1.{x² +y² =25{y² -x=5y²=5+xx²+5+x=25x²+x-20=0D=1+80=81x₁=
-1-9 = -5 y²=5-5 2 y² =0 y=0x₂ =
-1+9 =4 y²=5+4 2 y²=9 y₁=3 и у₂= -3Ответ: (-5; 0) (4; 3) (4; -3)2) {1/x + 1/y =5/6 {1/x - 1/y =1/6Складываем уравнения системы:1/x + 1/x = 5/6 + 1/62/x = 1x=21/2 + 1/y = 5/61/y= 5/6 - 1/21/y= 5/6 - 3/61/y=2/6 y=3Ответ: (2; 3)3.{1/(x-1) + 2/(y+1) = 1 ¹/₆{3/(x-1) - 1/(y+1) = 1 ¹/₆ | умножим на "2"{1/(x-1) + 2/(y+1) = 7/6{6/(x-1) - 2/(y+1)=14/6ОДЗ: х≠1 у≠ -1Складываем уравнения:
7 =
21х-1 6x-1 =
7*6 21x-1 =2x=3
1 +
2 = 7/6 3-1 y+1
2 =
7 -
1 y+1 6 2
2 =
4 y+1 6y+1 =
2*6 4y+1=3y=2Ответ: (3; 2)4) {2x+y²=3 {3x+y⁴=4y²=3-2xy⁴=(3-2x)² =9-12x+4x²3x+9-12x+4x²=44x²-9x+5=0D=81-80=1x₁=
9-1 = 1 y²=3-2 8 y²=1 y₁=1 и y₂= -1x₂ =
9+1 =1.25 y² = 3-2*1.25=3-2.5=0.5 8 y₁=√0.5 y₂= -√0.5Ответ: (1; 1) (1; -1) (1,25; √0,5) (1,25; -√0,5)5.{x+y+xy=7{x²+y²+xy=13 (x+y)²=x²+y²+2xy(x+y)² -2xy =x²+y²{x+y+xy=7{(x+y)² -2xy+xy=13{x+y+xy=7{(x+y)² -xy=13Пусть x+y=a xy=b.{a+b=7{a² -b=13b=7-aa²-7+a=13a² +a-20=0D=1+80=81a₁=
-1-9 = -5 b=7+5=12 2a₂=
-1+9 = 4 b=7-4=3 2При a= -5 и b=12:{x+y=-5{xy=12x=-5-y(-5-y)y=12-5y-y²-12=0y²+5y+12=0D=25-48<0нет решенийПри а=4 и b=3:{x+y=4{xy=3x=4-y(4-y)y=34y-y²-3=0y²-4y+3=0D=16-12=4y₁=
4-2 = 1 x=4-1=3 2y₂=
4+2=3 x=4-3=1 2Ответ: (3; 1) (1; 3)