Question

How do you solve using gaussian elimination or gauss-jordan elimination, `2x-3y-z=2`, `-x+2y-5z=-13`, `5x-y-z=-5`?

Answer 1

Taken you almost to the end. I will let you finish it off.
Check my working!!!

Explanation:

`color(green)([(2,-3,-1,|,2),(-1,2,-5,|,-13),(5,-1,-1,|,-50)])`

`R_1-:2 color(white)("ddd")R_3-:5`
`color(white)("dddddd")darr`

`color(green)([(1,-3/2,-1/2,|,1),(-1,2,-5,|,-13),(1,-1/5,-1/5,|,-10)])`

`R_1+R_2 color(white)("ddd")R_3+R_2`
`color(white)("dddddd")darr`

`color(green)([(0,1/2,-11/2,|,1),(-1,2,-5,|,-13),(0,9/5,-26/5,|,-10)])`

`R_1xx2 color(white)("ddd")R_3xx5`
`color(white)("dddddd")darr`

`color(green)([(0,1,-11,|,2),(-1,2,-5,|,-13),(0,9,-26,|,-50)])`

`R_2-2R_1 color(white)("ddd")R_3-9R_1`
`color(white)("dddddddd")darr`

`color(green)([(0,1,-11,|,2),(-1,0,-27,|,-17),(0,0,-125,|,-68)])`

`R_2-:(-1)color(white)("ddd")R_3-:(-125)`
`color(white)("dddddddd")darr`

`color(green)([(0,1,-11,|,2),(1,0,27,|,17),(0,0,1,|,+68/125)])`

You should be able to finish this now.

PLEASE MAKE SURE THERE ARE NO MISTAKES. Not checked!