For the following system of equation determine the value of k for which the given system of equation has a unique solution.
$2x-3y=1$
$kx+5y=7$

#### Solution

The given system of equations is
$2x-3y-1=0$
$kx+5y-7=0$
It is of the form $a_1x+b_1y+c_1=0$
$a_2x+b_2y+c_1=0$
where, $a_1=2, b_1=-3, c_1=-1$ and $a_2=k, b_2=5, c_2=-7$
For a unique solution, we must have
$\dfrac{a_1}{a_2}\neq \dfrac{b_1}{b_2}$ i.e., $\dfrac{2}{k}\neq \dfrac{-3}{5}\Rightarrow k\neq \dfrac{-10}{3}$
So, the given system of equations is consistent with a unique solution for all values of k other than $-10/3$.