let a=1/alpha , b=1/beta , c=1/gamma
=>
4a-b+c=12 => c=12+b-4a
2a+2b+3c=1
5a-2b+6c=22
=>
2a+2b+3*(12+b-4a)=1
5a-2b+6*(12+b-4a)=22
=>
2a+2b+36+3b-12a=1
5a-2b+72+6b-24a=22
=>
-10a+5b=-35
-19a+4b=-50
=>
-2a+b=-7 => b=-7+2a
-19a+4b=-50
=>
-19a+4*(-7+2a)=-50
=>
-19a-28+8a=-50
=>
-11a=-22 => a=2
=>
b=-7+2*(2)=-3
=>
c=12+(-3)-4*(2)=1
=>
2=1/alpha => alpha=1/2
-3=1/beta => beta=-1/3
1=1/gamma => gamma=1
check:
4/(1/2)-1/(-1/3)+1=8+3+1=12
2/(1/2)+2/(-1/3)+3/1=4-6+3=1
5/(1/2)-2/(-1/3)+6/1=10+6+6=22