solve-simultaneously-x-3-y-3-35-x-4-y-4-97-

Question Number 17985 by tawa tawa last updated on 13/Jul/17

solve simultaneously .

x^{3} + y^{3} = 35

x^{4} + y^{4} = 97

Answered by mrW1 last updated on 13/Jul/17

let u = x + y and xy = v

(x + y) (x^{2} + y^{2} + 2 xy - 3 xy) = 35

(x + y) [{(x + y)}^{2} - 3 xy] = 35

u (u^{2} - 3 v) = 35

v = \frac{u^{2} - \frac{35}{u}}{3} = \frac{u^{3} - 35}{3 u}

{(x^{2})}^{2} + {(y^{2})}^{2} + {2 x}^{2} y^{2} - {2 x}^{2} y^{2} = 97

{(x^{2} + y^{2})}^{2} - {2 x}^{2} y^{2} = 97

{(x^{2} + y^{2} + 2 xy - 2 xy)}^{2} - {2 x}^{2} y^{2} = 97

{({(x + y)}^{2} - 2 xy)}^{2} - {2 x}^{2} y^{2} = 97

{(u^{2} - 2 v)}^{2} - {2 v}^{2} = 97

{(u^{2} - 2 \frac{u^{3} - 35}{3 u})}^{2} - 2 {(\frac{u^{3} - 35}{3 u})}^{2} = 97

{(u^{3} + 70)}^{2} - 2 {(u^{3} - 35)}^{2} = {873 u}^{2}

u^{6} + {140 u}^{3} + 4900 - {2 u}^{6} + {140 u}^{3} - 2450 = {873 u}^{2}

u^{6} - {280 u}^{3} + {873 u}^{2} - 2450 = 0

\Rightarrow u = - 1.391 or 5

\Rightarrow v = 9.0322 or 6

x + y = u

xy = v

x^{2} + y^{2} - 2 xy + 4 xy = u^{2}

{(x - y)}^{2} = u^{2} - 4 v

x - y = \pm \sqrt{u^{2} - 4 v}

\Rightarrow x = \frac{u \pm \sqrt{u^{2} - 4 v}}{2}

\Rightarrow y = \frac{u \mp \sqrt{u^{2} - 4 v}}{2}

with u = - 1.391 and v = 9.0322

there is no real solution since

u^{2} - 4 v < 0

with u = 5 and v = 6

\Rightarrow x + y = 5

\Rightarrow x - y = \pm \sqrt{25 - 24} = \pm 1

\Rightarrow x = \frac{5 \pm 1}{2} = 3, 2

\Rightarrow y = \frac{5 \mp 1}{2} = 2, 3

\Rightarrow (x, y) = (3, 2) or (2, 3)

Commented by tawa tawa last updated on 13/Jul/17

God bless you sir . i really appreciate .