Question

a) 2x-y=3
9x+y=19

b) 2x-7y= -8
-2x+9y=12

c) 4x-7y=11
6x+7y= - 1

d) 2x+5y=0
-x-4y=3

e) 5x-y=6
4x+3y=1

f) x-2y= -2
2x-7y=-1

g) 4x-y=11
7x-5y=16

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Answer 1

Răspuns:

a)

2x-y=3

9x+y=19

_______ +

11x / = 22 =>x=2

2x-y=3 |•(-9)

9x+y=19 |•2

-18x+9y=(-27)

18x+2y=38

___________ +

/ 11y=11 =>y=1

b)

2x-7y=(-8)

-2x+9y=12

________ +

/ 2y=4 =>y=2

2x-7y=(-8) |•9

-2x+9y=12 |•7

18x-63y=(-72)

-14x+63y=84

___________ +

4x / = 12 =>x=3

c)

4x-7y=11

6x+7y=(-1)

________ +

10x / = 10 =>x=1

4x-7y=11 |•(-3)

6x+7y=(-1) |•2

-12x+21y=(-33)

12x+14y=(-2)

___________ +

/ 35y = (-35) =>y=(-1)

d)

2x+5y=0

-x-4y=3 |•2

2x+5y=0

-2x-8y=6

_______ +

/ -3y=6 =>y=(-2)

2x+5y=0 |•4

-x-4y=3 |•5

8x+20y=0

-5x-20y=15

_________ +

3x / = 15 =>x=5

e)

5x-y=6 |•3

4x+3y=1

15x-3y=18

4x+3y=1

________ +

19x / = 19 =>x=1

5x-y=6 |•(-4)

4x+3y=1 |•5

-20x+4y=(-24)

20x+15y=5

___________ +

/ 19y = (-19) =>y=(-1)

f)

x-2y=(-2) |•(-2)

2x-7y=(-1)

-2x+4y=4

2x-7y=(-1)

________ +

/ -3y = 3 =>y=(-1)

x-2y=(-2) |•(-7)

2x-7y=(-1) |•2

-7x+14y=14

4x-14y=(-2)

_________ +

-3x / = 12 =>x=(-4)

g)

4x-y=11 |•(-5)

7x-5y=16

-20x+5y=(-55)

7x-5y=16

____________ +

-13x / = (-39) =>x=3

4x-y=11 |•(-7)

7x-5y=16 |•4

-28x+7y=(-77)

28x-20y=64

___________ +

/ -13y = (-13) =>y=1

Answer 2

Răspuns:

Explicație pas cu pas:

• Metoda reducerii

a)

{ 2 x - y = 3

{ 9 x + y = 19

___________ adun relatiile

11 x = 22 =>    x = 22 : 11 =>   x = 2

2 x - y = 3

2 × 2 - y = 3

4 - y = 3

y = 4 - 3    =>   y = 1

Solutii   ( x; y) =  ( 2; 1 )

__________________________________________

b)

{  2 x - 7 y = - 8

{ - 2 x + 9 y = 12

_____________ adun relatiile

/   2 y = - 8 + 12

2 y = 4   =>     y = 4 : 2 =>  y = 2

2 x - 7 y = - 8

2 x - 7 × 2 = - 8

2 x - 14 = - 8

2 x = - 8 + 14

2 x = 6   =>   x = 6 : 2    =>   x = 3

Solutii   ( x; y) = ( 3; 2)

__________________________

c)

{  4 x - 7 y = 11

{  6 x + 7 y = - 1

___________ adun relatiile

10 x = 11 + ( - 1 )

10 x = 10   =>      x = 10 : 10 =>    x = 1

4 x - 7 y = 11

4 - 7 y = 11

- 7 y = 11 - 4

- 7 y = 7 =>    y = - 1

Solutii ( x; y) = ( 1;  - 1 )

___________________________

d)

{ 2 x + 5 y = 0

{  - x - 4 y =3   [ × 2

_________________

{ 2 x + 5 y = 0

{- 2 x - 8 y = 6

____________ adun relatiile

/  - 3 y = 6  =>   y = - 2

- x - 4 y = 3

- x - 4 × ( - 2 ) = 3

- x + 8 = 3

- x = 3 - 8   =>    x = 5

Solutii   ( x; y ) = ( 5; - 2)

_______________________________________________

e)

{ 5 x - y = 6   [ × 3

{ 4 x + 3 y = 1

____________

{ 15 x - 3 y = 18

{  4 x + 3 y = 1

____________ adun relatiile

19 x = 19   =>      x = 19 : 19 =>   x = 1

5 x - y = 6

5 - y = 6

- y = 6 - 5 =>    y = - 1

Solutii   ( x; y)  = ( 1; - 1)

__________________________________________________

f)

{  x - 2 y = - 2   [ × 2

{ 2 x - 7 y = - 1

____________

{ 2 x - 4 y = - 4

{ 2 x - 7 y = - 1

___________ scad relatiile

/ - 4 y - ( - 7 y ) = - 4 - ( - 1 )

- 4 y + 7 y = - 4 + 1

3 y = - 3   =>    y = - 1

x - 2 y = - 2

x + 2 = - 2

x = - 2 - 2      =>   x = - 4

Solutii  ( x; y) = ( - 4;  - 1 )  

_____________________________________________________

g)

{ 4 x - y = 11        [  × - 5

{ 7 x - 5 y = 16

____________

{  - 20 x + 5 y= - 55

{     7 x - 5 y = 16

____________ adun relatiile

- 13 x = - 39   =>    x = 3

4 x - y = 11

12 - y = 11

y = 12 - 11    =>      y = 1

Solutii  ( x; y) = ( 3; 1)

Verific:

4 x - y = 4×3 - 1 = 12 - 1 = 11 √

7 x - 5 y = 7 ×3 - 5 ×1 = 21 - 5 = 16 √