# 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!

• Where is this monster coming from ? Are you sure about the $5$ in the top equation ? – Yves Daoust Sep 14 '15 at 15:51
• Would a numerical solution suffice? – Rory Daulton Sep 14 '15 at 17:33
• where did this problem come from? it looks pretty interesting!! – Chinny84 Sep 16 '15 at 15:42
• It comes from my professor. Can you tell me this system of exponential equations has $x=?, y=?$ – kimtahe6 Sep 16 '15 at 15:48
• I guess that your professor tricked you. I see no easy way to handle those equations by hand. A graphical calculator able to deal with the implicit equations can help. – Yves Daoust Sep 18 '15 at 14:59

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 :

• I think so, I'll ask my professor again. – kimtahe6 Sep 20 '15 at 15:58
• @ JJacquelin: Can you tell me what software used to draw the graph as above? Can you post software or link of software? Thank you very much. – kimtahe6 Jan 14 '16 at 5:50
• @kimtahe6 : I used an home made software which was written with Borland-DELPHI. en.wikipedia.org/wiki/Delphi_(programming_language). Using it supposes to be familar with the Dephi progamming language. – JJacquelin Jan 14 '16 at 13:50
• Wow! You are very good. I learned program pascal but Delphi...It's the first time I have heard to it. – kimtahe6 Jan 14 '16 at 16:00
• > the system of equations has no real solution. Is there a way to calculate complex solutions? – MCCCS May 30 '17 at 14:18