What is the METHOD to solve a system of three equations?

When presented with 3 equations, to solve for $a,b$ and $c$ for me becomes a guessing game. If I have just two equations for example $3a + 2b = 5$ and $6a – 7b = 8$, you could easily solve for $a$ and $b$ through elimination or substitution. However, when I get three equations for example $4a + 2b + c = -3$, $a + b + c = -1$ and $a + b + c = -1$, it becomes a guessing game.

I usually start with elimination for a system of three equations but my biggest problem is substituting or elimination to get the same equation hence going around in circles. I then do blind/random substitutions to try and eliminate one variable, but I have no idea what I am doing or if it will work although I do get there in the end. Is this normal? Is there a set of steps or method which will be able to solve 3 for three unknowns without guessing randomly?