 Solution

The coefficient matrix has determinantA square linear system has a unique solution for every right-hand side precisely when its coefficient matrix is an invertible matrix. Here a unique solution exists exactly when $p\ne 1$, for every real $t$.

For an explicit Gaussian elimination, subtract twice the second equation from the third, then use the second equation, obtaining $x=2z+2t-t^2$ and $y=(t^2-t)/2-3z$. The first equation determinesand the preceding formulas determine $x,y$. When $p=1$ this last coefficient vanishes; the next parts determine compatibility rather than dividing by zero.