Question

sa se rezolve sistemele
1.{x+y-2=0
x^2-2x+y=0

2.{x+y=3
x-y=1

3.{2x+2y=16
xy=12

4.x+y= -6
xy=8

Answer 1

1. x=2-y
x^2=(2-y)^2=4+y^2-4y
4+y^2-4y-2(2-y)+y=0
y^2-4y+2y+4-4=0
y^2-2y=0
y(y-2)=0
are 2 solutii:
I. y=0, x=2-0=2
II. y=2, x=2-2=0

2. x+y=3
    x-y=1
adun cele 2 ecuatii
2x=4, x=4:2=2
2+y=3, y=3-2=1

3. 2(x+y)=16, x+y=16:2=8, x=8-y
    xy=12
    y(8-y)=12
    8y-y^2-12=0
inmultesc cu -1
y^2-8y+12=0
y^2-2y-6y+12=0
y(y-2)-6(y-2)=0
(y-2)(y-6)=0
sunt 2 solutii:
I. y=2, x=8-2=6
II. y=6, x=8-6=2

4. x=-6-y
y(-6-y)=8
-6y-y^2-8=0
inmultesc cu -1
y^2+6y+8=0
y^2+2y+4y+8=0
y(y+2)+4(y+2)=0
(y+2)(y+4)=0
sunt 2 solutii:
y= - 2 si x= -6- (-2)=-6+2=-4
y=-4 si x=-6-(-4)=-6+4=-2