Lovely congruence

$\begin{cases} 7x + 3y \equiv 10 \text{ (mod 16)} & ...(1) \\ 2x + 5y \equiv 9 \text{ (mod 16)} & ...(2) \end{cases}$

$\begin{cases} (1) \times 5: & 35x + 15y \equiv 50 \text{ (mod 16)} & ...(3) \\ (2) \times 3: & 6x + 15y \equiv 27 \text{ (mod 16)} & ...(4) \end{cases}$

$\begin{aligned} (3)-(4): \quad 29x & \equiv 23 \text{ (mod 16)} \\ 13x & \equiv 7 \text{ (mod 16)} \\ \implies x & \equiv 3 \end{aligned}$

$\begin{aligned} (1): \quad 21 + 3y & \equiv 10 \text{ (mod 16)} \\ 5 + 3y & \equiv 10 \text{ (mod 16)} \\ 3y & \equiv 5 \text{ (mod 16)} \\ \implies y & \equiv 7 \end{aligned}$

$\implies x + y \equiv 3 + 7 \equiv \boxed{10} \text{ (mod 16)}$