Question

If the system of equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = 12 is inconsistent then λ = __________________.

Answer 1

The system of equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = 12 is inconsistent.

$\therefore ∆=\left|\begin{array}{ccc}1& 1& 1\\ 1& 2& 3\\ 1& 2& \lambda \end{array}\right|=0$

$\Rightarrow 1\left(2\lambda -6\right)-1\left(\lambda -3\right)+1\left(2-2\right)=0$

$\Rightarrow 2\lambda -6-\lambda +3=0$

$\Rightarrow \lambda -3=0$

$\Rightarrow \lambda =3$

Also, for λ = 3,

${∆}_y=\left|\begin{array}{ccc}1& 6& 1\\ 1& 10& 3\\ 1& 12& 3\end{array}\right|\ne 0$ and ${∆}_z=\left|\begin{array}{ccc}1& 1& 6\\ 1& 2& 10\\ 1& 2& 12\end{array}\right|\ne 0$

Thus, the value of λ is 3.

If the system of equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + λz = 12 is inconsistent then λ = ___3___.