Elementary Ways to Solve System of Exponential Equation

Hint:

$$(3^x)^2+(12\cdot3^y)3^x-405=0$$

The discriminant is $$(12\cdot3^y)^2+4\cdot405=16\cdot3^{2y+2}+3^4\cdot20=4\cdot3^2(4\cdot9^y+45)$$

For rational $3^x,$ we need $$(2\cdot3^y)^2+45$$ to be perfect square $=d^2, d\ge0$(say)

$$\implies45=d^2-(2\cdot3^y)^2=(d+2\cdot3^y)(d-2\cdot3^y)\le(d+2\cdot3^y)^2$$

$$\implies d+2\cdot3^y\ge\sqrt{45}>6$$

Again, $d+2\cdot3^y$ must divide $45,$ hence can be one of $$\{9,15,45\}$$

From here we can find $3^y$ and $d$ and hence $3^x$