SOLVING LINEAR EQUATIONS USING GAUSSIAN ELIMINATION METHOD

Solving Linear Equations Using Gaussian Elimination Method :

Here we are going to see some practice questions on solving linear equations using gaussian elimination method.

Gaussian Elimination Method :

In this method, we transform the augmented matrix of the system of linear equations into row-echelon form and then by back-substitution, we get the solution

Solving Linear Equations Using Gaussian Elimination Method - Practice questions

Question 1 :

Solve the following systems of linear equations by Gaussian elimination method:

(i) 2x − 2y + 3z = 2, x + 2y − z = 3, 3x − y + 2z = 1

Solution :

The equivalent system is written by using the echelon form:

2x - 2y + 3z  =  2 ------(1)

-6y + 5z  =  -4------(2)

-5z  =  -20

z  =  -20/(-5)  =  4

By applying the value of z in (2), we get

-6y + 5(4)  =  -4

-6y + 20  =  -4

-6y  =  -4 - 20

-6y  =  -24

y  =  -24/(-6)  =  4

By applying the value of y and z in (1), we get

2x - 2(4) + 3(4)  =  2

2x - 8 + 12  =  2

2x + 4  =  2

2x  =  2 - 4  =  -2

x  =  -1

Hence the solution is (-1, 4, 4)

(ii) 2x + 4y + 6z = 22, 3x + 8y + 5z = 27, − x + y + 2z = 2

Solution :

2x + 4y + 6z  =  22 ------(1)

2y - 4z  =  -6 ------(2)

22z  =  44

z  =  2

By applying the value of z in (2), we get

2y - 4(2)  =  -6

2y - 8  =  -6

2y  =  -6 + 8

2y  =  2

y  =  1

By applying the values of y and z in (1), we get

2x + 4(1) + 6(2)  =  22

2x + 4 + 12  =  22

2x  =  22 - 16

2x  =  6

x  =  3

Hence the solution is (3, 1, 2)

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