Question

20 Rezolvați prin metoda substituției următoarele sisteme de ecuații:​

Answer 1

a) {2x + 3y = 5 + 2y {x = 5/2 - 1/2y

{4x + 2y = 7 + x {4x + 2y = 7 + x

4(5/2 - 1/2y) + 2y = 7 + 5/2 - 1/2y

10 = 19/2 - 1/2y

20 = 19 - y

y = 19 - 20 = -1

x = 5/2 - 1/2y = 5/2 - 1/2 × (-1) = 5/2 + 1/2× 1

x = 5/2 + 1/2 = 6/2 = 3

___________________________

b) {3(x - 1) + y = -5 {y = -2 - 3x

{2x + 2(y + 1) = 6 {2x + 2(y + 1) = 6

2x + 2 × (-2 - 3x + 1) = 6

2x - 2 - 6x = 6

-4x - 2 = 6

-4x = 6 + 2

-4x = 8

x = 8 : (-4) = -2

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

___________________________

c) {x + 2 = 16 - 4y {x + 2 = 16 - 4y

{-3x = 4y - 34 {4y = -34 - 3x

x + 2 = 16 - (34 - 3x)

x + 2 = -18 + 3x

x - 3x = -18 - 2

-2x = -20

x = -20 : (-2) = 10

4y = -34 - 3x

4y = -34 - 3 × 10

4y = 4

y = 4 : 4 = 1

____________________________

d) {4(x + 5) - 2 = 5(y + 3) {4(x + 5) - 2 = 5(y + 3)

{-x + 2(y + 4) = 11 {x = - 3 + 2y

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

8 + 8y - 2 = 5y + 15

6 + 8y = 5y + 15

8y - 5y = 15 - 6

3y = 9

y = 9 : 3 = 3

x = - 3 + 2y = -3 + 2 × 3 = -3 + 6 = 3

____________________________

e) {2(x + 4) = -3(y - 5) {2x + 8 = -3y + 15

{x = 2(y + 1) - 16 {x = 2y - 14

2(2y - 14) + 8 = -3y + 15

4y - 20 = -3y + 15

4y + 3y = 15 + 20

7y = 35

y = 35 : 7 = 5

x = 2y - 14 = 2 × 5 - 14 = 10 - 14 = -4

____________________________

f) {2y = 4 - 3x

{-3x = y + 1

2y = 4 + y + 1

2y = 5 + y

2 y - y = 5

y = 5

-3x = y + 1

-3x = 5 + 1

-3x = 6

x = 6 : (-3) = -2