x/y-y/x=5/6 and x^2-y^2=5 find x and y
Answers
Answer:
......
Step-by-step explanation:
x
=0 ;y
=0
\begin{gathered} \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. \\y= \frac{6}{x} \\x^2- \frac{36}{x^2} =5 \\x^4-5x^2-36=0 \\u=x^2;\ u \geq 0 \\u^2-5u-36=0 \\D=25+144=169=13^2 \\u_1= \frac{5+13}{2}=9 \\u_2= \frac{5-13}{2} \ \textless \ 0 \\x^2=9 \\x_1=3 \\x_2=-3 \\y_1= \frac{6}{3}=2 \\y_2= \frac{6}{-3} =-2 \end{gathered}
{
x
2
−y
2
=5
y
x
−
x
y
=
6
5
{
x
2
−y
2
=5
xy
x
2
−y
2
=
6
5
{
x
2
−y
2
=5
xy
5
=
6
5
{
x
2
−y
2
=5
xy=6
y=
x
6
x
2
−
x
2
36
=5
x
4
−5x
2
−36=0
u=x
2
; u≥0
u
2
−5u−36=0
D=25+144=169=13
2
u
1
=
2
5+13
=9
u
2
=
2
5−13
\textless 0
x
2
=9
x
1
=3
x
2
=−3
y
1
=
3
6
=2
y
2
=
−3
6
=−2
Ответ: (3;2) и (-3;-2)