Question

Rozwiąż układy równań metodą przeciwnych współczynników.

w załaczniku są te rownania.

jak cos : zbior zadania klasa 2 gimnazjum. 14 na 43.

Answer 1

a)

2x-y = 4

x+2y =7/*(-2)

 

2x-y = 4

-2x -4y = -14

-------------------- +

 

-5y = -10/:(-5)

y = 2

 

x+2*2 =7

x = 3

 

x-3

y=2

 

b)

3x+y =-1 /*(-3)

4x+3y =2

 

-9x -3y = 3

4x+3y =2

--------------- +

-5x =5 /:(-5)

x =-1

 

3*(-1) +y = -1

-3+y =-1

y = 2

 

x=-1

y=2

 

c)

5x-y =1,5 /*6

3x+6y =4,2

 

30x -6y = 9

3x+6y = 4,2

------------------- +

33x = 13,2 /:33

x = 0,4

 

5*0,4 -y = 1,5

-y = -0,5 /*(-1)

y = 0,5

 

x=0,4

y=0,5

 

d)

6x+2y =7 /*2

9x-4y = 0

 

12x +4y = 14

9x -4y =0

-----------------+

 

21x = 14 /:21

x = 2/3

 

12*2/3 +4y = 14

8 +4y =14

4y = 6 /:4

y = 1 i 1/2

 

x=2/3

y = 1 i 1/2

 

e)

x+4y =1

x-8y =-1 /*(-1)

 

x+4y =1

-x+8y =1

---------------- +

12y = 2 /:12

y = 1/6

 

x+ 4*1/6 = 1

x = 1/3

 

x=1/3

y=1/6

 

f)

4x = -5y +10

2x = 6y+18

 

4x+5y = 10

2x-6y =18 /*(-2)

 

4x+5y = 10

-4x +12y = -36

------------------------ +

17y = -26 /:17

y = - 26/17

y = - 1 i 9/17

 

4x = -5 *(-26/17) +10

4x = 7 i 11/17+10

4x = 299/17 /:4

x = 299/68

x = 4 i 27/68

 

x = 4 i 27/68

y = -1 i 9/17

 

g)

12y -x = 9

3y-1/3x =1 /*(-3)

 

12y -x =9

-9y +x = -3

-------------------- +

3y = 6 /:3

y=2

 

12*2 -x =9

-x = -15 /*(-1)

x = 15

 

x=15

y=2

 

h)

6x+7y = 0

-2x+3y =0 /*3

 

6x+7y =0

-6x +9y =0

------------------- +

 

16y = 0

y = 0

 

6x +7*0 =0

6x=0

x=0

 

x=0

y=0

 

i)

3x+2y =41/*4

5x-8y =17

 12x +8y = 164

5x-8y = 17

---------------------- +

17x = 181 /:17

x = 10 i 11/17

 

3 * 181/17 + 2y = 41

543/17 +2y = 41

2y = 9 i 1/17 /:2

y = 154/34

y = 4 i 18/34

y = 4 i 9/17

 

x= 10 i 11/17

y = 4 i 9/17

 

Answer 2

a)

{2x-y=4   /·2

{x+2y=7

 

{4x-2y=8

{x+2y=7

-------------------- +

5x=15

x=3

3+2y=7

2y=4

y=2

{x=3

{y=2

 

b)

{3x+y=-1   /·(-3)

{4x+3y=2

 

{-9x-3y=3

{4x+3y=2

-------------------- +

-5x=5

x=-1

3·(-1)+y=-1

-3+y=-1

y=2

{x=-1

{y=2

 

c)

{5x-y=1,5   /·6

{3x+6y=4,2

 

{30x-6y=9

{3x+6y=4,2

-------------------- +

33x=13,2

x=0,4

5·0,4-y=1,5

2-y=1,5

-y=-0,5

y=0,5

{x=0,4

{y=0,5

 

d)

{6x+2y=7   /·2

{9x-4y=0

 

{12x+4y=14

{9x-4y=0

-------------------- +

21x=14

x=2/3

12·2/3+4y=14

8+4y=14

4y=6

y=1,5

{x=2/3

{y=1,5

 

e)

{x+4y=1

{x-8y=-1   /·(-1)

 

{x+4y=1

{-x+8y=1

-------------------- +

12y=2

y=1/6

x+4·1/6=1

x+2/3=1

x=1/3

{x=1/3

{y=1/6

 

f)

{4x=-5y+10

{2x=6y+18

 

{4x+5y=10

{2x-6y=18   /·(-2)

 

{4x+5y=10

{-4x+12y=-36

-------------------- +

17y=-26

y=-1 i 9/17

4x+5·(-1 i 9/17)=10

4x-7 i 11/17=10

4x=17 i 11/17

x=4 i 7/17

{x=4 i 7/17

{y=-1 i 9/17

 

g)

{12y-x=9

{3y-1/3x=1   /·(-3)

 

{12y-x=9

{-9y+x=-3

-------------------- +

3y=6

y=2

12·2-x=9

24-x=9

-x=-15

x=15

{x=15

{y=2

 

h)

{6x+7y=0

{-2x+3y=0   /·3

 

{6x+7y=0

{-6x+9y=0

-------------------- +

16y=0

y=0

6x+0=0

6x=0

x=0

{x=0

{y=0

 

i)

{3x+2y=41   /·4

{5x-8y=17

 

{12x+8y=164

{5x-8y=17

-------------------- +

17x=181

x=10 i 11/17

3·10 i 11/17+2y=41

31 16/17+2y=41

2y=9 i 1/17

y=4 i 9/17

{x=10 i 11/17

{y=4 i 9/17