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X + Y + Z = Equation 1

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X + Y + Z = Equation 2

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X + Y + Z = Equation 3

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HOW TO USE: Simply enter new numbers.

To validate your answer substitute the values of x, y, and z to your original equation 1 , equation 2, and equation 3. The answer must satisfy all the equations or with very small error.

Answer, X = Answer, Y = Answer, Z =

Result of Pivot 1, which means making column 1 (X) follow the pattern below.

Column 1 (X)	Column 2 (Y)	Column 3 (Z)
1	C	C
0	1	C
0	0	1

rule number 1 definition.

Result of Pivot 1, eliminating x and x in column 1.

X + Y + Z = Equation 1

Reason Operation 2. Both sides of equal signs in Equation 1 multiply by 1/3.

3/3 * x	+ 2/3 * y	- 1/3 * z	=	1 * 1/3	Equation 1.1
Simplify to get new result
1 x	2/3 y	- 1/3 z	=	1/3	Equation 1.1
1x + 0.667 y - 0.3333 z = 0.3333

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X + Y + Z = Equation 2. Eliminating x

Reason Operation 3. A multiple of one equation maybe added to another equation. Multiply equation 1.1 by - 2 then add to Equation 2

-2 x	- 4/3 * y	+ 2/3 * z	=	- 2/3	Equation 1.1
2 x	- 2 y	+ 4 z	=	2	Equation 2
0 x	- 10/3 y	+ 14/3 z	=	-8/3	Equation 2.1
0 x	- 10 y	+ 14 z	=	-8	Equation 2.1 * 3 simplify

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X + Y + Z = Equation 3 . Eliminating x

Reason Operation 3. Add equation 1.1 to Equation 3

1 x	+ 2/3 y	- 1/3 z	=	1/3	Equation 1.1
-1 x	+ 0.5 y	- z	=	0	Equation 3
0 x	+ 7/6 y	- 4/3 z	=	1/3	Equation 2.1
0 x	+ 3.5 y	- 4 z	=	1	Equation 3.1 simplify

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Result of Pivot 2, which means making column 2 follow the pattern [ CY, 1, 0 ] where C (0.6667) is the constant of Y.

Result of Pivot 2 eliminating in equation 3 above.
Then converting in equation 2 to one (1) in column 2.

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X + Y + Z = Equation 1

X + Y + Z = Equation 2

Reason Operation 2. Both sides of Equation 2.1 multiply by 1/10.

0 x	+ 10/10 * y	- 14/10 * z	=	8/10	Equation 2.1
0 x	1 y	- 1.4 z	=	0.80	Equation 2.2

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X + Y + Z = Equation 3

Reason Operation 3. Multiply Equation 2.2 by -3.5 and Add equation 3.1

0 x	- 3.5 * y	- 3.5 * (-1.4 z)	=	- 3.5 * 0.8	Equation 2.2
0 x	-3.5 y	+ 4.899 z	=	-2.80	Equation 2.3
0 x	+ 3.5 y	- 4 z	=	1	Equation 3.1
0 x	+ 0 y	+ 0.8999 z	=	-1.80	Equation 2.3 + Eq. 3.1

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ANSWER: Z = .
Using back substitution , substitute Z = in equation 2 above,
you will get Y = .
Finally substitute Z = , Y = in equation 1 below,
you will get X = .

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Important definition to remember about row echelon. The process of reducing augmented matrix in row echelon form is called Gaussian elimination

1. The first nonzero entry in each row is always 1
2. A row with more leading zero entries compare to the previous row must be located below.
3. A row with all zero entries must be below the rows having nonzero entries.

Gaussian elimination will not work properly if one of the above definition is violated.

Operation 1 - The order in which any two equations are written may be interchanged.

Operation 2 - Both sides of the equal sign of the equation may be multiplied by the same nonzero real number.

Operation 3 - A multiple of one equation may be added to another equation.