Question

341 (1), 343 (1) Заранее благодарю)

Answer 1

$1); ; left { {{2sqrt{3y+x}-sqrt{6y-x}; =x} atop {sqrt{3y+x}+sqrt{6y-x}; =3y}} right. ; oplus ; left { {{3sqrt{3y+x}=3y+x} atop {sqrt{3y+x}+sqrt{6y-x}=3y}} right. ; left { {{3sqrt{3y+x}-(3y+x)=0} atop {sqrt{6y-x}=3y-sqrt{3y+x}}} right. \\a); ; 3sqrt{3y+x}-(3y+x)=0\\sqrt{3y+x}cdot (3-sqrt{3y+x})=0; ; Rightarrow \\sqrt{3y+x}=0; ; ili; ; ; sqrt{3y+x}=3\\3y+x=0; ; ; ili; ; ; 3y+x=9\\x=-3y; ; ; ili; ; ; x=9-3y$

$b); ; left { {{x=-3y} atop {sqrt{6y-3}=3y-0}} right. ; left { {{x=-3y} atop {6y-3=9y^2; ,; y textgreater 0}} right. ; left { {{x=-3y} atop {9y^2-6y+3=0}} right. \\3y^2-2y+1=0; ,\\D/4=1-3=-2 textless 0; ; to ; ; 3y^2-2y+1ne 0; ; (3y^2-2y+1 textgreater 0)\\net; reshenij\\c); ; left { {{x=9-3y} atop {sqrt{6y-x}=3y-3}} right. ; left { {{x=9-3y} atop {6y-(9-3y)=(3y-3)^2}} right. ; left { {{x=9-3y} atop {9y-9=9y^2-18y+9}} right. \\ left { {{x=9-3y} atop {9y^2-27y+18=0}} right. ; left { {{x=9-3y} atop {y^2-3y+2=0}} right. ; left { {{x_1=6; ,; x_2=3} atop {y_1=1; ,; y_2=2}} right. \\(6,1); ,; ; (3,2)$

$Proverka:\\(6,1):; ; left { {{2sqrt{9}-sqrt{0}=6} atop {sqrt{9}+sqrt{0}=3}} right. \\(3,2):; ; left { {{2sqrt{9}-sqrt{9}=3} atop {sqrt{9}+sqrt{9}=6}} right.\\Otvet:; ; (6,1); ,; (3,2).$

$2); ; left { {{x^2+y^2=5} atop {x+y=2}} right. ; left { {{x^2+y^2=5} atop {(x+y)^2=4}} right. ; left { {{x^2+y^2=5} atop {x^2+y^2+2xy=4}} right. ; left { {{x^2y^2=5} atop {5+2xy=4}} right. \\ left { {{x^2+y^2=5} atop {xy=-frac{1}{2}}} right. ; left { {{x^2+frac{1}{4x^2}=5} atop {y=-frac{1}{2x}}} right. ; left { {{4x^4-20x^2+1=0} atop {y=-frac{1}{2x}}} right. \\4x^4-20x^2+1=0\\t=x^2 geq 0; ,; ; ; 4t^2-20t+1=0; ,; ; D/4=100-4=96$

$t_1=frac{10-4sqrt6}4}= frac{5-2sqrt6}{2}=2,5-sqrt6 ; ,; ; t_2= frac{5+2sqrt6}{2} = 2,5+sqrt6 \\ x^2=2,5+sqrt6; ; to ; ; ; x=pm sqrt{2,5+sqrt6}=pm frac{sqrt{5+2sqrt6}}{sqrt2 }\\y=-frac{1}{pm 2sqrt{2,5+sqrt6}}; ; to \\y_1=-frac{sqrt2}{-2sqrt{5+2sqrt6}}= frac{1}{sqrt{2(5+2sqrt6)}} = frac{1}{sqrt{10+4sqrt6}} = frac{sqrt{10-4sqrt6}}{sqrt{100-96}}= frac{sqrt{10-4sqrt6}}{2} \\y_2=-frac{sqrt2}{2sqrt{5+2sqrt6}}=- frac{sqrt{10-4sqrt6}}{2}$

$x^2=frac{5-2sqrt6}{2}; ; to ; ; x_{3,4}=pm frac{sqrt{5-2sqrt6}}{sqrt2} ; ; to \\y_{3,4}=mp frac{1}{sqrt2cdot sqrt{5-2sqrt6}}=mp frac{sqrt{10+4sqrt6}}{2}$

$Otvet:; ; Big ( -frac{sqrt{5+2sqrt6}}{sqrt2} ; ;; frac{sqrt{10-4sqrt6}}{2} Big ); ,; Big ( frac{sqrt{5+2sqrt6}}{sqrt2} ; ;; -frac{sqrt{10-4sqrt6}}{2} Big ); ,$

$Big ( frac{sqrt{5-2sqrt6}}{sqrt2};-frac{sqrt{10+4sqrt6}}{2}Big ); ,; Big (-frac{sqrt{5-2sqrt6}}{sqrt2}; ;; frac{sqrt{10+4sqrt6}}{2}Big ); .$