Find the value of k for which each of the following system of equations have no solution :

Solution:

GIVEN:

$k x-5 y=2$

$6 x+2 y=7$

To find: To determine for what value of k the system of equation has no solution 

We know that the system of equations

$a_{1} x+b_{1} y=c_{1}$

$a_{2} x+b_{2} y=c_{2}$

For no solution

$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}}$

Here,

$\frac{k}{6}=\frac{-5}{2} \neq \frac{2}{7}$

$\frac{k}{6}=\frac{-5}{2}$

$2 k=-30$

$k=-15$

Hence for $k=-15$ the system of equation have infinitely many solutions.