#System

Описание: #System of Linear Equations – Gauss Elimination & Gauss–Jordan Method:~

1) System of Linear Equations:~
@Definition

A system of linear equations is a set of two or more linear equations with the same variables that are solved simultaneously.

2) Gauss Elimination Method (Definition)

The Gauss Elimination Method (also called Forward Elimination Method) is a procedure used to solve a system of linear equations by:

Converting the system into an augmented matrix

Transforming the matrix into upper triangular form

Solving the equations using back substitution

Steps:

Write the augmented matrix.

Use row operations to make all elements below the main diagonal zero.

Solve the last equation first.

Substitute backward to find other variables.

✔ It gives the solution after back substitution.

3) Gauss–Jordan Method (Definition)

The Gauss–Jordan Method is an extension of Gauss Elimination. It transforms the matrix into Reduced Row Echelon Form (RREF).

Steps:

Convert the system into an augmented matrix.

Use row operations to make:

All elements below and above the pivot zero.

All pivot elements equal to 1.

The matrix becomes an identity matrix on the left side.

✔ The solution is obtained directly without back substitution.