If we have three equations in three variables,
$\begin{aligned}
a_1 x + b_1 y + c_1 z &= d_1 \\
a_2 x + b_2 y + c_2 z &= d_2 \\
a_3 x + b_3 y + c_3 z &= d_3
\end{aligned}$
then $\Delta = \begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix}$, $\Delta_1 = \begin{vmatrix} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \end{vmatrix}$, $\Delta_2 = \begin{vmatrix} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \end{vmatrix}$, $\Delta_3 = \begin{vmatrix} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \end{vmatrix}$.
If the set of equations do not have a solution then $\Delta = 0$ and at least one among $\Delta_1$, $\Delta_2$, $\Delta_3$ is not equal to zero.
$\Delta = \begin{vmatrix} 2 & 3 & -1 \\ 1 & 1 & 1 \\ 1 & -1 & \lambda \end{vmatrix} = 0$
$\Rightarrow |\lambda| = 7 \Rightarrow \lambda = \pm 7 \ldots 1$
Case 1: $\lambda = 7$
The system of equations are
Comparing the coefficients of and we get,
For these values of and , coefficients of and the constant values are matching. So the set of equations will have infinitely many solutions.
Case 2:
The system of equations are
Comparing the coefficients of and we get,
For these values of and , coefficients of and the constant values are not matching. So the set of equations will have no solutions.