Question

Помогите решить 2, 3 и 5. Заранее спасибо.

Answer 1

$2); ; left { {{x^2=11x+6y} atop {y^2=6x+11y}} right. ; Rightarrow ; ; x^2-y^2=5x-5y\\(x-y)(x+y)-5(x-y)=0\\(x-y)(x+y-5)=0; ; Rightarrow ; ; x-y=0; ili; ; x+y-5=0\\a); ; x=y; to ; ; left { {{x^2=11x+6y} atop {x=y}} right. ; left { {{x^2=11x+6x} atop {x=y}} right. ; left { {{x^2-17x=0} atop {x=y}} right. ; left { {{x(x-17)=0} atop {x=y}} right. \\ left { {{x_1=0; ili; x_2=17} atop {y_1=0; ili; y_2=17}} right.$

$b); ; x=5-y; to ; ; left { {{x=5-y} atop {y^2=6(5-y)+11}} right. ; left { {{x=5-y} atop {y^2=30-6y+11y}} right. ; left { {{x=5-y} atop {y^2-5y-30=0}} right. \\y^2-5y-30=0; ,; ; D=25+4cdot 30=145; ,; ; y_{1,2}=frac{5pm sqrt{145}}{2}\\x_1=5-frac{5-sqrt{145}}{2}=frac{5+sqrt{145}}{2} ; ,; ; x_2=5-frac{5+sqrt{145}}{2}=frac{5-sqrt{145}}{2}\\Otvet:; ; (0;0); ,; (17;17); ,; (frac{5-sqrt{145}}{2};frac{5+sqrt{145}}{2}); ,; (frac{5+sqrt{145}}{2};frac{5-sqrt{145}}{2}); .$

$3); ; left { {{4x^2+7xy-2y^2=0} atop {3x^2-5xy+y^2=-1}} right. \\4x^2+7xy-2y^2=0, |:y^2ne 0; ; ,; ; 4cdot (frac{x}{y})^2+7cdot frac{x}{y}-2=0\\t= frac{x}{y}; ,; ; 4t^2+7t-2=0; ,; D=49+32=81; ,\\t_1=frac{-7-9}{8}=-2; ,; t_2=frac{-7+9}{8}=frac{1}{4}\\a); ; frac{x}{y}=-2; ,; ; x=-2y; ,; ; 3cdot 4y^2-5ycdot (-2y)+y^2=-1\\23y^2=-1; ,; ; y^2=-frac{1}{23}; Rightarrow ; ; yin varnothing ; ; (t.k.; ; y^2 geq 0)$

$b); ; frac{x}{y}=frac{1}{4}; ,; ; y=4x; ,; ; 3x^2-5xcdot 4x+16x^2=-1\\-x^2=-1; ,; ; x^2=1; Rightarrow ; ; x_{1}=-1; ,; ; x_2=1\\y_1=-4; ,; x_2=4\\Otvet:; ; (-1;-4); ,; ; (1;4); .\\5); ; left { {{x^2+y^2=13,} atop {x+y+xy=11;}} right. ; ; ; ; ; (x+y)^2=x^2+y^2+2xy; Rightarrow \\x^2+y^2=(x+y)^2-2xy; ,quad left { {{(x+y)^2-2xy=13,} atop {x+y+xy=11;}} right. ; ,; ; u=x+y; ,; v=xy \\left { {{u^2-2v=13} atop {u+v=11}} right. ; left { {{u^2-2(11-u)=13} atop {v=11-u}} right. ; left { {{u^2+2u-35=0} atop {v=11-u}} right.$

$u^2+2u-35=0; ,; ; u_1=-7; ,; ; u_2=5; ; (teorema; Vieta)\\v_1=11-(-7)=18; ,; ; v_2=11-5=6\\a); ; left { {{x+y=-7} atop {xy=18}} right. ; left { {{y=-7-x} atop {x(-7-x)=18}} right. ; left { {{y=-7-x} atop {x^2+7x+18=0}} right. \\x^2+7x+18=0; ,; ; D=-23 textless 0; Rightarrow ; ; xin varnothing \\b); ; left { {{x+y=5} atop {xy=6}} right. ; left { {{y=5-x} atop {x(5-x)=6}} right. ; left { {{y=5-x} atop {x^2-5x+6=0}} right. ; left { {{y_1=3; ,; y_2=2} atop {x_1=2; ,; x_2=3}} right. \\Otvet:; ; (2;3); ,; ; (3;2); .$