The value of k for which the system of equations x + 2y − 3 = 0

The given system of equations are

$x+2 y-3=0$

$5 x+k y+7=0$

For the equations to have no solutions,

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

$\frac{1}{5}=\frac{2}{k}=\frac{-3}{7}$

If we take

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

$\frac{1}{5}=\frac{2}{k}$

$k=10$

Therefore the value of k is10.

Hence, correct choice is $a$.