Answer to Question #180110 in Linear Algebra for Oliver

Question

Solve the following question by using Gauss elimination. And Gauss Jordan method.

2x + 7y + z = 14

X + 3y - z = 2

X + 7y + 12z = 45

Answer 1

1. Solution by Gayss-Jordan's method

x + 3y - z = 2

2x + 7y + z = 14

x + 7y + 12z = 45

Subtract the first equation, multiplied by 2, from the second one and multiplied by 1 - from the third.

x + 3y - z = 2

y + 3z = 10

4y + 13z = 43

Subtract the second equation, multiplied by 4, from the third one and multiplied by 3 - from the first.

x - 10z = -28

y + 3z = 10

z = 3

Subtract the third equation, multiplied by 3, from the second one, add the third equation, multiplied by 10, to the first one:

x = 2

y = 1

z = 3

2. Solution by using Gauss elimination.

x + 3y - z = 2

2x + 7y + z = 14

x + 7y + 12z = 45

From the first equation we obtain: x= 2 - 3y + z

Now eleminate this variable in the other equations:

2(2 - 3y + z)+ 7y + z = 14 implies y + 3z = 10

(2 - 3y + z) + 7y + 12z = 45 implies 4y + 13z = 43

Wrom the equation y + 3z = 10 we obtain y=10-3z. Then

4(10-3z) + 13z = 43 implies z=3.

y=10-3z=10-9=1

x= 2 - 3y + z=2-3+3=2

Answer. x = 2, y = 1, z = 3