The set of homoegeneous equations: $tx+\left(t+1\right)y+\left(t-1\right)z=0;$  $\left(t+1\right)x+ty+\left(t+2\right)z=0;$  $\left(t-1\right)x+\left(t+2\right)y+tz=0$  has non-trivial solution for  

Determinant of coefficients $=\left|\begin{array}{c}tt+1t-1\\ t+1tt+2\\ t-1t+2t\end{array}\right|=\left|\begin{array}{c}t1-1\\ t+1-11\\ t-131\end{array}\right|$  $=\left|\begin{array}{c}t1-1\\ 2t+100\\ 2t-140\end{array}\right|=-4\left(2t+1\right)$  For non-trivial solution $t=-\frac{1}{2}.$