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Gaussin elimination method or Gauss-Jordan elimination method.HELP!











































The system of equations
x+2y−z=2
x+z=0
2x−y−z=−9
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
Gaussian elimination method
by: Staff


The question:

The system of equations

x + 2y - z = 2
x + z = 0
2x - y – z = -9

has a unique solution.

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

x=
y=
z=



The answer:

The Gaussian elimination method USES ROW OPERATIONS on a matrix to (1) reduce an augmented matrix to Row Echelon Form, then (2) continues on to reduce the Row Echelon Form to “reduced” Row Echelon Form.

This is an example of an augmented matrix in Row Echelon Form:

1 3 7 : k₁
0 5 2 : k₂
0 0 3 : k₃

This is an example of an augmented matrix in “reduced” Row Echelon Form:

1 0 0 : k₄
0 1 0 : k₅
0 0 1 : k₆

This is also an example of an augmented matrix in “reduced” Row Echelon Form:

1 8 0 : k₄
0 1 1 : k₅
0 0 1 : k₆

The system of equations in your problem statement is:

x + 2y - z = 2
x + z = 0
2x - y – z = -9

The augmented matrix for this system of equations is:

1 2 -1 : 2
1 0 1 : 0
2 -1 -1 : -9

The row operations used by the Gaussian elimination method are:

1 2 -1 : 2
1 0 1 : 0
2 -1 -1 : -9

Add (-1 * row1) to row2

1 2 -1 : 2
0 -2 2 : -2
2 -1 -1 : -9

Add (-2 * row1) to row3

1 2 -1 : 2
0 -2 2 : -2
0 -5 1 : -13

Divide row2 by -2

1 2 -1 : 2
0 1 -1 : 1
0 -5 1 : -13

Add (5 * row2) to row3

1 2 -1 : 2
0 1 -1 : 1
0 0 -4 : -8

Divide row3 by -4

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

Add (1 * row3) to row2

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

Add (1 * row3) to row1

1 2 0 : 4
0 1 0 : 3
0 0 1 : 2

Add (-2 * row2) to row1

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

Converting the final matrix back into equation form:

x = -2
y = 3
z = 2

the final answer to your question is:

x = -2
y = 3
z = 2


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

x + 2y - z = 2

-2 + 2*3 – 2 = 2, correct


x + z = 0

-2 + 2 = 0, correct


2x - y – z = -9

2*(-2) - 3 – 2 = -9, 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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