Решите, то, что справа заранее благодарю !

А)
$left { {{ frac{x}{y} + frac{y}{x} =- frac{5}{2} } atop { x^{2} - y^{2} = frac{13}{4} }} right.$
Замена $frac{x}{y} =t; frac{y}{x} = frac{1}{t}; x=t*y$
$left { {{t+ frac{1}{t} = -frac{5}{2} } atop {(t*y)^2-y^2=y^2*(t^2-1)= frac{13}{4} }} right.$

2t^2 + 2 = -5t
2t^2 + 5t + 2 = 0
(t + 2)(2t + 1) = 0
t1 = x/y = -2; x = -2y
t2 = x/y = -1/2; x = -y/2

1) y^2*(t^2 - 1) = 13/4
y^2*(2^2 - 1) = 3y^2 = 13/4
y^2 = 13/12
$y1=- sqrt{ frac{13}{12} }; x1=-2y= sqrt{ frac{13*4}{12} }= sqrt{ frac{13}{3} }$
$y2= sqrt{ frac{13}{12} }; x2=-2y=- sqrt{ frac{13*4}{12} }=-sqrt{ frac{13}{3} }$

2) y^2*(t^2 - 1) = 13/4
y^2*((1/2)^2 - 1) = -3/4*y^2 = 13/4
y^2 < 0, Решений нет

б)
$left { {{x^{2} + y^{2} = 68} atop {frac{x}{y} - frac{y}{x} =frac{17}{4} }} right.$
Замена $frac{x}{y} =t; frac{y}{x} = frac{1}{t}; x=t*y$
$left { {{t- frac{1}{t} = frac{17}{4} } atop {(t*y)^2+y^2=y^2*(t^2+1)= 68 }} right.$

4t^2 - 4 = 17t
4t^2 - 17t - 4 = 0
D = 289 - 4*4(-4) = 289 + 64 = 353
1) t1 = (17 - √353)/8 = x/y
$t^2+1= frac{(17- sqrt{353})^2 }{64}+1= frac{289+353+64-34 sqrt{353} }{64} = frac{353-17 sqrt{353}}{32}$

$y^2*(t^2+1)=y^2*frac{353-17 sqrt{353}}{32}=68$
$y^2= frac{68*32}{353-17 sqrt{353}}= frac{64*34}{353-17 sqrt{353}}$
$x=t*y=(17 - sqrt{353})/8*y$

2) t2 = (17 + √353)/8 = x/y
Решается точно также

в)
$left { {{|x|+|y|=6} atop {x^2-y^2=24}} right.$
1) Пусть x < 0, y < 0. Тогда |x| = -x, |y| = -y
$left { {{-x-y=6} atop {x^2-y^2=24}} right.$
$left { {{x+y=-6} atop {(x+y)(x-y)=24}} right.$
$left { {{x+y=-6} atop {x-y=-4}} right.$
$left { {{x=-5} atop {y=-1}} right.$

2) Пусть x < 0, y > 0. Тогда |x| = -x, |y| = y
$left { {{-x+y=6} atop {x^2-y^2=24}} right.$
$left { {{x-y=-6} atop {(x+y)(x-y)=24}} right.$
$left { {{x-y=-6} atop {x+y=-4}} right.$
$left { {{x=-5} atop {y=1}} right.$

3) Пусть x > 0, y < 0. Тогда |x| = x, |y| = -y
$left { {{x-y=6} atop {x^2-y^2=24}} right.$
$left { {{x-y=6} atop {(x+y)(x-y)=24}} right.$
$left { {{x-y=6} atop {x+y=4}} right.$
$left { {{x=5} atop {y=-1}} right.$

4) Пусть x > 0, y > 0. Тогда |x| = x, |y| = y
$left { {{x+y=6} atop {x^2-y^2=24}} right.$
$left { {{x+y=6} atop {(x+y)(x-y)=24}} right.$
$left { {{x+y=6} atop {x-y=4}} right.$
$left { {{x=5} atop {y=1}} right.$

г)
$left { {{|x|-|y|=4} atop {x^2+y^2=41}} right.$
1) Пусть x < 0, y < 0. Тогда |x| = -x, |y| = -y
$left { {{-x+y=4} atop {x^2+y^2=41}} right.$

$left { {{x^2-2xy+y^2=16} atop {x^2+y^2=41}} right.$
Подставляем 2 ур-ние в 1 ур-ние
$41-2xy=16$
$xy= frac{41-16}{2}= frac{25}{2} =12,5$

$left { {{x-y=-4} atop {xy=12,5}} right.$
$left { {{x=y-4} atop {y(y-4)=12,5}} right.$

y^2 - 4y - 12,5 = 0
2y^2 - 8y - 25 = 0
D/4 = 4^2 - 2*(-25) = 16 + 50 = 66 = (√66)^2
y1 = (4-√66)/2; x1 = y-4 = (-4-√66)/2
y2 = (4+√66)/2; x2 = y-4 = (-4+√66)/2

2) Пусть x < 0, y > 0. Тогда |x| = -x, |y| = y
$left { {{-x-y=4} atop {x^2+y^2=41}} right.$

$left { {{x^2+2xy+y^2=16} atop {x^2+y^2=41}} right.$
Подставляем 2 ур-ние в 1 ур-ние
$41+2xy=16$
$xy= frac{-41+16}{2}= frac{-25}{2} =-12,5$

$left { {{x+y=-4} atop {xy=-12,5}} right.$
$left { {{x=-y-4} atop {y(-y-4)=-12,5}} right.$

-y^2 - 4y + 12,5 = 0
2y^2 + 8y - 25 = 0
D/4 = 16 + 2*25 = 16 + 50 = 66 = (√66)^2
y3 = (-4-√66)/2; x3 = -y-4 = (-4+√66)/2
y4 = (-4+√66)/2; x4 = -y-4 = (-4-√66)/2

3) Пусть x > 0, y < 0. Тогда |x| = x, |y| = -y
$left { {{x+y=4} atop {x^2+y^2=41}} right.$

$left { {{x^2+2xy+y^2=16} atop {x^2+y^2=41}} right.$
Подставляем 2 ур-ние в 1 ур-ние
$41+2xy=16$
$xy= frac{-41+16}{2}= frac{-25}{2} =-12,5$

$left { {{x+y=4} atop {xy=-12,5}} right.$
$left { {{x=-y+4} atop {y(-y+4)=-12,5}} right.$

-y^2 + 4y + 12,5 = 0
2y^2 - 8y - 25 = 0
D/4 = 4^2 - 2*(-25) = 16 + 50 = 66 = (√66)^2
y5 = (4-√66)/2; x5 = -y+4 = (4+√66)/2
y6 = (4+√66)/2; x6 = -y+4 = (4-√66)/2

4) Пусть x > 0, y > 0. Тогда |x| = x, |y| = y
$left { {{x-y=4} atop {x^2+y^2=41}} right.$

$left { {{x^2-2xy+y^2=16} atop {x^2+y^2=41}} right.$
Подставляем 2 ур-ние в 1 ур-ние
$41-2xy=16$
$xy= frac{41-16}{2}= frac{25}{2} =12,5$

$left { {{x-y=4} atop {xy=12,5}} right.$
$left { {{x=y+4} atop {y(y+4)=12,5}} right.$

y^2 + 4y - 12,5 = 0
2y^2 + 8y - 25 = 0
D/4 = 4^2 - 2*(-25) = 16 + 50 = 66 = (√66)^2
y7 = (-4-√66)/2; x7 = y+4 = (4-√66)/2
y8 = (-4+√66)/2; x8 = y+4 = (4+√66)/2