Question

3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1

Answer 1

Given: 3x − y + 2z = 3
2x + y + 3z = 5
x − 2y − z = 1

$$\begin{gathered} D=\left|\begin{array}{ccc}3& -1& 2\\ 2& 1& 3\\ 1& -2& -1\end{array}\right| \\ =3\left(-1+6\right)+1\left(-2-3\right)+2\left(-4-1\right) \\ =0 \\ {D}_{1=}\left|\begin{array}{ccc}3& -1& 2\\ 5& 1& 3\\ 1& -2& -1\end{array}\right| \\ =3\left(-1+6\right)+1\left(-5-3\right)+2\left(-10-1\right) \\ =-15 \\ {D}_2=\left|\begin{array}{ccc}3& 3& 2\\ 2& 5& 3\\ 1& 1& -1\end{array}\right| \\ =3\left(-5-3\right)-3\left(-2-3\right)+2\left(2-5\right) \\ =-15 \\ {D}_3=\left|\begin{array}{ccc}3& -1& 3\\ 2& 1& 5\\ 1& -2& 1\end{array}\right| \\ =3\left(1+10\right)+1\left(2-5\right)+3\left(-4-1\right) \\ =-15 \end{gathered}$$

Here, D is zero, but D₁, D₂ and D₃​ are non-zero. Thus, the system of linear equations is inconsistent.