Question

Find the value of ${}^{\prime }{\lambda }^{\prime }$ for which the system
$3x+\lambda y+1=0$
$2x+y-9=0$ will have
(i) unique solution
(ii) no solution

Answer 1

The given system of equations are
$3x+\lambda y+1=0$ .......(i)
$2x+y-9=0$ .....(ii)
Here we have
$a_1=3,b_1=\lambda$ and $c_1=1$
$a{a}_2=2,b_2=1$ and $c_2=-9$
(i) Condition for unique solution
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
$\Rightarrow \frac{3}{2}\ne \frac{\lambda }{1}\Rightarrow \lambda \ne \frac{3}{2}$
Hence, for $\lambda \ne \frac{3}{2},$ the system has a unique solution.
(ii) Condition for no solution
$\frac{a_1}{a_2}=\frac{b_1}{b_2}\ne \frac{c_1}{c_2}$
$\Rightarrow \frac{3}{2}=\frac{\lambda }{1}\ne \frac{1}{-9}$
$\Rightarrow \lambda =\frac{3}{2}$
Hence, for $\lambda =\frac{3}{2},$ the system has no solution.