Question

How to solve this 3 equations

Answer 1

I mean any tricks to solve

Answer 2

The second equation....that is indeed yz + yz right?

Answer 3

wait do we add those

Answer 4

\[\huge y ^{2}+yz+yx=-12\]

Answer 5

\[\huge x ^{2}+xy+xz=18\]

Answer 6

\[\huge z ^{2}+zx+zy=30\]

Answer 7

yeah smells like (x+y+z)^2
but how are ou gona find x,y,z values ? are you supposed to solve them in integers ?

Answer 8

yeah
the question is silent on that

Answer 9

By adding we get:-
\[\huge (x ^{2}+y ^{2}+z ^{2})+2(yz+zx+yx)=36\]

Answer 10

is this some kind of expansion

Answer 11

you can factor each into e.g. y(x+y+z)= -12
if you divide them you can get ratios, and put e.g. y and z in terms of x

Answer 12

\[\huge x+y+z=6\]

Answer 13

@phi i didn't get u

Answer 14

But how is x+y+z=6 helping

Answer 15

can it also be -6

Answer 16

\(\pm 6\)

Answer 17

example:
x/y = -3/2 so y= -⅔ x
x/z = 3/5 so z= 5/3 x

use those in
x(x+y+z)= 18

Answer 18

yeah

Answer 19

Yeah I believe there are 2 separate solutions for this

Answer 20

yeah there are 2 sollutions given

Answer 21

factor y out of the first equation
x out of the second
z out of the third

what do you get ?

Answer 22

or

x(x+y+z) = 18

pluging the two x+y+z valyes to get the two solutions for x

Answer 23

cool!

Answer 24

Nice application of the formula

Answer 25

i mean identity

Answer 26

THANKS ALL!!

Answer 27

i think 3 independent equations and 3 unknowns cannot have infinitely many solutions ! i was thinking there might be infiniftely many real solutions earlier which is just wrong as wolfram confirms it :

http://www.wolframalpha.com/input/?i=solve+y%5E2%2Byz%2Byx+%3D+-12%2C+x%5E2%2Bxy%2Bxz%3D18%2Cz%5E2%2Bzx%2Bzy%3D30

Answer 28

3 equations for 3 unknowns ,

Answer 29

just to finish this approach
example:
x/y = -3/2 so y= -⅔ x
x/z = 3/5 so z= 5/3 x

use those in
x(x+y+z)= 18
x(-⅔ x + 5/3 x)= 18
2x^2= 18
x= ± 3

Answer 30

they are like 3 surfaces intersecting one another exactly at two points or something

Answer 31

my brain was first thinking of finding out x in terms of x,y,z and then substitute that chunk of shi.t into the second equation Xd

Answer 32

I have learnt something that whenever there are some big looking equations , there is always a hidden trick to solve them

Answer 33

and
y= -⅔ x gives y= ± 2 (but the -2 goes with the +3 x solution)
z= 5/3 x gives z=± 5

(3,-2,5) and
(-3,2,-5)

Answer 34

yes they r same differ in sign though i got them