Algebra 2 - solving a system of three equations with three variables.. !

i've attached an image here. i've been trying to solve this problem for over 30 minutes and neither the class or textbook makes sense to me. i am looking for an explanation. you in advance.

Ya need to eliminate 1 variable from 2 equations then eliminate one variable from those 2

ok

we see the last equation already has y eliminated

so we need to eliminate y from one of the other equation

ok we got
x+3y-z=2 and
4x+2y+5z=1
let's elimiate from top equation

multiply top equation by 2 and 2nd by -3
2x+6y-2z=4
-12x-6y-15z=-3
add them

2x+6y-2z=4
-12x-6y-15z=-3 +
-10x+0y-17z=1

-10x-17z=1

use that equation and the last one

3x+z=12 and
-10x-17z=1

eliminate z in first equation
multiply top eqquation by 17
51x+17z=204

add to other equation
51x+17z=204
-10x-17z=1 +
41x+0z=205

41x=205
divide both sides by 41
x=5

sub back in other equation

3x+z=12
3(5)+z=12
15+z=12
z=-3

sub back again

x+3y-z=2
5+3y-(-3)=2
5+3y+3=2
8+3y=2
3y=-6
y=-2

x=5
y=-2
z=-3

(x,y,z)
(5,-2,-3)

answer is 2nd option