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Gauss-Jordan elimination method - HELP - please











































The system of equations
2x−3y−z=0
−x+2y−5z=12
5x−y−z=10
has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.
x=
y=
z=

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Mar 26, 2011
Gauss-Jordan elimination method
by: Staff


The question:

The system of equations

2x−3y−z=0
−x+2y−5z=12
5x−y−z=10

has a unique solution. Find the solution using Gaussin elimination method or Gauss-Jordan elimination method.

x=
y=
z=



The answer:

The system of equations in your problem statement is:

2x - 3y - z = 0
-x + 2y - 5z = 12
5x - y - z = 10

The augmented matrix for this system of equations is:

2 -3 -1 : 0
-1 2 -5 : 12
5 -1 -1 : 10

The row operations used by the Gaussian elimination method are:

2 -3 -1 : 0
-1 2 -5 : 12
5 -1 -1 : 10

Divide row1 by 2

1 -3/2 -1/2 : 0
-1 2 -5 : 12
5 -1 -1 : 10

Add (1 * row1) to row2

1 -3/2 -1/2 : 0
0 1/2 -11/2 : 12
5 -1 -1 : 10

Add (-5 * row1) to row3

1 -3/2 -1/2 : 0
0 1/2 -11/2 : 12
0 13/2 3/2 : 10

Divide row2 by 1/2

1 -3/2 -1/2 : 0
0 1 -11 : 24
0 13/2 3/2 : 10

Add (-13/2 * row2) to row3

1 -3/2 -1/2 : 0
0 1 -11 : 24
0 0 73 : -146

Divide row3 by 73

1 -3/2 -1/2 : 0
0 1 -11 : 24
0 0 1 : -2

Add (11 * row3) to row2

1 -3/2 -1/2 : 0
0 1 0 : 2
0 0 1 : -2

Add (1/2 * row3) to row1

1 -3/2 0 : -1
0 1 0 : 2
0 0 1 : -2

Add (3/2 * row2) to row1

1 0 0 : 2
0 1 0 : 2
0 0 1 : -2


Converting the final matrix back into equation form:

x = 2
y = 2
z = -2

The final answer to your question is:

x = 2
y = 2
z = -2


Check the answer by substituting the numerical values of x, y and z into the original equations:

2x - 3y - z = 0

2*2 – 3*2 - (-2) = 0, correct


-x + 2y - 5z = 12

-2 + 2*2 – 5*(-2) = 12, correct


5x - y - z = 10

5*2 - 2 - (-2) = 10, correct


Since the numerical values of x, y, and z work in all three of the original equations, the solutions are correct.



Thanks for writing.


Staff
www.solving-math-problems.com


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