I'm teaching myself how to solve systems of equations using matrices and row operations. I get the basic mechanics of it (legal operations, etc.), but it seems like it's kind-of a crapshoot deciding where to begin, and choosing the "wrong" operation to start with can lead to a really difficult problem.
My question is: Are there any rules of thumb for deciding which operations to begin with and how to proceed?
For instance, given the system:
2x - 9y - z = -23 x + 9y - 5z= 2 3x + y + z = 24
What row operation would you begin with, and where would you go from there? I'm not asking you to do it for me, but for insight into your thought process regarding how to proceed.