From the first equation a=1-3c. Substitute this value for a in the other equations:
-5b+c-(1-3c)=-5b+c-1+3c=-5b+4c-1=-3, -5b+4c=-2;
2(1-3c)+3b+c=2-6c+3b+c=2+3b-5c=-3, 3b-5c=-5.
Now we have two equations and two unknowns:
-5b+4c=-2,
3b-5c=-5.
Multiply the first equation by 3 and second by 5:
-15b+12c=-6,
15b-25c=-25.
Now add: -13c=-31, 13c=31, c=31/13.
Since a=1-3c, a=1-3(31/13)=1-93/13=-80/13.
(3b=5c-5=5(31/13)-5=155/13-5=90/13, b=30/13.)
Solution is a=-80/13.