Question

Find the value $k$ for which each of the following systems of equations has a unique solution:

$4x-5y=k,2x-3y=12.$

Answer 1

The given system may be written as
$4x-5y-k=0$
$2x-3y-12=0$
The given system of equation is of the form
$a_{1}x+b_{1}y+c_1=0,a_{2}x+b_{2}y+c_2=0$
Where, $a_1=4,b_1=-5,c_1=-k$
$a_2=2,b_2=-3,c_2=-12$
For unique solution, we have
$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$
$\frac{4}{2}\ne \frac{-5}{-3}$
Since, $k$ value is independent of above condition.
So, $k$ can have any real values.
Therefore, the given system will have a unique solution for all real values of $k$.