The existence of the unique solution of the system of equations:

Question:

The existence of the unique solution of the system of equations:

$x+y+z=\lambda$

$5 x-y+\mu z=10$

 

$2 x+3 y-z=6$

depends on

(a) $\mu$ only

(b) $\lambda$ only

(c) $\lambda$ and $\mu$ both

 

(d) neither $\lambda$ nor $\mu$

Solution:

(a) $\mu$ only

For a unique solution, $|A| \neq 0$

$\Rightarrow\left|\begin{array}{ccc}1 & 1 & 1 \\ 5 & -1 & \mu \\ 2 & 3 & -1\end{array}\right| \neq 0$

$\Rightarrow 1(1-3 \mu)-1(-5-2 \mu)+1(15+2) \neq 0$

$\Rightarrow 1-3 \mu+5+2 \mu+17 \neq 0$

$\Rightarrow-\mu+23 \neq 0$

$\Rightarrow \mu \neq 23$

So, existence of a unique solution depends only on $\mu$.

Leave a comment

Close
faculty

Click here to get exam-ready with eSaral

For making your preparation journey smoother of JEE, NEET and Class 8 to 10, grab our app now.

Download Now