The values of λ and μ for which the system of linear equations x + y + z = 2 , x + 2   y + 3…

For infinite number of solutions, all determinants $D, D_1, D_2 \text{and} D_3$ must be zero and planes represented by these equations should not be parallel. So, we have,

$D=\left|\begin{array}{lll}1& 1& 1\\ 1& 2& 3\\ 1& 3& \lambda \end{array}\right|=0 \Rightarrow \lambda =5$

$D_1=\left|\begin{array}{lll}2& 1& 1\\ 5& 2& 3\\ \mu & 3& \lambda \end{array}\right|=0$

$D_2=\left|\begin{array}{lll}1& 2& 1\\ 1& 5& 3\\ 1& \mu & \lambda \end{array}\right|=0$

And $D_3=\left|\begin{array}{lll}1& 1& 2\\ 1& 2& 5\\ 1& 3& \mu \end{array}\right|=0 \Rightarrow \mu =8$