Is there a way of doing these?
That depends.
It's a good idea to ask for specific examples rather than really general questions. It's hard to answer statements like the one you wrote here.
The answer to your question depends on if you mean by hand vs. computer, only basic algebra versus logarithms, etc.
$\displaystyle e^x+3e^y=3$
$\displaystyle e^{2x}-9e^{2y}=6$
$\displaystyle (e^x)^2-(3e^y)^2=6$
$\displaystyle (e^x+3e^y)(e^x-3e^y)=6$
$\displaystyle 3(e^x-3e^y)=6$
$\displaystyle e^x-3e^y=2$
$\displaystyle 2e^x=5$
$\displaystyle e^x=\frac{5}{2}$
$\displaystyle x=ln (\frac{5}{2})$
$\displaystyle 6e^y=1$
$\displaystyle e^y=\frac{1}{6}$
$\displaystyle y=ln(\frac{1}{6})=-ln 6$