Solve Equations x+y+z=3,2x-y-z=3,x-y+z=9 using Gauss Elimination Back Substitution method

Solution:
Total Equations are `3`

`x+y+z=3 -> (1)`

`2x-y-z=3 -> (2)`

`x-y+z=9 -> (3)`

Converting given equations into matrix form

	`1`	`1`	`1`		`3`
	`2`	`-1`	`-1`		`3`
	`1`	`-1`	`1`		`9`

`R_2 larr R_2-2xx R_1`

=	`1` `1` `1` `3` `0` `-3` `-3` `-3` `1` `-1` `1` `9`

`R_3 larr R_3- R_1`

=	`1` `1` `1` `3` `0` `-3` `-3` `-3` `0` `-2` `0` `6`

`R_3 larr R_3-0.6667xx R_2`

=	`1` `1` `1` `3` `0` `-3` `-3` `-3` `0` `0` `2` `8`

`i.e.`

`x+y+z=3 ->(1)`

`-3y-3z=-3 ->(2)`

`2z=8 ->(3)`

Now use back substitution method
From (3)
`2z=8`

`=>z=(8)/(2)=4`

From (2)
`-3y-3z=-3`

`=>-3y-3(4)=-3`

`=>-3y-12=-3`

`=>-3y=-3+12=9`

`=>y=(9)/(-3)=-3`

From (1)
`x+y+z=3`

`=>1x+1(-3)+1(4)=3`

`=>1x+1=3`

`=>1x=3-1=2`

Solution using back substitution method.
`x=2,y=-3 and z=4`

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