{ x² + 4xy -2y² = 5x +5y ; 5x² -xy- y² -7x =7y . ⇔
{ x² + 4xy -2y² = 5(x +y) ; 5x² -xy- y² =7(x +y). ⇔
{x+y = (x² + 4xy -2y²) / 5 ; x +y =(5x² -xy- y²) / 7. ⇔
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(x² + 4xy -2y²) / 5 = (5x² -xy- y²) / 7 ⇔ 7(x² + 4xy -2y²) = 5(5x² -xy- y² ) ⇔
18x² -33xy +9y² =0 ⇔
3y² -11x *y + 6x² =0 ; .
y₁ = (11x -7x) / 6 = (2/3)x ;
y₂ = (11x +7x) / 6 = 3x .
a) y= (2/3)x .
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x² + 4xy -2y² = 5x +5y ⇔ x² + 4x*(2/3)x -2*(4/9)x² = 5x +5(2/3)x ⇔
9x² + 24x² -8x² = 45x +30x ⇔25x² = 75x ⇔25x(x-3) =0 ⇔ [ x₁ =0 ; x₂ =3 .
{ (0 ; 0) ; (3 ; 2) } .
б) y=3x .
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x² + 4xy -2y² = 5x +5y ⇔ x² + 4x*3x -2*9x² = 5x +5*3x ⇔
-5x² = 20x ⇔ x² = - 4x ⇔ x(x+4) =0 ⇔ [ x₁ =0 ; x₂ = - 4 .
{ (0 ; 0) ; (- 4 ; -12) } .
ответ : { (- 4 ; -12) ;(0 ; 0) ; (3 ; 2) }.
* * * по другому * * *
1 .если x = 0 ;
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{ -2y² = 5y ;- y² = 7y ⇔ { -2y² = 5y ;- 2y² =14y ⇒5y=14y⇔y=0.
(0;0).
если x ≠ 0
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3y² -11x *y + 6x² =0 || :x² || ⇒
3(y/x)² -11*(y/x) +6 =0 замена t = y/x ⇒3t² -11t +6 =0 и т.д.
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