The system of linear equations
\[x+y+z=2,\qquad 2x+3y+2z=5,\qquad 2x+3y+(a^2-1)z=a+1\]
Solutions · 1
Official solution
\[\Delta=\begin{vmatrix}1&1&1\\2&3&2\\2&3&a^2-1\end{vmatrix}\xrightarrow{R_3\to R_3-R_2}\begin{vmatrix}1&1&1\\2&3&2\\0&0&a^2-3\end{vmatrix}=a^2-3.\]
When Δ=0 (i.e. ∣a∣=3), the third plane is parallel to the (identical direction) second-minus-first combination, and the constant terms do not match, so the system is inconsistent.