Find when the equations $$\begin{cases}x + y - 2z = 0\\ax + by + cz = 0\\bx + cy + az = d\end{cases}$$ are consistent and solve them completely when they are consistent.
I have tried the conventional way to solve these equations by taking them into the echelon form but I get stuck when it comes to the part where I have to show the consistency of the system. Can someone please guide me as to how I should proceed with the problem?
Cramer's rule is perfectly suited for such a task. We have to solve: $$ \begin{pmatrix} 1 & 1 & -2 \\ a & b & c \\ b & c & a\end{pmatrix}\begin{pmatrix} x \\ y \\ z\end{pmatrix}=\begin{pmatrix}0 \\ 0 \\ d\end{pmatrix} $$ where the determinant of the matrix in the LHS is $D=(a+b+c)(a+c-2b)$.
Assuming that $D\neq 0$ and $d\neq 0$, explicit solutions are given by: $$ (x,y,z) = \frac{d}{D}\cdot\left(-(2b+c),(2a+c),(a-b)\right). $$