**Toolbox:**

- Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $ \Delta \neq 0$ thesystem is consistent and has a unique solution. Cramer's rule can be used. Case 2 : If $ \Delta = 0$, there are two / ( four) possibilities:
- 2a: If $ \Delta = 0$, and atleast one of the values $ \Delta_ x, \Delta_ y ( or \: \Delta_ z )$ is nonzero, the system is inconsistent and has no solution.
- 2b: If $ \Delta = 0\: and \: \Delta_x=\Delta_y ( = \Delta_z)=0$ and at least one element ( second order minor ) of $ \Delta$ is nonzero, the system is consistent with infinitely many solution. The system reduces to one equation ( two equations)
- 2c: If $ \Delta = 0$ and $ \Delta_x=\Delta_y = \Delta_z=0$ and all their 2 x 2 minors are 0, and $\Delta$ has atleast one nonzero element, then the system is consistent with infinitely many solutions. The system reduces to one equation.
- 2d: If $ \Delta = 0, \Delta_x=\Delta_y = \Delta_z=0$, all 2 x 2 minors of $ \Delta$ are zero but there is at least one 2 x 2 minor of $\Delta_x, \Delta_y \: or \: \Delta_z$ that is nonzero, then the system is inconsistent and has no solution.

Step 1

$ \Delta = \begin{vmatrix} 2 & -1 & 1 \\ 6 & -3 & 3 \\ 4 & -2 & 2 \end{vmatrix} = -\begin{vmatrix} 2 & 1 & 1 \\ 6 & 3 & 3 \\ 4 & 2 & 2 \end{vmatrix} = 0 \: ( \because C_2 \equiv C_3)$

The equations may or may not be consistent.

Step 2

$ \Delta_x = \begin{vmatrix} 2 & -1 & 1 \\ 6 & -3 & 3 \\ 4 & -2 & 2 \end{vmatrix} = 0 ( = \Delta)$

$ \Delta_y = \begin{vmatrix} 2 & 2 & 1 \\ 6 & 6 & 3 \\ 4 & 4 & 2 \end{vmatrix} = 0 ( \because C_1 \equiv C_2)$

$ \Delta_z = \begin{vmatrix} 2 & -1 & 2 \\ 6 & -3 & 6 \\ 4 & -2 & 4 \end{vmatrix} = 0 ( \because C_1 \equiv C_3)$

$ \Delta = \Delta_x =\Delta_y=\Delta_z=0$

All 2nd order minors of $ \Delta, \Delta_x, \Delta_y, \Delta_z$ vanish. $\Delta$ has atleast one nonzero element. The system reduces to one equation. It is consistent with infinitely many solutions. Let $ y = s, \: z = t, \: and\: t \in R$.

Then $ x = \large\frac{2+s-t}{2}$ from the first equation.

$ (x,y,z) =\bigg( \large\frac{2+s-t}{2}, s, t \bigg) \: \: s,t \in R$