Question

The existence of the unique solution of the system of equations:
x + y + z = λ
5x − y + µz = 10
2x + 3y − z = 6
depends on
(a) µ only
(b) λ only
(c) λ and µ both
(d) neither λ nor µ

Answer 1

(a) µ only
$$\begin{gathered} \text{For a unique solution,}\left|A\right|\text{≠ 0} \\ \Rightarrow \left|\begin{array}{ccc}1& 1& 1\\ 5& -1& \mu \\ 2& 3& -1\end{array}\right|\ne 0 \\ \Rightarrow 1\left(1-3\mu \right)-1\left(-5-2\mu \right)+1\left(15+2\right)\ne 0 \\ \Rightarrow 1-3\mu +5+2\mu +17\ne 0 \\ \Rightarrow -\mu +23\ne 0 \\ \Rightarrow \mu \ne 23 \\ \text{So, existence of a unique solution depends only on}\mathit{\text{μ.}} \end{gathered}$$