solving-the-following-system-of-equations-3x-y-x-3y-x-2-3y-z-y-3z-y-2-3z-x-z-3x-z-2-

Question Number 119921 by bramlexs22 last updated on 28/Oct/20

s o l v i n g t h e f o l l o w i n g s y s t e m o f e q u a t i o n s

{\begin{cases} \frac{3 x - y}{x - 3 y} = x^{2} \\ \frac{3 y - z}{y - 3 z} = y^{2} \\ \frac{3 z - x}{z - 3 x} = z^{2} \end{cases}

Answered by mindispower last updated on 28/Oct/20

\Rightarrow x = \frac{z^{3} - 3 z}{- 3 z^{2} + 1}

t g (Σ a_{k}) = \frac{Σ {(- 1)}^{k} S_{2 k + 1}}{Σ {(- 1)}^{k} S_{2 k}}, W i t h e S_{0} = 1

t g (a + b + c) = \frac{t g (a) + t g (b) + t g (c) - t g (a) t g (b) t g (c)}{1 - t g (a) t g (b) - t g (c) t g (b) - t g (a) t g (c)}

a = b = c

t g (3 c) = \frac{3 t g (c) - {t g}^{3} (c)}{1 - 3 {t g}^{2} (c)},

l e t z = t g (c), y = t g (a), x = t g (b)

\Leftrightarrow t g (a) = \frac{3 t g (b) - {t g}^{3} (b)}{1 - 3 {t g}^{2} (b)} = t g (3 b) \dots E q u a t i o n 1

\Leftrightarrow t g (c) = \frac{3 t g (a) - {t g}^{3} (a)}{1 - 3 {t g}^{2} (a)} = t g (3 a) \dots E (2)

\Leftrightarrow t g (b) = \frac{3 t g (c) - {t g}^{3} (c)}{1 - 3 {t g}^{2} (c)} = t g (3 c)

t g (a) = t g (3 b) \Rightarrow a = 3 b, a = k π + 3 b

t g (c) = t g (3 a) \Rightarrow c = s π + 3 a

t g (b) = t g (3 c) \Rightarrow b = 3 c + d π = 3 (s π + 3 a) + d π

b = (d + 3 s) π + 9 k π + 27 b

b = \frac{- k π}{3} - (\frac{d + 3 s}{27}) π, d, s, k \in Z

f i n d b, \Rightarrow a \Rightarrow c, t g (a) = y, t g (c) = z, t g (b) = x

Answered by MJS_new last updated on 28/Oct/20

one solution is x = y = z = \pm i

for the other solutions let

y = p x \land z = q x

\Rightarrow

{\begin{cases} \frac{p - 3}{3 p - 1} = x^{2} \\ \frac{3 p - q}{p - 3 q} = p^{2} x^{2} \\ \frac{3 q - 1}{q - 3} = q^{2} x^{2} \end{cases}

\Rightarrow

\frac{p - 3}{3 p - 1} = \frac{3 p - q}{p^{2} (p - 3 q)} = \frac{3 q - 1}{q^{2} (q - 3)}

we can eliminate one of the two unknown

and get a polynome of degree 13 with one

solution = 1 . the other 12 are \in R but I only

can approximate

q = \frac{p (p^{3} - 3 p^{2} - 9 p + 3)}{3 p^{3} - 9 p^{2} - 3 p + 1}

(p - 1) ((p^{12} + 1) - 20 (p^{11} + p) + 34 (p^{10} + p^{2}) + 700 (p^{9} + p^{3}) - 2705 (p^{8} + p^{4}) - 680 (p^{7} + p^{5}) + 9436 p^{6}) = 0

p_{1} = 1

p_{2} \approx - 5.87781

p_{3} \approx - 1.63530

p_{4} \approx - . 611508

p_{5} \approx - . 170131

p_{6} \approx . 0637272

p_{7} \approx . 199046

p_{8} \approx . 278216

p_{9} \approx . 320163

p_{10} \approx 3.12340

p_{11} \approx 3.59433

p_{12} \approx 5.02397

p_{13} \approx 15.6919

please do the rest for yourself

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