Question

1.resolva:

a){2x-y²=1
3x+y=4
b){ x+6y=3
x.y=-9
c){x-y=8
x+y²=14
d)x+y=5/6
x²+3y=5/4

Answer 1

a)
2x - y²=1
3x + y = 4

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

2x - y^2 = 1
2x - (4 - 3x)^2 = 1
2x - (16 - 2.4.3x + 9x^2) = 1
2x - (16 - 24x + 9x^2) = 1
2x - 16 + 24x - 9x^2 = 1
-9x^2 + 26x - 16 - 1 = 0
- 9x^2 + 26x - 17 = 0

a = - 9; b = 26; c = - 17
∆ = b^2-4ac
∆ = 26^2 - 4.(-9).(-17)
∆ = 676 + 36.(-17)
∆ = 676 - 612
∆ = 64

x = [ - b +/- √∆]/2a
x = [ - 26 +/- √64]/2.(-9)
x = [ - 26 +/- 8]/(-18)
x ' = [- 26 + 8]/(-18) = (-18)/(-18) = 1
x " = [- 26 - 8]/(-18) = -34/(-18) = 34/18 (:2)/(:2) = 17/9

x = 1
y = 4 - 3x
y = 4 - 3.1
y = 4 - 3
y = 1

x = 17/9
y = 4 - 3x
y = 4 - 3.17/9
y = 4 - 51/9
y = (9.4 - 51)/9
y = (36 - 51)/9
y = - 15/9

R.: (1 ; 1) ou (17/9 ; - 15/9)

b)
x + 6y = 3
x.y = - 9
x = 3 - 6y
x.y = - 9
(3 - 6y).y = - 9
3y - 6y^2 = - 9
0 = 6y^2 - 3y - 9
6y^2 - 3y - 9 = 0
a = 6; b = - 3; c = - 9
∆ = b^2 - 4ac
∆ = (-3)^2 - 4.6.(-9)
∆ = 9 + 24.9
∆ = 9 + 216
∆ = 225

y = [ - b +/- √∆]/2a
y = [ -(-3) +/- √225]/2.6
y = [3 +/- 15]/12
y' = [3 - 15]/12 = - 12/12 = - 1
y" = [3 + 15]/12 = 18/12 (:6)/(:6) = 3/2
y = - 1
x = 3 - 6y
x = 3 - 6.(-1)
x = 3 + 6
x = 9

y = 3/2
x = 3 - 6y
x = 3 - 6.3/2
x = 3 - 9
x = - 6

R.: ( - 6 ; 3/2) ou ( 9 ; - 1)

c)
x - y = 8
x + y²=14

x = 8 + y
x + y^2 = 14
8 + y + y^2 = 14
y^2 + y + 8 - 14 = 0
y^2 + y - 6 = 0
a= 1; b = 1; c = - 6

∆ = b^2-4ac
∆ = 1^2 - 4.1.(-6)
∆ = 1 + 24
∆ =25

y = [- b +/- √∆]/2a
y = [- 1 +/- √25]/2.1
y = [ - 1 +/- 5]/2
y = [ - 1 - 5]/2 = - 6/2 = - 3
y = [- 1 + 5]/2 = 4/2 = 2

y = - 3
x = 8 + y
x = 8 - 3
x = 5

y = 2
x = 8 + y
x = 8 + 2
x = 10

R.: (10 ; 2) ou (5 ; - 3)

d)
x + y = 5/6 (-3)
x² + 3y = 5/4

- 3x - 3y = - 15/6
x^2 + 3y = 5/4 (+)
--------------------------
x^2 - 3x = - 15/6 (:3)/(:3) + 5/4
x^2 - 3x = - 5/2 + 5/4
(4x^2 - 4.3x)/4 = (-2.5 + 5)/4
4x^2 - 12x = - 10 + 5
4x^2 - 12x = - 5
4x^2 - 12x + 5 = 0
a= 4; b = - 12; c = 5
∆ = b^2-4ac
∆ = (-12)^2 - 4.4.5
∆ = 144 - 80
∆ = 64

x = [ - b +/- √∆]/2a
x = [- (-12) +/- √64]/2.4
x = [12 +/- 8]/8
x' = [12 + 8]/8 =20/8(:4)/(:4) = 5/2
x" = [12 - 8]/8 = 4/8(:4)/(:4) = 1/2

x = 1/2
x + y = 5/6
1/2 + y = 5/6
y = 5/6 - 1/2
y = (5 - 3)/6
y = 2/6 (:2)/(:2)
y = 1/3

x = 5/2
x + y = 5/6
5/2 + y = 5/6
y = 5/6 - 5/2
y = (5 - 3.5)/6
y = (5-15)/6 = -10/6(:2)/(:2)= -5/3

R.: (1/2 ; 1/3) ou (5/2 ; - 5/3)