<s>[INST] How to solve this system of exponential equations? Solve the following system of equations ($x,y \in \Bbb R$):
$$\begin{cases} 
3^{x+3y-2} + 6\cdot 3^{y^2+4x-2} &=5^{5y-3x} + 2\cdot 3^{y^2-2y+1}\\
 1+2\sqrt{x+y-1} &=3\sqrt[3]{3y-2x}.
\end{cases}$$
I think about it but I still have no solution... :(
Since the second equation, I write $x+y \ge 1$ then $y \ge \dfrac{2}{3}x.$ So $y \ge \dfrac{2}{5}$. 
I rewrite the 1st equation:
\begin{align*}
 3^{x+3y-2} + 6\cdot 3^{y^2+4x-2} &=5^{5y-3x} + 2\cdot 3^{y^2-2y+1}\\
\iff 3^{x+3y-2} + 6\cdot 3^{y^2+4x-2}& \le 5^{5y-3x} -3^{5y-3x}+3^{5y-3x} + 2\cdot 3^{y^2+2y+1}\\
\iff (3^{x+3y-2}-3^{5y-3x})(1+2\cdot 3^{y^2+3x-3y+1})& \le 5^{5y-3x} -3^{5y-3x}\\
\iff (9^{2x-y-1}-1)\underset{>0}{\underbrace{(1+2\cdot 3^{y^2+3x-3y+1}})}&\le \left (\dfrac{5}{3}  \right )^{5y-3x}-1.
\end{align*}
Now, I have trouble.... Can anyone post the roots of this system of exponential equations.
I really appreciate if some one can help me. Thanks!
 [/INST] The curves corresponding to the functions $y(x)$ , computed by numerical calculus, are drawn on the figure below. 
Since they dosn't intersect, the system of equations has no real solution.
I guess that there is a mistake in the wording of the question or in copying the equations.
 
The more $x, y$ are large, the more the curves are distanced from one another :
 
</s>