Question

РЕШИТЕ ЛОГАРИФМИЧЕСКУЮ СИСТЕМУ УРАВНЕНИЙ С ОДЗ​, СРОЧНО

Answer 1

ОДЗ : x > 0 , y > 0

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

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