Question

Find the value of $k$ for which following system of equations has a unique solution:
$4x+ky+8=0$
$2x+2y+2=0$

Answer 1

For the system to have unique solution, lines must intersect each other.

i.e., If the lines $a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$ have the unique solution,

Then $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$

Given equations are,

$4x+ky+8=0$ and $2x+2y+2=0$ has a unique solution.

$\Rightarrow \frac{4}{2}\ne \frac{k}{2}$ [Since, $\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$]

$\Rightarrow 2k\ne 8$

$\Rightarrow k\ne \frac{8}{2}$

$\Rightarrow k\ne 4$

i.e., The value of $k$ is all the real numbers except $4$.