**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

**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)

After having gone through the stuff given above, we hope that the students would have understood, "Solving Linear Equations Using Gaussian Elimination Method".

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