Solve-simultaneously-3x-2-4xy-3y-2-3-i-x-2-y-2-3x-3y-4-ii-

Question Number 9367 by tawakalitu last updated on 02/Dec/16

Solve simultaneously .

{3 x}^{2} + 4 xy + {3 y}^{2} = 3 \dots \dots \dots (i)

x^{2} + y^{2} + 3 x + 3 y = 4 \dots \dots . (ii)

Answered by mrW last updated on 02/Dec/16

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

3 {(x + y)}^{2} - 2 xy = 3 \dots . . (i)

x^{2} + y^{2} + 2 xy + 3 (x + y) - 2 xy = 4

{(x + y)}^{2} + 3 (x + y) - 2 xy = 4 \dots . . (ii)

u = x + y

v = xy

{3 u}^{2} - 2 v = 3 \dots . . (i)

u^{2} + 3 u - 2 v = 4

{3 u}^{2} + 9 u - 6 v = 12

- (i) :

9 u - 4 v = 9 \dots . . (ii)

v = \frac{9 (u - 1)}{4}

{3 u}^{2} - 2 \times 9 (\frac{u - 1}{4}) = 3

{3 u}^{2} - \frac{9}{2} u + \frac{3}{2} = 0

u^{2} - \frac{3}{2} u + \frac{1}{2} = 0

(u - 1) (u - \frac{1}{2}) = 0

u = (1, \frac{1}{2})

v = (0, - \frac{9}{8})

x + y = u

y = u - x

xy = v

x (u - x) = v

x^{2} - ux + v = 0

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

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

for u = 1, v = 0 :

x = \frac{1 \pm 1}{2} = (1, 0)

y = (0, 1)

for u = \frac{1}{2}, v = - \frac{9}{8} :

x = \frac{\frac{1}{2} \pm \sqrt{{(\frac{1}{2})}^{2} + 4 \times \frac{9}{8}}}{2} = \frac{1 \pm \sqrt{19}}{4} = (\frac{1 + \sqrt{19}}{4}, \frac{1 - \sqrt{19}}{4})

y = (\frac{1 - \sqrt{19}}{4}, \frac{1 + \sqrt{19}}{4})

all solutions together :

x = (1, 0, \frac{1 + \sqrt{19}}{4}, \frac{1 - \sqrt{19}}{4})

y = (0, 1, \frac{1 - \sqrt{19}}{4}, \frac{1 + \sqrt{19}}{4})

Commented by tawakalitu last updated on 03/Dec/16

Thank you sir . God bless you .