Question

Rozwiąż dany układ metodą podstawiania, porządkując najpierw równania danego układu:
a) 2(3x-2y)-(x-4y)=20
-5(x+y)+2(x-y)=23

b) 7(2x+3y)-6(x+4y)=2
3(-2x+y)-5(3x-2y)=5

c)-3(2x-y)+2(3x-1,5y)=x+4
2(y-x)-2x=4x+5

Answer 1

Odpowiedź:

a)

2(3x - 2y) - (x - 4y) = 20

- 5(x + y) + 2(x - y) = 23

6x - 4y - x + 4y = 20

- 5x - 5y + 2x - 2y = 23

5x = 20 ⇒ x = 20/5 ⇒ x = 4

- 3x - 7y = 23

- 3 * 4 - 7y = 23

- 12 - 7y = 23

- 7y = 23 + 12

- 7y = 35

7y = - 35

y = - 35/7 = - 5

Odp: x = 4 , y = - 5

b)

7(2x + 3y) - 6(x + 4y) = 2

3(- 2x + y) - 5(3x - 2y) = 5

14x + 21y - 6x - 24y = 2

- 6x + 3y - 15x + 10y = 5

8x - 3y = 2 ⇒ 8x = 2 + 3y ⇒ x = (2 + 3y)/8

- 21x + 13y = 5

- 21 * (2 + 3y)/8 + 13y = 5 | * 8

- 21(2 + 3y) + 104y = 40

- 42 - 63y + 104y = 40

- 42 + 41y = 40

41y = 40 + 42

41y = 82

y = 82/41 = 2

x = (2 + 3y)/8 = (2 + 3 * 2)/8 = (2 + 6)/8 = 8/8 = 1

Odp: x = 1 , y = 2

c)

- 3(2x - y) + 2(3x - 1,5y) = x + 4

2(y - x) - 2x = 4x + 5

- 6x + 3y + 6x - 3y = x + 4

2y - 2x - 2x = 4x + 5

0 = x + 4

2y - 4x = 4x + 5

x + 4 = 0

- 4x - 4x + 2y = 5

x + 4 = 0 ⇒ x = - 4  

- 8x + 2y = 5

- 8 * (- 4) + 2y = 5

32 + 2y = 5

2y = 5 - 32

2y = - 27

y = - 27/2 = - 13 1/2

Odp: x = - 4 , y = - 13 1/2

⇒ znaczy "to"