Solve Equations 2x+3y-z=5,3x+2y+z=10,x-5y+3z=0 using Gauss Elimination Back Substitution method

Solution:
Total Equations are `3`

`2x+3y-z=5 -> (1)`

`3x+2y+z=10 -> (2)`

`x-5y+3z=0 -> (3)`

Converting given equations into matrix form

	`2`	`3`	`-1`		`5`
	`3`	`2`	`1`		`10`
	`1`	`-5`	`3`		`0`

`R_2 larr R_2-1.5xx R_1`

=	`2` `3` `-1` `5` `0` `-2.5` `2.5` `2.5` `1` `-5` `3` `0`

`R_3 larr R_3-0.5xx R_1`

=	`2` `3` `-1` `5` `0` `-2.5` `2.5` `2.5` `0` `-6.5` `3.5` `-2.5`

`R_3 larr R_3-2.6xx R_2`

=	`2` `3` `-1` `5` `0` `-2.5` `2.5` `2.5` `0` `0` `-3` `-9`

`i.e.`

`2x+3y-z=5 ->(1)`

`-2.5y+2.5z=2.5 ->(2)`

`-3z=-9 ->(3)`

Now use back substitution method
From (3)
`-3z=-9`

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

From (2)
`-2.5y+2.5z=2.5`

`=>-2.5y+2.5(3)=2.5`

`=>-2.5y+7.5=2.5`

`=>-2.5y=2.5-7.5=-5`

`=>y=-5xx-2/5=2`

From (1)
`2x+3y-z=5`

`=>2x+3(2)-1(3)=5`

`=>2x+3=5`

`=>2x=5-3=2`

`=>x=(2)/(2)=1`

Solution using back substitution method.
`x=1,y=2 and z=3`

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