<s>[INST] Simultaneous equations involving indices Please can someone help me solve this. I saw it in a text but I have tried to solve all to no avail.
$$3^x + 9^{2y} = 27\\2^x + 4^{-y} = \frac18 $$
Find 2$x$ + 3$y$
 [/INST] \begin{align}
   3^x + 9^{2y} &= 27\\
   3^{4y} &= 3^3 - 3^x \\
   3^{4y} &< 3^3 \\
   y &< 0.75 \\
\hline
   2^x + 4^{-y} &= \frac18 \\
   2^{-2y} &= 2^{-3} - 2^x \\
   2^{-2y} &< 2^{-3} \\
   -2y &< -3 \\
   y &> 1.5
\end{align}
Since we can't have both $y < 0.75$ and $y > 1.5$, there is no solution.

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