Gauss Elimination Method part 2

Posted by Unknown on 10:09 PM with No comments
The Gaussian elimination method is used to solve systems of three of more equations. There are two ways to implement this method.

First Method

1.Transform the equations of a system so that in each equation there will be one unknown less than the previous.
sistema
2.Take the equation with the coefficient of x: 1 or −1, as the first equation. If not possible with x, do with y or z, and rearrange the order of the unknowns:
sistema
3. Perform the elimination method with the 1st and 2nd equation to eliminate the term of x in the 2nd equation. Then, in the second equation, place the result of the operation:
E'2 = E2 − 3E1
sistema
4. Perform the same with the 1st and 3rd equation to eliminate the term of x:
E'3 = E3 − 5E1
sistema
sistema
5. Perform the elimination method with the 2nd and 3rd equation:
E''3 = E'3 − 2E'2
sistema
6. So, we obtain another equivalent system:
sistema
7.Solve the system:
z = 1
− y + 4 ·1 = −2        y = 6
x + 6 −1 = 1          x = −4

Second Method

1.Transform the system into a matrix, in which place the coefficients of the variables and the independent terms (separated by a straight line).
matriz matriz

Example:

3x +2y + z = 1 cierre
5x +3y +4z = 2
x + y - z = 1
2.Change the position of row 3 into where row 1 is and shift the other rows down accordingly:
operations
3.Subtract row 2 by three times the value of row 1 and subtract row 3 by five times the value of row 1.
4.Subtract row 3 by twice the value of row 2.
operations
operations
operations
Consistent Independent System
operations

sistema
operations
operations
operations
operations
operations
Consistent Dependent System
operations
operations
operations

sistema
operations
operations
operations
operations
Inconsistent System
Examples:

1sistema
soluciones
operations
soluciones
2sistema
soluciones
operations
soluciones
soluciones
soluciones

3sistema
operations
operations
operations

4sistema
operations
operations
operations
operations
operations
operations
operations

5sistema
operations
operations
operations
operations
operations
operations

6sistema
sistema