Solve-simultaneously-x-2-y-2-61-equation-i-x-3-y-3-91-equation-ii-

Question Number 16600 by tawa tawa last updated on 24/Jun/17

Solve simultaneously

x^{2} + y^{2} = 61 \dots \dots \dots \dots . . equation (i)

x^{3} - y^{3} = 91 \dots \dots \dots \dots . . equation (ii)

Commented by prakash jain last updated on 24/Jun/17

x = \sqrt{61} \cos u, y = \sqrt{61} \sin u

61^{3 / 2} \cos^{3} u - 61^{3 / 2} \sin^{3} u = 91

(\cos u - \sin u) (\cos^{2} u + \sin^{2} u - \cos u \sin u)

= \frac{91}{61^{3 / 2}}

(\cos u - \sin u) (1 - \cos u \sin u) = . 191

\cos (\frac{π}{4} - u) (2 - \sin 2 u) = . 191 \times 2 \sqrt{2} = . 54

2 \cos (\frac{π}{4} - u) - \cos (\frac{π}{4} - u) \sin 2 u = . 54

c o n t i n u e

Commented by tawa tawa last updated on 24/Jun/17

Am with you sir . God bless you sir .

Commented by Tinkutara last updated on 24/Jun/17

Hit and trial method :

y^{2} = 61 - x^{2}

Let x, y \in N .

Then y^{2} = 60, 57, 52, 45, 36, 25, 12 .

But x > y . So x = 6 and y = 5 .

Commented by tawa tawa last updated on 24/Jun/17

Am with you sir . God bless you sir .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 26/Jun/17

Commented by mrW1 last updated on 26/Jun/17

geogebra is very powerful!

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 26/Jun/17

p l e a s e c o r r e c t y o u r f i n a l a n s w e r

f r o m : (6, - 5) \Rightarrow (6, 5) . c x c u s e m e .

Commented by mrW1 last updated on 26/Jun/17

certainly you are right . thanks!

Answered by mrW1 last updated on 26/Jun/17

x^{2} + y^{2} - 2 xy + 2 xy = 61

{(x - y)}^{2} + 2 xy = 61

x^{3} - y^{3} = (x - y) (x^{2} + xy + y^{2}) = 91

(x - y) (61 + xy) = 91

let u = x - y and v = xy

u^{2} + 2 v = 61 \dots (1)

u (61 + v) = 91 \dots (2)

v = \frac{61 - u^{2}}{2}

u (61 + \frac{61 - u^{2}}{2}) = 91

u (\frac{183 - u^{2}}{2}) = 91

u^{3} - 183 u + 182 = 0

(u - 1) (u - 13) (u + 14) = 0

\Rightarrow u = (- 14, 1, 13)

\Rightarrow v = \frac{61 - {(- 14)}^{2}}{2} = - \frac{135}{2}

\Rightarrow v = \frac{61 - {(1)}^{2}}{2} = 30

\Rightarrow v = \frac{61 - {(13)}^{2}}{2} = - 54

\Rightarrow v = (- \frac{135}{2}, 30, - 54)

x - y = u \dots (3)

xy = v

{(x + y)}^{2} = {(x - y)}^{2} + 4 xy = u^{2} + 4 v = 61 + 2 v ⩾ 0

v ⩾ - \frac{61}{2}

\Rightarrow for real roots the only solution is

u = 1

v = 30

x + y = \pm \sqrt{61 + 2 v} \dots (4)

from (3) and (4) :

\Rightarrow x = \frac{u \pm \sqrt{61 + 2 v}}{2}

\Rightarrow y = \frac{- u \pm \sqrt{61 + 2 v}}{2}

with (u, v) = (1, 30)

x = \frac{1 \pm \sqrt{61 + 2 \times 30}}{2} = \frac{1 \pm 11}{2} = 6, - 5

y = \frac{- 1 \pm \sqrt{61 + 2 \times 30}}{2} = \frac{- 1 \pm 11}{2} = 5, - 6

\Rightarrow real solution is :

(x, y) = (6, 5) or (- 5, - 6)

Commented by tawa tawa last updated on 24/Jun/17

God bless you sir . i really appreciate .

Commented by tawa tawa last updated on 24/Jun/17

God bless you sir . i really appreciate .

Commented by b.e.h.i.8.3.4.1.7@gmail.com last updated on 25/Jun/17

p e r f e c t! & c x c e l l e n t .

Commented by RasheedSoomro last updated on 25/Jun/17

e^{x} cellent!