Modular Equation 1

$4x + 5y \equiv 9 \mod 23 \\ \implies 4x + 5y = 23k + 9 \\ \implies 20x + 25y = 115k + 45 \ (1)$ $5x + 11y \equiv 9 \mod 23 \\ \implies 5x + 11y = 23h + 9 \\ \implies 20x + 44y = 92h + 36 \ (2) \\ \\ (1) - (2) \ \implies -19y = 115k - 92h + 9$ $\text{Applying} \mod 23 \\ \implies 4y \mod 23 \equiv 9 \mod 23 \\ (4y - 9) \mod 23 = 0 \\ \implies 4y - 9 = 23a \implies 4y = 23a + 9 \\ \implies y = 8 , a = 1 \\ \implies y = 8 + 23p \ \text{or in terms of mod, } \ y = 8 \mod 23$ $4x + 5y = 23k + 9 \\ \implies 44x + 55y = (23*11)k + 99 \ (3)$ $5x + 11y = 23h + 9 \\ \implies 25x + 55y = (23*5)h + 45 \ (4) \\ \\ (3) - (4) \ \implies 19x = (23)(11k - 5h) + 54$ $\text{Applying} \mod 23 \\ \implies -4x \mod 23 \equiv 54 \mod 23 \equiv 8 \mod 23 \\ \implies x \equiv -2 \mod 23 \equiv 21 \mod 23$ $\text{Since} \ x \mod 23 \equiv 21 \ , \ y \mod 23 \equiv 8 \\ \\ \implies \boxed{(x+y) \mod 23 \equiv 29 \mod 23 \equiv 6}$