Question

x^2-y^2=16
xy=15
Решите систему уравнений

Answer 1

x=15/y
225/y²-y²=16
225-y^4-16y²=0
y≠0
y²=a
a²+16a-225=0
D=256+900=1156
√D=34
a1=(-16-34)/2=-25⇒y²=-25 нет решения
a2=(-16+34)/2=9⇒y²=9⇒y=+-3
x1=15/(-3)=-5
x2=15/3=5
(-5;-3);(5;3)

Answer 2

$\left \{ {{ x^{2} - y^{2} =16} \atop {xy=15} } \right. \\ \left \{ {{ x^{2} - y^{2} =16} \atop {y= \frac{15}{x} } } \right. \\ x^{2} - (\frac{15}{x})^{2} =16 \\ x^{2} - \frac{225}{ x^{2} } =16$

Замена:
$x^{2} = t \\ t - \frac{225}{t} = 16 \\ t^{2} - 16t - 225 = 0 \\ t=25, t = - 9 \\ \\ x^{2} =25 =\ \textgreater \ x=5, x=-5$
$y= \frac{15}{5} = 3, y= \frac{15}{-5} = -3$

ОТвет: ( 5;3 ),  ( -5;-3 ).