I'm having trouble wrapping my head around this problems, and others similar to it. I can typically solve systems of linear equations, but some give me trouble, especially dealing with unknown constants.
I would really love help trying to understand better the methods of solving these types of problems so I can move forward in my studies more effectively. Onto the problem.
Give restrictions on a, b, and c such that the linear system is consistent.
The system is as follows:
\begin{Bmatrix} x - 2y + 4z = a \\ 2x + y - z = b \\ 3x - y + 3z = c \\ \end{Bmatrix}
I know that the solution is "consistent for all a, b, and c such that 0 = c - a - b" but I have never been able to arrive at the solution entirely on my own.
The closest I've gotten on my own, done just a few minutes before the posting of this, was reaching a point where the system is as follows \begin{Bmatrix} x - 2y + 4z = a \\ 5y - 9z = -2a + b \\ -4z = -a - b + c \\ \end{Bmatrix}
Unfortunately I'm not totally sure where to go from here, though I can see the beginnings of the solution in equations three. The steps I've taken to reach this point are as follows:
-2E1 + E2 -> E2
-3E1 + E3 -> E3
-E3 -> E3 (Probably superfluous)
-E2 + E3 -> E3
Thank you in advance for any help.
Somewhere along the line I screwed up some simple math. The row operations I performed are sound, but the I should not have wound up with 4z. After starting from scratch I realized that my math was off.