a - 2b + c = -6 (1)
a = -5c - 12 (2)
-a + 6b + 4c = 3 (3)
Putting (2) into (1) & (3) gives
(-5c -12) - 2b + c = -6 (5)
-(-5c - 12) + 6b +4c =3 (6)
Neatening it up gives
-2b - 4c = 6 (7)
6b + 9c = -9 (8)
Using simulataenous equations, we want to cancel out either b or c. b in this case is easier. Multiplying (7) makes that possible
(7) * 3
gives
-6b - 12c = 18 (9)
Adding (8) & (9) gives
-3c = 9
c = -3
Now we know c we can use that to calculate b in one of the previous equations.
Using (7)
-2b -4(-3) = 6
-2b = -6
b = 3
Now, b & c are known, we can use these in any equation with a to get the final unknown.
Using (1)
a - 2(3) + (-3) = -6
a = 3
Plugging a,b&c back into each (1) (2) & (3) to check you get the right answer is also a good sanity check that you've not made any mistakes in your previous working!