Multiply each equation by 12:
(1) 4x+6y-2z=48
(2) 3x-9y+6z=18 (⇒x-3y+2z=6)
(3) 6x-8y-3z=-64
(4)=(2)+3(1)=15x+9y=162 (⇒5x+3y=54)
(5)=(2)+2(3)=15x-25y=-110
(6)=(4)-(5)=34y=272, y=8⇒5x+24=54, 5x=30, x=6⇒6-24+2z=6, 2z=24, z=12.
CHECK
Substitute for x, y, z in original equations:
⅓x+½y-⅙z=2+4-2=4✔️
¼x-¾y+½z=3/2-6+6=3/2✔️
½x-⅔y-¼z=3-16/3-3=-16/3✔️
This confirms the solution: (x,y,z)=(6,8,12).