SOLVING LINEAR EQUATIONS USING GAUSSIAN ELIMINATION METHOD

In mathematics, Gaussian elimination method is known as the row reduction algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients.

To reduce the augmented matrix to row - echelon form you should follow the following steps :

Step 1 :

Locate the leftmost column that does not consist entirely of zeros.

Step 2 :

Interchange the top row with another row, if necessary, to bring a nonzero entry to the top of the column found in Step 1.

Step 3 :

If the entry that is now at the top of the column found in Step 1 is b, multiply the first row by 1/b in order to introduce a leading 1.

Step 4 :

Add suitable multiples of the top row to the rows below so that all entries below the leading 1 become zeros.

Step 5 :

Now cover the top row in the matrix and begin again with Step 1 applied to the submatrix that remains. Continue in this way until the entire matrix is in row - echelon form.

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

Example 1 :

2x - 2y + 3z = 2

x + 2y - z = 3

3x - y + 2z = 1

Solution :

The last matrix is in row - echelon form. The corresponding reduced system is :

x + 2y - z = 3 ----(1)

-6y + 5z = -4 ----(2)

-5z = -20 ----(3)

In (3), solve for z.

-5z = -20

Divide both sides by -5.

z = 4

Substitute z = 4 in (2).

-6y + 5(4) = -4

-6y + 20 = -4

Subtract 20 from both sides.

-6y = -24

Divide both sides by -6.

y = 4

Substitute y = 4 and z = 4 in (1).

x + 2(4) - 4 = 3

x + 8 - 4 = 3

x + 4 = 3

x = -1

Therefore the solution of the system is

x = -1, y = 4 and z = 4

Example 2 :

2x + 4y + 6z = 22

3x + 8y + 5z = 27

-x + y + 2z = 2

Solution :

The last matrix is in row - echelon form. The corresponding reduced system is :

x + 2y + 3z = 11 ----(1)

2y - 4z = -6 ----(2)

22z = 44 ----(3)

In (3), solve for z.

22z = 44

Divide both sides by 22.

z = 2

Substitute z = 2 in (2).

2y - 4(2) = -6

2y - 8 = -6

Add 8 to both sides.

2y = 2

Divide both sides by 2.

y = 1

Substitute y = 1 and z = 2 in (1).

x + 2(1) + 3(2) = 11

x + 2 + 6 = 11

x + 8 = 11

Subtract 8 from both sides.

x = 3

Therefore the solution of the system is

x = 3, y = 1 and z = 2

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

SOLVING LINEAR EQUATIONS USING GAUSSIAN ELIMINATION METHOD

Recent Articles

Writing Quadratic Functions in Standard Form

Factoring Quadratic Trinomials

Factoring Trinomials Worksheet