I am having a hard time finding clear information on this topic. I would like to know wether these systems are consistent or not, and also the number of solutions on them.
Basically, we have five different geometries for plotting three planes. I came up, after checking some sources, with these consistent/not consistent answers. Also, I stated the number of solutions in each of them. I am not sure these are correct.
Since none of the sources seemed to state that clear enough, I would like to check with the community. Here are my answers for these:
Additional question:
Is it possible to guarantee that, if the determinant formed with the coefficients of x, y and z is not equal to zero, we will have only one solution, placing it on the first case? Or does that mean it could be cases #1 or #2?
Thank you.
Yes, they are correct. If the determinant is nonzero, we are in Case #1, since the solution is then given by $$\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{pmatrix}^{-1}\begin{pmatrix}d_1\\d_2\\d_3\end{pmatrix}.$$