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

Solution:
Total Equations are `3`

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

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

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

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Select the equations `(1)` and `(2)`, and eliminate the variable `z`.

`2x+3y-z=5`	` xx 1->`		``	`2x`	`+`	`3y`	`-`	`z`	`=`	`5`	``
		+
`3x+2y+z=10`	` xx 1->`		``	`3x`	`+`	`2y`	`+`	`z`	`=`	`10`	``
			``	`5x`	`+`	`5y`			`=`	`15`	` -> (4)`

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Select the equations `(2)` and `(3)`, and eliminate the variable `z`.

`3x+2y+z=10`	` xx 3->`		``	`9x`	`+`	`6y`	`+`	`3z`	`=`	`30`	``
		−
`x-5y+3z=0`	` xx 1->`		``	`x`	`-`	`5y`	`+`	`3z`	`=`	`0`	``
			``	`8x`	`+`	`11y`			`=`	`30`	` -> (5)`

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Select the equations `(4)` and `(5)`, and eliminate the variable `x`.

`5x+5y=15`	` xx 8->`		``	`40x`	`+`	`40y`			`=`	`120`	``
		−
`8x+11y=30`	` xx 5->`		``	`40x`	`+`	`55y`			`=`	`150`	``
					`-`	`15y`			`=`	`-30`	` -> (6)`

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Now use back substitution method
From (6)
`-15y=-30`

`=>y=(-30)/(-15)=2`

From (4)
`5x+5y=15`

`=>5x+5(2)=15`

`=>5x+10=15`

`=>5x=15-10=5`

`=>x=(5)/(5)=1`

From (2)
`3x+2y+z=10`

`=>3(1)+2(2)+1z=10`

`=>1z+7=10`

`=>1z=10-7=3`

Solution using Elimination method.
`x=1,y=2,z=3`

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