Question

Решите, плиз, систему уравнений
2^((x+y)/3)+2^((x+y)/6)=6
2*x^2+10*y^2=9*x*y

Answer 1

Ответ:

$\left\{\begin{array}{l}2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6\\2x^2+10y^2=9xy\end{array}\right\\\\\\a)\ \ 2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6\ \ ,\ \ \ t=2^{\frac{x+y}{6}}>0\ \ ,\ \ t^2+t-6=0\ \ ,\ \ t_1=-3<0\ ,\ t_2=2\\\\2^{\frac{x+y}{6}}=2^1\ \ ,\ \ \dfrac{x+y}{6}=1\ \ ,\ \ x+y=6\ \ ,\ \ \underline{y=6-x}\\\\b)\ \ 2x^2+10y^2-9xy=0\ \ \to \ \ \ 2x^2+10(6-x)^2-9x(6-x)=0\ \ ,\\\\2x^2+10(36-12x+x^2)-54x+9x^2=0\\\\21x^2-174x+360=0\ \ \to \ \ \ 7x^2-58x+120=0\ ,\ \ D/4=1\ \ ,$

$x_1=\dfrac{29-1}{7}=4\ \ ,\ \ x_2=\dfrac{29+1}{7}=\dfrac{30}{7}=4\dfrac{2}{7}\\\\y_1=6-x_1=6-4=2\ \ ,\ \ y_2=6-x_2=6-\dfrac{30}{7}=\dfrac{12}{7}\\\\Otvet:\ \Big(4\, ;\, 2\, \Big)\ \ ,\ \ \Big(\ \dfrac{30}{7}\ ;\ \dfrac{12}{7}\ \Big)\ .$