Solution

The pair of linear equations
$$2x+5y=k$$
and $$ kx+15y=18$$
Here,$$ a_{1}=2,b_{1}=5,c_{1}=-k$$
and $$ a_{2}=k,b_{2}=15,c_{2}=-18$$

The system of equation has infinitely many solutions if
$$\displaystyle \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$$

$$\Rightarrow \displaystyle \frac{2}{k}=\frac{5}{15}=\frac{-k}{-18}$$

$$\Rightarrow \displaystyle \frac{2}{k}=\frac{1}{3} =\frac{k}{18} $$

Taking,
$$\displaystyle \frac{2}{k}=\frac{1}{3} \Rightarrow k=6$$

Taking,
$$\displaystyle \frac{1}{3} =\frac{k}{18} \Rightarrow k=6$$

So, answer is $$k=6$$