Gauss Elimination Method part 2

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.

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:

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

4. Perform the same with the 1st and 3rd equation to eliminate the term of x:

E'₃ = E₃ − 5E₁

5. Perform the elimination method with the 2nd and 3rd equation:

E''₃ = E'₃ − 2E'₂

6. So, we obtain another equivalent system:

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).

Example:

3x	+2y	+ z	=	1
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:

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.

Examples:

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5

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