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Gauss Elimination Method

A methodical algebraic approach for resolving linear equation systems is Gauss Elimination. It makes it simpler to determine the values of unknown variables by methodically reducing a given system to an upper triangular matrix.

This approach is frequently used in domains where systems of equations naturally occur, such as data modelling, computer science, physics, and engineering.

 

Table of Content

Definition and Objective

Gauss Elimination is a matrix-based process that uses basic row operations to convert a system of equations into row echelon form (REF).

The primary goal is to simplify the system so that back substitution can be used to solve variables with ease.

The new system has the same solutions as the original because the method maintains the original system's solution set.

 

Prerequisites

Prior to studying or using Gauss Elimination, you should comprehend:

  • Simple matrix operations (scalar multiplication, addition)

  • Principles of linear equation systems

  • The definition and formation of an augmented matrix

  • How row operations are carried out and interpreted:

    • Swapping rows

    • Using a non-zero scalar to multiply a row

    • One row is added to or subtracted from another.

 

Step-by-Step Procedure

Create an augmented matrix from the system of equations.

Employ simple row operations to:

  • If necessary, set the first row's leading coefficient (pivot) to 1.

  • Make all values below the pivot zero.

  • For the following pivot, proceed to the second row and repeat the procedure.

  • Keep going until the matrix is in row echelon or upper triangular form.

Apply back substitution after the matrix is in REF:

  • Solve for the final variable starting with the last equation.

  • To find the remaining variables, substitute back into the earlier equations.

Types of Solutions

  • A Special Solution

occurs when the system has a consistent and autonomous structure and every variable can be solved.

  • No Answer

There are irregularities in the system. A row like [0 0 0 | 5] might appear, indicating that 0 = 5, which is not possible.

  • Unbounded Solutions

occurs when a number of equations are linear combinations of one another. Free variables are indicated by rows like [0 0 0 | 0].

 

Example Problems

Example 1: Solve the system

2x + y = 5
4x + 3y = 11

Step 1: Write the augmented matrix

[ 2   1 |  5 ]
[ 4   3 | 11 ]

Step 2: Eliminate below the pivot

Row2 = Row 2 - 2 × Row1
[ 2   1 |  5 ]
[ 0   1 |  1 ]

Step 3: Back-substitution

From Row 2: y = 1
From Row 1: 2x + 1 = 5 → x = 2

Solution: x = 2, y = 1

 

Example 2: Solve the system

x + 2y + z = 9
2x + 5y + z = 20
3x + 6y + 2z = 29

Step 1: Augmented Matrix

[ 1   2   1 |  9 ]
[ 2   5   1 | 20 ]
[ 3   6   2 | 29 ]

Step 2: Eliminate entries below pivot in column 1

Row2 = Row 2 - 2 × Row1
Row3 = Row 3 - 3 × Row1
[ 1   2   1 |  9 ]
[ 0   1  -1 |  2 ]
[ 0   0  -1 |  2 ]

 

Step 3: Back-substitution

From Row 3: -z = 2 → z = -2
From Row 2: y - (-2) = 2 → y = 0
From Row 1: x + 0 - 2 = 9 → x = 11

Solution: x = 11, y = 0, z = -2

Example 3: Solve the system

x + y + z = 6
2x + 3y + 5z = 19
4x + 5y + 6z = 26

Step 1: Augmented Matrix

[ 1   1   1 |  6 ]
[ 2   3   5 | 19 ]
[ 4   5   6 | 26 ]

Step 2: Eliminate entries below pivot in column 1

Row2 = Row 2 - 2 × Row1
Row3 = Row 3 - 4 × Row1
[ 1   1   1 |  6 ]
[ 0   1   3 |  7 ]
[ 0   1   2 |  2 ]

Step 3: Eliminate below pivot in column 2

Row3 = Row3 - Row2
[ 1   1   1 |  6 ]
[ 0   1   3 |  7 ]
[ 0   0  -1 | -5 ]

Step 4: Back-substitution

From Row 3: -z = -5 → z = 5
From Row 2: y + 3(5) = 7 → y = -8
From Row 1: x - 8 + 5 = 6 → x = 9

Solution: x = 9, y = -8, z = 5

 

Common Mistakes and How to Avoid Them

  • not using row operations consistently for the augmented portion as well as the other elements in a row.

  • making arithmetic mistakes when manipulating rows.

  • disregarding a zero pivot and failing to switch to a row with a non-zero element below it.

  • not completely reducing the matrix prior to back substitution.

  • failing to re-enter the original equations to confirm the solution.

 

Practice Questions

  • Solve five 3x3 systems using Gauss Elimination and identify the type of solution in each case.

  • Change one coefficient in a system and observe how the solution changes.

  • Create an inconsistent system and identify the row that proves it.

  • Use Gauss Elimination to solve a system where one variable is missing in one or more equations.

 

Real-Life Applications

  • Electrical circuit analysis involves applying Kirchhoff's laws to solve for voltage or current.

  • 3D modelling and transformations are done with computer graphics.

  • For figuring out the stresses and forces in support beams, use structural engineering.

  • Economics and Input-Output Models: Illustrating how economic systems are interdependent.

  • Finding coefficients in statistical models is a component of data fitting and regression analysis.

Conclusion

To solve equation systems in an organised, algorithmic manner, Gauss Elimination is necessary.

Converting the matrix to row echelon form and using back substitution should always be the goal.

Get comfortable recognising pivot points and strategically switching rows as necessary.

Recognise the effects of zero rows and how they affect the characteristics of solutions

To become more proficient with the approach, go over problems of various sizes and kinds on a regular basis.

 

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Frequently Asked Questions on Gauss Elimination Method

1. What is the Gaussian elimination method?

Gaussian Elimination is a method used to solve systems of linear equations. It involves performing row operations on an augmented matrix to transform it into row echelon form, from which the solutions to the equations can be found using back substitution. This method simplifies complex linear systems and is widely used in algebra and computational mathematics.

 

2. What is the method of Gauss formula?

The Gauss formula in the context of linear equations refers to the algorithmic steps of Gaussian Elimination. The method includes:

  • Representing equations in matrix form.

  • Using row operations to create zeros below the pivot elements.

  • Reducing the matrix to upper triangular (row echelon) form.

  • Applying back substitution to find the unknown variables.

Note: This is different from Gauss’s summation formula, which calculates the sum of the first nnn natural numbers:
Sum = n(n + 1)/2

 

3. How to solve an equation by Gauss elimination method?

To solve a system using Gauss Elimination:

  • Step 1: Write the augmented matrix of the system.

  • Step 2: Use elementary row operations to reduce the matrix to row echelon form.

  • Step 3: Once in upper triangular form, apply back substitution to solve for the variables.

This method provides a clear and structured approach to solving both 2x2 and larger 3x3 systems.

 

4. What is the elimination method formula?

There is no single “formula” for the elimination method, but the technique involves these general steps:

  • Multiply one or both equations to align coefficients.

  • Add or subtract the equations to eliminate one variable.

  • Solve the resulting equation for the remaining variable.

  • Substitute the value back to find the other variable(s).

This technique is also part of Gaussian Elimination when done with matrix operations.

 

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