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I need to analytically solve this system of non-linear equations: \begin{cases} x + (y-1)^2 + 2z^2 = 10 \\ x^2 + y^2 + 3z = 28 \\ 2x + y + z = 12 \end{cases} I solve this system numerically with Gradient Descent method and with different initial approximations obtained the following solutions: \begin{array}{c|lll} \text{1} & x = 4 & y = 3 & z = 1 \\ \hline \text{2} & x \approx 4.75151 & y \approx 0.94466 & z \approx 1.62509 \end{array} I tested the solutions with different software, such as Wolfram and Python libraries, and they gave about the same result.

But no matter how I try to solve a seemingly simple system analytically, I can't do it, because the solutions are too complicated. Perhaps I am doing something wrong, and I would be glad to get even if not a solution, at least tips on how to find it.

Thanks!

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  • $\begingroup$ @gpmath, thank you for answer! This solution seemed strange to me and I decided to double-check the system. I made a small mistake by forgetting the degree in first equation, sorry. $\endgroup$
    – Chonk
    Apr 30, 2023 at 13:09
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    $\begingroup$ Your solution 2 is wrong, correct is $\{x\to 4.715115361323142,y\to 0.9446809049263751,z\to 1.625088372427342\}$ $\endgroup$
    – gpmath
    Apr 30, 2023 at 13:12
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    $\begingroup$ Exact form for $x\approx 4.715115361323142$: wolframalpha.com/… $\endgroup$
    – Robert Z
    Apr 30, 2023 at 13:19
  • $\begingroup$ @gpmath Yes, y has accuracy problems, thanks for the note, but this is more about the my implementation of the numerical method of solving the system, not about an analytical solution $\endgroup$
    – Chonk
    Apr 30, 2023 at 13:23
  • $\begingroup$ @Chonk The exact analytic solution is given in my link above. It is the root of cubic polynomial with integer coefficients $\endgroup$
    – Robert Z
    Apr 30, 2023 at 13:37

1 Answer 1

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$(3)$ gives $z=-2 x-y+12$

$3 \times (2)-(1)$ gives $y=\frac{-5 x^2+77 x-255}{8 x-41}$

Plug in $(1)$ to obtain $$\frac{3 (x-4) \left(89 x^3-1334 x^2+6809 x-11777\right)}{(8 x-41)^2}=0$$

Solbing the cubic equation using the hyperbolic solution for only one real root gives $$x=\frac{2}{267} \left(667-\sqrt{38447} \sinh \left(\frac{1}{3} \sinh^{-1}\left(\frac{9080939}{76894 \sqrt{38447}}\right)\right)\right)$$ which is $$4.7151153613231415366265466208961077638519627948872$$

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  • $\begingroup$ Thank you very much, it was very helpful in finding an analytical solution! $\endgroup$
    – Chonk
    May 1, 2023 at 11:38

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