Solve the system. (If there are infinitely many solutions, enter INFINITELY MANY. If there is no solution, enter NO SOLUTION.)

Question

Solve the system. (If there are infinitely many solutions, enter INFINITELY MANY. If there is no solution, enter NO SOLUTION.)

sqrt 3x - sqrt 2y= 3 sqrt 3

7sqrt2x + sqrt 3y= sqrt 2

Answer 1

Squares both sides:

(3x) - 2 * sqrt(3x)*sqrt(2y) + (2y) = 27

49(2x) + 2 * (7 * sqrt(2x))(sqrt(3y)) + (3y) = 2

Simplifies:

3x - 2 * sqrt(3x) sqrt(2y) + 2y = 27

98x + 14 *sqrt(2x)sqrt(3y) + 3y = 2

Isolates radicals:

-2 sqrt(3x) sqrt(2y) = -3x -2y+27

13 sqrt(2x)sqrt(2y) = -98x -3y + 2

squares both sides again:

4 (3x)(2y) = [-3x -2y+27]^2

160(2x)(2y) = [-98x - 3y + 2]^2

simplifies:

24xy = [-3x -2y+27]^2

640xy = [-98x - 3y + 2]^2

common denominator is 1920:

1920xy = 80[-3x -2y+27]^2

1920xy = 3[-98x - 3y + 2]^2

80[-3x -2y+27]^2 = 3[-98x - 3y + 2]^2

80[-3x -2y+27][-3x -2y+27] = -3[-98x - 3y + 2][-98x - 3y + 2]

FOIL:

left side:

80[ 9x^2 + 6xy -81x

+6xy +4y^2 - 54y

-81x -54y + 729 ]

right side:

-3[ 9604x^2 +294xy - 196x

294xy +9y^2 - 6y

-196x - 6y + 4]

simplifies:

80[ 9x^2 + 12xy -162x + 4y^2 -108y + 729] = -3[ 9604x^2 + 588xy - 392x +9y^2 -12y +4]

distributive:

720x^2 + 960xy - 12960x + 320y^2 - 8640y + 58320 =

-28812x^2 -1764xy +1176x - 27y^2 + 36y -12

29532x^2 + 2724xy -14136x + 347y^2 +8676y +58332 = 0

29532x^2 +x( 2724y - 14136) +(59332+8676y+247y^2) = 0

This equation is quadratic in terms of x....

A= 29532

B = ( 2724y - 14136)

C = (59332+8676y+247y^2)