Linear Systems &Gaussian

Elimination

Aju George SCET KODAKARA

Contents

o Linear Equationo Back substitutiono Gaussian Elimination methodo Solving of linear equation using

Gaussian Elimination methodo Some examples o Bibliography

What is linear equation?

An equation contains variables that gives a straight line when plotted on a graph.

Linear equations can have one or more variables. An example of a linear equation with three variables, x, y, and z, is given by: ax + by + cz + d = 0, where a, b, c, and d are constants and a, b, and c are non-zero

What is back substitution method?

• The process of solving a linear system of equations that has been transformed into row-echelon form or reduced row-echelon form. The last equation is solved first, then the next-to-last, etc

• For Example ;X-2y+z=4Y+6z=-1

Z=2

Gaussian elimination method• Gaussian elimination (also known as

row reduction) is an algorithm for solving systems of linear equations.

• To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible

• Using these operations, a matrix can always be transformed into an upper triangular matrix, and in fact one that is in row echelon form.

For Example;

How to solve?Example 1

x + 5y= 7-----(1)−2x − 7y = −5.-----(2)

Step 1

572

751

Make the Equation into matrix form

Ie, x + 5y= 7-----(1)−2x − 7y = −5.-----(2)

becomes

Step 2

572

751

930751

Find suitable elementary transformation method to form a upper triangular matrix

In this problem “Add twice Row 1 to Row 2”

930751

310751

Multiply Row 2 by 1/3.

This matrix gives Y=3

By back substitution ,We know that y=3From equation (1)

x + 5y= 7X+5*3=7X=7-15

X=-8

Some problems • Use Gaussian elimination to solve the

system of linear equations2x2 + x3 = −8

x1 − 2x2 − 3x3 = 0−x1 + x2 + 2x3 = 3

• Use Gaussian elimination to solve the system of linear equations

x1 − 2x2 − 6x3 = 122x1 + 4x2 + 12x3 = −17x1 − 4x2 − 12x3 = 22..

Bibilography• https

://en.wikipedia.org/wiki/Gaussian_elimination

• https ://math.dartmouth.edu/archive/m23s06/public_html/handouts/row_reduction_examples.pdf

• http ://www.purplemath.com/modules/systlin6.html