Question

1.Rozwiąż podane układy równań metodą podstawiania a)2(x+4)+y=x 2x+3y=-22 b)3x+1-2(y-3)=7 3x-y=-2 c)x-2y=5 2x+3y=17 2.Rozwiąż podane układy równań metodą przeciwnych współczynników a)x+y=3 3x-2y=-1 b)-2x+3y=-2 3(x-1)+2(1-y)=0

Answer 1

1.

a)

2 (x + 4) + y = x

2x + 3y = -22

 

2x + 8 + y = x

2 (2x + 8 + y) + 3y = -22

 

2x + 8 + y = x

4x + 16 + 2y + 3y = -22

 

2x + 8 + y = x

4x + 5y = -22 - 16

 

8 + y = x - 2x

5y = -38 - 4x

 

8 + y = -x

y = -38/5 - 4/5x

 

8 + (-38/5) - 4/5x = -x

y = -38/5 - 4/5x

 

8 - 7 3/5 = -x + 4/5x

y = -38/5 - 4/5x

 

2/5 = -1/5x

y = - 38/5 - 8/5

 

x = -2

y = -30/5

 

x = -2

y = -6

 

b)

3x + 1 - 2 (y - 3) = 7

3x - y = -2

 

3x + 1 - 2y + 6 = 7

3x - y = -2

 

3x = 7 - 1 - 6 + 2y

3x - y = -2

 

3x = 2y

2y - y = -2

 

3x = 2y

y = -2

 

3x = -4

y = -2

 

c)

x  - 2y = 5

2x + 3y = 17

 

x = 5 + 2y

2 (5 + 2y) + 3y = 17

 

x = 5 + 2y

10 + 4y + 3y = 17

 

x = 5 + 2y

7y = 17 - 10

 

x = 5 + 2y

y = 1

 

x = 7

y = 1

 

 

 

2.

a)

x + y = 3     / · (-3)

3x - 2y = -1

 

      -3x - 3y = -9

      3x - 2y = -1

+____________

 

-3y - 2y = -9 - 1

-5y = -10

y = 2

 

 

x + 2 = 3

3x - 4 = -1

 

x = 1

y = 2

 

b)

-2x + 3y = -2

3(x-1) + 2(1-y) = 0

 

-2x + 3y = -2

3x - 3 + 2 - 2y = 0

 

-2x + 3y = -2        / : 2

3x - 2y = 1        / : 3

 

   -x + 3/2y = -1

    x - 2/3y = 1/3

+____________

 

3/2y - 2/3y = 1/3 - 1

9/6y - 4/6y = 2/3

5/6y = 2/3

5y = 4

y = 4/5

 

-2x + 3(4/5) = -2

3(x-1) + 2(1 - 4/5) = 0

 

-2x + 12/5 = -2

3x - 1 + 2 - 8/5 = 0

 

-2x = -2 - 12/5

 

2x = 2 + 12/5

2x = 22/5

x = 11/5

x = 2 1/5

 

y = 4/5

x = 2 1/5