Question:
The set of values of $k$ for which the system of equations $x+y+z=2,2 x+y-z=3,3 x+2 y+k z=4$ has a unique solution, is__________
Solution:
The system of equations $x+y+z=2,2 x+y-z=3$ and $3 x+2 y+k z=4$ has a unique solution.
$\therefore \Delta=\left|\begin{array}{ccc}1 & 1 & 1 \\ 2 & 1 & -1 \\ 3 & 2 & k\end{array}\right| \neq 0$
$\Rightarrow 1(k+2)-1(2 k+3)+1(4-3) \neq 0$
$\Rightarrow k+2-2 k-3+1 \neq 0$
$\Rightarrow k \neq 0$
Thus, the set of values of $k$ for which the given system of equations has a unique solution is $R-\{0\}$.
The set of values of k for which the system of equations x + y + z = 2, 2x + y − z = 3, 3x + 2y + kz = 4 has a unique solution, is __R − {0}__.