Question

Find the values of x and y:
x+y+xy=-5
x^2+y^2+x^2y^2=49

Answer 1

Can someone please help me?

Answer 2

yes
x+y = -5 - xy
square both sides, what do u get ?

Answer 3

x^2+y^2=25-x^2y^2

Answer 4

\((a+b)^2 = a^2+2ab+b^2 \)

Answer 5

so, (x+y)^2 =..?

Answer 6

x^2+y^2+2xy

Answer 7

what about (-5-xy)^2
?

Answer 8

25+x^2y^2+10xy

Answer 9

yes,
so you get
x^2+2xy+y^2 = 25 + x^2y^2 + 10xy
right ?

Answer 10

Yes

Answer 11

now from 2nd equation
x^2+y^2+x^2y^2=49
we get the value of x^2+y^2 as 49 - x^2y^2

Answer 12

so,
x^2+2xy+y^2 = 25 + x^2y^2 + 10xy
changes to
49 -x^2y^2 + 2xy = 25 +x^2y^2 +10xy

this equation is a Quadratic in 'xy'
like if you put p=xy

49 - p^2 +2p = 25 + p^2 +10p
can you solve this quadratic in 'p' ?
to get 2 values of p (which are actually 2 values of xy)

Answer 13

ooooohhhhh

Answer 14

let me give you steps , and see whether you get it

49 -25 +2p - 10 p - p^2-p^2 = 0
so,
24 - 8p -2p^2 = 0
which is
p^2 +4p -12 = 0

Answer 15

that quatic can be factored easily to get 2 values of p
p^2 + 6p - 2p -12 = 0
p(p+6) - 2 (p+6) = 0
(p-2)(p+6) = 0
p=2, -6
so the 2 values of xy are 2 and -6

Answer 16

**quadratic

Answer 17

so, x^2y^2 has 2 values as +4 and +36

Answer 18

using this :
x+y = -5 - xy
we can get 2 values of x+y too

Answer 19

and if we know
x+y and xy, we can get x and y easily
try it, if you don't get i'll help later (then you also ask if you have doubts in the above explanation)

Answer 20

Umm thanks. I'll try it again.

Answer 21

hello

Answer 22

Hello

Answer 23

could you get the values ?
which step are you stuck ?