Question

For what value(s) of $\lambda$, does the pair of linear equations $\lambda x+y={\lambda }^{2}andx+\lambda y=1$ have a unique solution?

Answer 1

The given pair of linear equations is
$\lambda x+y={\lambda }^2$ and $x+\lambda y=1$
For a unique solution,
$\frac{a_1}{a_2}\ne \frac{b1}{b2}$

Here, $a_1=\lambda ,b_1=1$ and $a_2=1,b_2=\lambda$

$\Rightarrow \frac{\lambda }{1}\ne \frac{1}{\lambda }$

$\Rightarrow {\lambda }^2\ne 1\Rightarrow \lambda \ne \pm 1$

So, for all real values of $\lambda$ except $\pm$ 1, the given pair of linear equations will have a unique solution.