Question

РЕШИТЕ ЛОГАРИФМИЧЕСКУЮ СИСТЕМУ УРАВНЕНИЙ С ОДЗ​, СРОЧНО

Answer 1

ОДЗ : x > 0 , y > 0

$left { {{frac{1}{x}-frac{1}{y}=frac{2}{15}} atop {log_{3}x+log_{3}y=1+log_{3}5}} right.\\left { {{frac{y-x}{xy} =frac{2}{15} } atop {log_{3}xy=log_{3}3+log_{3}5}} right.\\left { {{xy=7,5(y-x)} atop {log_{3}xy=log_{3}15 }} right.$

$left { {{xy=7,5(y-x)} atop {xy=15}} right. \\left { {{xy=15} atop {7,5(y-x)=15}} right. \\left { {{xy=15} atop {y-x=2}} right.\\left { {{y=x+2} atop {x(x+2)=15}} right. \\left { {{y=x+2} atop {x^{2}+2x-15=0 }} right.\\left { {{y=x+2} atop {left[begin{array}{ccc}x_{1}=3 \x_{2} =-5<0-neydend{array}right }} right.\\left { {{x=3} atop {y=3+2=5}} right. \\Otvet:boxed{(3;5)}$