Question

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

Answer 1

The given system of equations is
$2x-3y-1=0$
$kx+5y-7=0$

It is of the form

$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_1=0$

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

$\frac{a_1}{a_2}\ne \frac{b_1}{b_2}$ i.e., $\frac{2}{k}\ne \frac{-3}{5}\Rightarrow k\ne \frac{-10}{3}$

So, the given system of equations is consistent with a unique solution for all values of k other than $-10/3$.