Question

Помогите пожалуйста решить систему уравнений

Answer 1

$\left \{ {{ \frac{x}{y} - \frac{y}{x} = \frac{5}{6} } \atop { x^{2} - y^{2}=5 }} \right.\\\\ \left \{ {{ \frac{ x^{2} - y^{2} }{xy} = \frac{5}{6} } \atop { x^{2} - y^{2}=5 }} \right.\\\\ \left \{ {{ \frac{5}{xy} = \frac{5}{6} } \atop { x^{2} - y^{2}=5 }} \right.\\\\ \left \{ {{xy=6} \atop {x^{2}- y^{2} =5}} \right. \\\\ \left \{ {{x= \frac{6}{y} } \atop {( \frac{6}{y}) ^{2} - y^{2} =5 }} \right.\\\\ \left \{ {{x= \frac{6}{y} } \atop {y ^{4} -5 y^{2}+36=0 }} \right.$
Сделаем замену:
y² = m ≥ 0 ,  тогда y⁴ = m²
m² + 5m - 36 = 0
D = 5² - 4 * 1 * ( - 36) = 25 + 144 = 169 = 13²
$m _{1} = \frac{-5+13}{2}=4\\\\m _{2} = \frac{-5-13}{2} =-9$
m₂ = - 9 - не подходит
y² = 4
y₁ = 2         y₂ = - 2
$\left \{ {{y _{1} =2} \atop {x _{1} = \frac{6}{2} }} \right. \\\\ \left \{ {{y _{1} =2} \atop { x_{1} =3}} \right.\\\\ \left \{ {{ y_{2} =-2} \atop {x _{2} = \frac{6}{-2} }} \right. \\\\ \left \{ {{y _{2} =-2} \atop {x _{2} =-3}} \right.$
Ответ: (3 , 2) ,(- 3 , - 2)