Question

Find the value(s) of $k$ for which the pair of linear equations $kx+y=k^2$ and $x+ky=1$ have infinitely many solutions.

Answer 1

Pair of linear equations $a,x+b,y=c_1,a_{2}x+b_{2}y=c_2$

has infinitely many solutions if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$

Given : $kx+y=k^2$ and $x+ky=1$ has infinitely many solutions

$\Rightarrow \frac{k}{1}=\frac{1}{k}=\frac{k^2}{1}$ ...(i)

Compassing $\frac{k}{1}=\frac{1}{k}$, we get $K^2=1$

$\Rightarrow k=\pm 1$

But $k\ne -1$ because it does not satisfy (i)

$\therefore k=1$