System of exponential equations

Let $a = 5^x$ and $b = 3^y$.

1. $a - b = 13438 \implies b = a - 13438$
2. $\frac{a}{5} + 3b = 9686 \implies a + 15b = 48430$

Substitute $b$ into the second equation: $a + 15(a - 13438) = 48430$ $16a - 201570 = 48430$ $16a = 250000 \implies a = 15625 = 5^6 \implies x = 6$ $b = 15625 - 13438 = 2187 = 3^7 \implies y = 7$

Thus $x + y = 6 + 7 = 13$.