Let A = $\begin{bmatrix} 1 & -3 &2\\ -2 & 5 &-1 \\ 3&-3 &-12 \end{bmatrix}$ and b = $\begin{bmatrix} b_{1}\\ b_{2}\\ b_{3} \end{bmatrix}$.
Determine if the equation Ax = b is consistent for all possible $b_{1}, b_{2}, b_{3}$. If the equation is not consistent for all possible $b_{1}, b_{2}, b_{3}$, give a description of the set of all b for which the equation is consistent.
I know that the solution is that the equation is consistent for all $b_{1}, b_{2}, b_{3}$ satisfying $9b_{1}+6b_{2}+b_{3}=0$ via an answer key but I don't understand how to get there. I found that the reduced row echelon form of the matrix is $\begin{bmatrix} 1 &0 &-7 \\0 &1 &-3 \\0 &0 &0 \end{bmatrix}$, but if anything, doesn't that tell me that the ratio between $b_{1}$ and $b_{3}$ is $-7$?
To extrapolate this to more general problems, since it seems I lack an understanding of systems of equations like these, what does $A$ tell me about the system as a whole? I thought that $A$ told me about the ratio between the coefficients to $x$ that satisfy $Ax=b$, if that makes sense. Why am I wrong, or what am I missing?
The augmented matrix
$$\left[\begin{array}{ccc|c} 1 & -3 &2 & b_1\\ -2 & 5 &-1 & b_2\\ 3&-3 &-12 &b_3\end{array} \right]$$
is row reduced to$$ \left[\begin{array}{ccc|c} 1 & -3 &2 & b_1\\ 0 & -1 &3 &2b_1+ b_2\\ 0&0 &0 &9b_1+6b_2+b_3\end{array} \right]$$
Since the first three elements of the third row are $0$, for consistency you need the last row to be completely $0$ so you have $$9b_1+6b_2+b_3=0$$