solve-for-x-y-z-R-x-2-yz-1-y-2-xz-2-z-2-xy-3-

Question Number 49188 by behi83417@gmail.com last updated on 04/Dec/18

s o l v e f o r x, y, z \in R .

x^{2} + yz = 1

y^{2} + xz = 2

z^{2} + xy = 3

Commented by tanmay.chaudhury50@gmail.com last updated on 05/Dec/18

e x c e l l e n t p r o b l e m \dots s t i l l t o f i n d t h e g o a l o f

x y z \dots

Answered by ajfour last updated on 04/Dec/18

l e t y = p x, z = q x

\Rightarrow x^{2} (1 + p q) = 1

x^{2} (p^{2} + q) = 2

x^{2} (q^{2} + p) = 3

i f p q \neq - 1 t h e n

p^{2} + q = 2 + 2 p q

q^{2} + p = 3 + 3 p q

\Rightarrow q = \frac{p^{2} - 2}{2 p - 1}

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

i f p \neq \frac{1}{2}, t h e n

{(p^{2} - 2)}^{2} + (p - 3) {(2 p - 1)}^{2} = 3 p (2 p - 1) (p^{2} - 2)

\Rightarrow

p^{4} - 4 p^{2} + 4 + 4 p^{3} - 16 p^{2} + 13 p - 3

- 6 p^{4} + 3 p^{3} + 12 p^{2} - 6 p = 0

\Rightarrow 5 p^{4} - 7 p^{3} + 8 p^{2} - 7 p - 1 = 0

\dots . .

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