Question

найти кол-во решений системы уравнений
{ x^2+y^2=5
{ 3xy+2x=2y

Answer 1

$left { {{x^2+y^2=5} atop {3xy+2x=2y}} right. ; left { {x^2+y^2=5} atop {3xy=2y-2x}} right. ; left { {{x^2+y^2=5} atop {3xy=2cdot (y-x)}} right. ; left { {{x^2+y^2=5} atop {9x^2y^2=4(y-x)^2}} right. \\9x^2y^2=4y^2+4x^2-8xy\\9x^2y^2=4(x^2+y^2)-8xy\\9x^2y^2=4cdot 5-8xy\\t=xy; ,; ; 9t^2+8t-20=0; ,; D/4=196; ,; t_1=-2,; t_2=frac{10}{9}\\ 1); left { {{x^2+y^2=5} atop {xy=-2}} right. quad ili quad 2);left { {{x^2+y^2=5} atop {xy=frac{10}{9}}} right.$

$1); ; left { {{x^2+2xy+y^2=5+2cdot (-2)} atop {xy=-2}} right. ; left { {{(x+y)^2=1} atop {xy=-2}} right. ; left { {{x+y=pm 1} atop {xy=-2}} right. \\a) left { {x+{y=1} atop {xy=-2}} right. ; left { {{y=1-x} atop {x(1-x)=-2}} right. ; left { {{y=1-x} atop {x^2-x-2=0}} right. ; left { {{y_1=2,; y_2=-1} atop {x_1=-1},; x_2=2} right. \\b); left { {{x+y=-1} atop {xy=-2}} right. ; left { {{y=-1-x} atop {x(-1-x)=-2}} right.$

$left { {{y=-1-x} atop {x^2+x-2=0}} right. ; left { {{y_1=1,; y_2=-2} atop {x_1=-2,; x_2=1}} right. \\2); ; left { {{x^2+2xy+y^2=5+2cdot frac{10}{9}} atop {xy=frac{10}{9}}} right. ; left { {{(x+y)^2=frac{65}{9}} atop {xy=frac{10}{9}}} right. ; left { {{x+y=pm frac{sqrt{65}}{3}} atop {xy=frac{10}{9}}} right.$

$a); left { {{x+y=frac{sqrt{65}}{3}} atop {xy=frac{10}{9}}} right. ; left { {{y=frac{sqrt{65}}{3}-x} atop {x(frac{sqrt{65}}{3}-x)=frac{10}{9}}} right.$

$x^2-frac{sqrt{65}}{3}x+frac{10}{9}=0, |cdot 9\\9x^2-3sqrt{65}x+10=0\\D=9cdot 65-4cdot 9cdot 10=225; ,; sqrt{D}=15\\x_1=frac{3sqrt{65}-15}{18}= frac{sqrt{65}-5}{6} ; ,; ; x_2= frac{sqrt{65}+5}{6} \\y_1=frac{sqrt{65}}{3}- frac{sqrt{65}-5}{6} = frac{sqrt{65}+5}{6} ; ,; ; y_2=frac{sqrt{65}}{3}- frac{sqrt{65}+5}{6}= frac{sqrt{65}-5}{6}$

$b); left { {{x+y=-frac{sqrt{65}}{3}} atop {xy=frac{10}{9}}} right. left { {{y=-frac{sqrt{65}}{3}-x} atop {xy=frac{10}{9}} right.$ 

$x(-frac{sqrt{65}}{3}-x)=frac{10}{9}\\x^2+frac{sqrt{65}}{3}x+frac{10}{9}=0, |cdot 9\\9x^2+3sqrt{65}x+10=0; ,; D=9cdot 65-4cdot 9cdot 10=225\\x_1= frac{-3sqrt{65}-15}{18}= frac{-sqrt{65}-5}{6}; ,; x_2= frac{-sqrt{65}+5}{6} \\y_1=-frac{sqrt{65}}{3}-frac{-sqrt{65}-5}{6}= frac{-sqrt{65}+5}{6} ; ,; y_2=-frac{sqrt{65}}{3}- frac{-sqrt{65}+5}{6}= frac{-sqrt{65}-5}{6}$

$Otvet:; (-1,2); ,; (2,-1); ,; (-2,1); ,; (1,-2); ,\\left( frac{sqrt{65}-5}{6} , frac{sqrt{65}+5}6} right); ,; left ( frac{sqrt{65}+5}{6} , frac{sqrt{65}-5}{6} right ); ,$

$left ( frac{-sqrt{65}-5}{6} , frac{-sqrt{65}+5}{6} right ); ,; left ( frac{-sqrt{65}+5}{6} , frac{-sqrt{65}-5}{6}right )$