Solve the following system of equations without using logarithms:

$$
y^{x}=64, \quad y^{\frac{x+1}{x-1}}=16
$$
Our equations can still be written as:

$$
y=2^{\frac{6}{x}}, y=2^{4 \cdot \frac{x-1}{x+1}}
$$

and thus

$$
\frac{6}{x}=\frac{4 x-4}{x+1}
$$

or

$$
2 x^{2}-5 x-3=0
$$

from which

$$
x_{1}=3, \quad x_{2}=-\frac{1}{2}, \quad \text { therefore } \quad y_{1}=4, \quad y_{2}=\frac{1}{4096}
$$

(Zoltán Reich, Zenta.)

The problem was also solved by: L. Bánó, P. Ehrenstein, P. Füstös, Z. Harsányi, P. Heimlich, Gy. Jánosy, J. Kiss, E. Makó, J. Martini, S. Pichler, A. Róth, E. Sárközy, Gy. Schlesinger, Gy. Schwarz, Gy. Schuster, A. Stagl, J. Szabó, I. Szécsi, B. Tóth, Gy. Weber.