This document discusses two direct methods for solving linear equation systems: Gauss-Jordan elimination and the Aitken method. It introduces linear systems of equations and outlines the key steps of the Gauss-Jordan method. Examples of applying Gauss-Jordan elimination to systems with two and three equations are provided. The document also provides an overview of the Aitken method and works through two examples of using the Aitken method to solve systems of equations.

1. Solving linear equation systems using direct
methods:
Gauss Jordan and Aitken methods
Presented by: Sulaiman M. KARIM
Student No: 4013031617
E-mail: Sulaiman_musaria@yahoo.Com

2. Outlines
 INTRODUCTION
 METHODS OF SOLUTION
 GAUSS - JORDAN METHOD
LINEAR SYSTEM OF EQUATIONS
 GAUSS - JORDAN METHOD THEORY
EXAMPLES
AITKEN METHOD
EXAMPLES

3. A system of equations is a collection of two or more
equations with a same set of unknowns (x1,x2,… xn).
In solving a system of equations, we try to find
values for each of the unknowns ,that will satisfy
every equation in the system. The equations in the
system can be linear or non-linear. Here we are
interested in solving linear equations systems using
two direct methods:
1- Gauss Jordan method
2- Aitkem method
Introduction

4. There are two classes of methods for solving linear
systems equations:-
Class 1: Direct methods : The most known methods are:-
* Gauss - Elimination method,
* Gauss - Jordan method,
* Matrix inversion method,
* Aitken method.
2- Indirect methods (or iterative methods):
* Jacobi’s method .
* Gauss - Seidel method .
Methods of solution

5.  Gauss elimination method (with back substitution) is
named after Carl Gauss how formulated this method in
1810. Gauss – Jordan method is an extension of this
methods proposed by W. Jordan in 1888. Using a sequence
of row reduction techniques, a system of linear equations
can be resolved without back substitutions.
Gauss – Jordan method provides a direct method for
obtaining the solution of linear system of equations.
The methods of Gauss-Jordan and Gauss elimination can
look almost identical, the former requires approximately
50% fewer operations.
Gauss - Jordan method

6. Linear system of equations

7. Gauss - Jordan method theory

8. Example (1): system of two equations

9. Example (2): system of three equations

10. Example (2): system of three equations

11. Example (3): system of three equations

12. Example (3): system of three equations

13. 2- Aitken Method

14. 2- Aitken Method

15. 2- Aitken Method

16. Aitken Method- Example (1)

17. Aitken Method- Example (1)

18. Aitken Method- Example (1)

19. Aitken Method- Example (2)

20. Aitken Method- Example (2)

21. Aitken Method- Example (2)