The solution should satisfy the appropriate equation.
So, let us investigate each option by substituting $x=5$ & $y=2$
Option $A$, $x+2y=7$.
L.H.S. $=5+2\times 2=9\ne$ R.H.S.
$\therefore$ It is not the appropriate equation.
Option $B$, $5x+2y=7$
L.H.S. $=5\times 5+2\times 2=29\ne$ R.H.S.
$\therefore$ It is not the appropriate equation.
Option $C$, $x+y=7\Rightarrow$ L.H.S. $=5+2=7=$ R.H.S
$\therefore$ It  is the appropriate equation.
Option $D$, $5x+y=7\Rightarrow$ L.H.S. $=5\times 5+2=27\ne$ R.H.S.
$\therefore$ It is not the appropriate equation.

Hence, option $C$ is correct.