Если tg(l+пи/4)=-1/3.найдите значение ctg2l.

a) (2x - y)(x + 2y) = -3

1) 2x - y = 3 => y = 2x - 3

x + 2y = -1 => x + 2(2x - 3) = - 1 =>

x + 4x - 6 = -1 => 5x = 5 => x = 1

y = 2x - 3 = 2 - 3 = -1.

первое решение (x, y) = (1, -1)

2) 2x - y = - 3 => y = 2x + 3

x + 2y = 1 => x +2(2x + 3) = 1 =>

x + 4x + 6 = 1 => 5x = - 5 => x = -1

y = 2x + 3 = -2 + 3 = 1.

второе решение (x, y) = (-1, 1)

3) 2x - y = 1 => y = 2x - 1

x + 2y = -3 => x + 2(2x - 1) = -3 =>

x + 4x - 2 = -3 => 5x = -1 => x = - 1/5 - нецелое.

4) 2x - y = -1 => y = 2x + 1

x + 2y = 3 => x + 2(2x + 1) = 3 =>

x + 4x + 2 = 3 => 5x = 1 => x = 1/5 - нецелое.

всего два решения (x, y) = (1, -1) и (x, y) = (-1, 1)

б) 2x² + xy - y² = -3

x² - y² + x² + xy = -3

(x - y)(x + y) + x(x + y) = -3

(x + y)(x - y + x) = -3

(x + y)(2x - y) = -3

1) x + y = 3 => y = 3 - x

2x - y = -1 => 2x - 3 + x = -1 =>

3x - 3 = -1 => 3x = 2 => x = 2/3 - нецелое

2) x + y = 1 => y = 1 - x

2x - y = -3 => 2x - 1 + x = -3 =>

3x -1 = -3 => 3x = -2 => x = -2/3 - нецелое

3) x + y = -3 => y = -3 - x

2x - y = 1 => 2x +3 + x = 1 =>

3x + 3 = 1 = 3x = -2 => x = -2/3 - нецелое

4) x + y = -1 = y = -1 - x

2x - y = 3 => 2x +1 + x = 3 =>

3x + 1 = 3 => 3x = 2 => x = 2/3 - нецелое.

второе уравнение не имеет решений в целых числах.

a){y+2x=-5{x^2+y^2=25

 

{y=-5-2x

{x^2+(-5-2x)^2=25

x^2+4x^2+10x+25=25

5x^2+10x=0

5(x^2+2x)=0

x^2+2x=0

x(x+2)=0

x=0

x+2=0

x=-2

y=-5-2*-2=-9

y=-5-2*0=-5

ответ   (0 -2)   (-9 -5)

 

б)

{y=x^2+6x+7{y-2x=4

 

  y=4+2x

y=x^2+6x+7

 

4+2x=x^2+6x+7

x^2+4x+3=0

d=16-4*1*3=v4=2

x=-4+2/2=-1

x2=-6/2=-3

  y=4+2*-1=-2

  y2=4+2*-3=-2

 

ответ (-1 -3) (-2 -2) 

 

в)

{xy-2y-4x=-5

{x-3y=-10

 

x=-10+3y

(-10+3y)y-2y-4(-10+3y)=-5

  -10y+3y^2-2y+40-12y=-5

3y^2-24y+45=0

  d=4

y=3

y=5

x=-10+3*3=-1

x=-10+3*5 =5

ответ   (3 5 )   (   -1   5)