If and when a linear system has exactly three solutions

HINT: The system of linear equations can have only 0, 1 or infinitely many solutions. Hence no calculations are needed.

Using row reduction you can show that $x=7/4$, $y=-25/8$ and $z=-15/8$. You can use even Cramer rule to compute the unique solution using that the matrix $A$ given by $$A=\left(\begin{array}{ccc} 2&-1&3\\ 1&4&-2\\ 3&1&-1\end{array}\right)$$ and $\det(A)=-32\neq 0$ and this implies that the system has unique solution for any vector of independent terms.