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!
Best Answer
$(3)$ 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 $$4.7151153613231415366265466208961077638519627948872$$