System of 2 equations x^y = y^x

System of 2 equations x^y = y^x

System of 2 equations:
$$\begin{tabular}{|l}x^{y} = y^{x}\\x^{3} = y^{2} \end{tabular}$$
Math Tutor

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Re: System of 2 equations x^y = y^x

The system seems to be easy but noone can solve it!
Me too
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Re: System of 2 equations x^y = y^x

$$log_{x} x^{y}=log_{x} y^{x}$$(x>0 from the second equation) $$\Leftrightarrow y=x.log_{x}y$$ (1)
Similarly from the second equation $$log_{x}x^{3}=log_{x}y^{2} \Leftrightarrow log_{x}y=\frac{3}{2}$$ (2).
From (1) and (2) $$y=\frac{3x}{ 2}$$. Then again from the second equation of the system we have $$x^{3}=\frac{9}{4 }x^{2}$$ and it is easy then...

shemet

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Re: System of 2 equations x^y = y^x

Thank you shemet for the solution
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