Question

Розв'яжіть систему рівнянь: {x + 1 = 5 (y + 2) {3 (2x-5) -4 (3y - 4) = 19

Answer 1

Ответ:

Способ подстановки:

{x+1=5(y+2)

{3(2x-5)-4(3y-4)=19

{x=5y+9

{3(2x-5)-4(3y-4)=19

3(2(5y+9)-5)-4(3y-4)=19

y=-2

x=5*(-2)+9

x=-1

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,-2)

Метод исключения переменных:

{x+1=5(y+2)

{3(2x-5)-4(3y-4)=19

{x-5y=9

{6x-12y=18

{x-5y=9

{-x+2y=-3

-3y=6

y=-2

x-5*(-2)=9

x=-1

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,2)

Метод крамера:

{x+1=5(y+2)

{3(2x-5)-4(3y-4)=19

{x-5y=9

{6x-12y=18

$d = \frac{ |1| }{ |6| } \frac{ | - 5| }{ | - 12| }$

$d_{1} = \frac{ |9| }{ |18| } \frac{ | - 5| }{ | - 12| }$

$d_{2} = \frac{ |1| }{ |6| } \frac{ |9| }{18}$

D=18

D_1=-18

D_2=-36

D≠0

Находим x,y по формулам:

$x = \frac{ d_{1} }{d}$

$y = \frac{ d_{2} }{d}$

x=-1

y=-2

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,-2)

Метод сравнения:

{x+1=5(y+2)

{3(2x-5)-4(3y-4)=19

{x+1=5y+10

{3(2x-5)-4(3y-4)=19

{x+1=5y+10

{6x-15-4(3y-4)=19

{x+1=5y+10

{6x-15-12y+16=19

{x+1-5y=10

{6x-15-12y+16=19

{x-5y=10-1

{6x-15-12y+16=19

{x-5y=10-1

{6x+1-12y=18

{x-5y=9

{6x+1-12y=19

{x-5y=9

{6x-12y=19-1

{x=9+5y

{6x-12y=19-1

{x=9+5y

{6x-12y=18

{x=9+5y

{6x=18+12y

{x=9+5y

{x=3+2y

9+5y=3+2y

y=-2

x=3+2*(-2)

x=-1

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,-2)

Обратная матрица:

{x-5y=9

{3(2x-5)-4(3y-4)=19

{x-5y=19

{6x-12y=18

[1 -5]

[6 -12]

*[x]

[y]

=[9]

[18]

A=[1 -5]

[6 -12]

A^-1=[-2/3 5/18]

[-1/3 1/18]

[x]

[y]

=[-2/3 5/18]

[-1/3 1/18]

*[9]

[18]

[x]

[y]

=[-1]

[-2]

x=-1

y=-2

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,-2)

Метод Гаусса Жордана:

{x+1=5(y+2)

{3(2x-5)-4(3y-4)=19

{x-5y=9

{3(2x-5)-4(3y-4)=19

{x-5y=9

{6x-12y=18

[1 -5| 9]

[6 -12| 18]

[1 -5| 9]

[1 - 2| 3]

[1 -5| 9]

[0 3| - 6]

[1 - 5| 9]

[0 1| -2|

[1 0| -1]

[0 1| -2]

{x=-1

{y=-2

(x,y)=(-1,-2)

{-1+1=5(-2+2)

{3(2*(-1)-5)-4(3*(-2)-4)=19

{0=0

{19=19

(x,y)=(-1,-2)