, b)

print("The solution is x =", x[0], ", y =", x[1], ", z =", x[2])

This code implements the Gaussian elimination method with back-substitution to solve a system of linear equations. It takes the augmented matrix A and the constant matrix b as input and returns the solution x.

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Q: What is Gaussian Elimination?

A: Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix into upper triangular form using elementary row operations.

Q: What is Back-Substitution?

A: Back-substitution is a method for solving systems of linear equations by substituting the values of the variables into the equations in reverse order.

Q: How Does Gaussian Elimination with Back-Substitution Work?

A: Gaussian elimination with back-substitution works by first transforming the augmented matrix into upper triangular form using elementary row operations. Then, the values of the variables are substituted into the equations in reverse order to find the solution.

Q: What are the Advantages of Gaussian Elimination with Back-Substitution?

A: The advantages of Gaussian elimination with back-substitution include:

Efficient: Gaussian elimination with back-substitution is an efficient method for solving systems of linear equations.
Accurate: Gaussian elimination with back-substitution is an accurate method for solving systems of linear equations.
Easy to Implement: Gaussian elimination with back-substitution is easy to implement in computer programs.

Q: What are the Disadvantages of Gaussian Elimination with Back-Substitution?

A: The disadvantages of Gaussian elimination with back-substitution include:

Sensitive to Round-Off Errors: Gaussian elimination with back-substitution is sensitive to round-off errors.
Not Suitable for Large Systems: Gaussian elimination with back-substitution is not suitable for large systems of linear equations.

Q: When to Use Gaussian Elimination with Back-Substitution?

A: Gaussian elimination with back-substitution should be used when:

The System of Linear Equations is Small: Gaussian elimination with back-substitution is suitable for small systems of linear equations.
The System of Linear Equations is Well-Conditioned: Gaussian elimination with back-substitution is suitable for well-conditioned systems of linear equations.

Q: How to Implement Gaussian Elimination with Back-Substitution in a Computer Program?

A: Gaussian elimination with back-substitution can be implemented in a computer program using the following steps:

Define the Augmented Matrix: Define the augmented matrix A and the constant matrix b.
Transform the Augmented Matrix into Upper Triangular Form: Transform the augmented matrix into upper triangular form using elementary row operations.
Perform Back-Substitution: Perform back-substitution to find the values of the variables.

Q: What are the Applications of Gaussian Elimination with Back-Substitution?

A: The applications of Gaussian elimination with back-substitution include:

Physics: Gaussian elimination with back-substitution is used to solve systems of linear equations that describe the motion of objects in physics.
Engineering: Gaussian elimination with back-substitution is used to solve systems of linear equations that describe the behavior of electrical circuits, mechanical systems, and other engineering systems.
Computer Science: Gaussian elimination with back-substitution is used to solve systems of linear equations that arise in computer graphics, machine learning, and other areas of computer science.

Q: What are the Limitations of Gaussian Elimination with Back-Substitution?

A: The limitations of Gaussian elimination with back-substitution include:

Sensitive to Round-Off Errors: Gaussian elimination with back-substitution is sensitive to round-off errors.
Not Suitable for Large Systems: Gaussian elimination with back-substitution is not suitable for large systems of linear equations.

Q: How to Choose the Best Method for Solving a System of Linear Equations?

A: The best method for solving a system of linear equations depends on the size and condition of the system. Gaussian elimination with back-substitution is suitable for small systems of linear equations, while other methods such as LU decomposition and QR decomposition are suitable for large systems of linear equations.

Q: What are the Future Directions of Gaussian Elimination with Back-Substitution?

A: The future directions of Gaussian elimination with back-substitution include:

Improving the Efficiency of the Method: Improving the efficiency of Gaussian elimination with back-substitution by using more efficient algorithms and data structures.
Developing New Applications: Developing new applications of Gaussian elimination with back-substitution in fields such as physics, engineering, and computer science.

Q: How to Get Started with Gaussian Elimination with Back-Substitution?

A: To get started with Gaussian elimination with back-substitution, follow these steps:

Learn the Basics of Linear Algebra: Learn the basics of linear algebra, including vectors, matrices, and linear transformations.
Learn the Basics of Gaussian Elimination: Learn the basics of Gaussian elimination, including the elementary row operations and the transformation of the augmented matrix into upper triangular form.
Practice with Examples: Practice with examples of Gaussian elimination with back-substitution to gain a deeper understanding of the method.

Q: What are the Resources Available for Learning Gaussian Elimination with Back-Substitution?

A: The resources available for learning Gaussian elimination with back-substitution include:

Textbooks: Textbooks on linear algebra and numerical analysis that cover Gaussian elimination with back-substitution.
Online Courses: Online courses on linear algebra and numerical analysis that cover Gaussian elimination with back-substitution.
Tutorials: Tutorials on Gaussian elimination with back-substitution that provide step-by-step instructions and examples.
Software: Software packages that implement Gaussian elimination with back-substitution, such as MATLAB and Python.