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#1 2017-01-17 02:47:25

Mario23
Member
Registered: 2017-01-03
Posts: 27

Need help with these equations

Hey guys,here I am with another exercise I can't solve.
Find all the triplets of distinct real numbers (x,y,z) for which:




I remember this problem as an exercise for integers.There I would use that xyz=1 but how do I do it for reals?

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#2 2017-01-17 03:23:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Need help with these equations

Hi;

You will need a bit more to get a solution. Is there a constraint you left out?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2017-01-17 04:06:13

Mario23
Member
Registered: 2017-01-03
Posts: 27

Re: Need help with these equations

Well the only thing I did not include is that x,y and z are distinct from one another

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#4 2017-01-17 04:27:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Need help with these equations

That will not be good enough:

Here are 100 solutions, I am sure there are lots more...

{{x -> -(5645/10002), y -> -(10002/4357), z -> 4357/5645}, {x -> 1334,
   y -> -(1/1335), z -> -(1335/1334)}, {x -> -(3199/5001),
  y -> -(5001/1802), z -> 1802/3199}, {x -> 8015, y -> -(1/8016),
  z -> -(8016/8015)}, {x -> 7170, y -> -(1/7171),
  z -> -(7171/7170)}, {x -> 16, y -> -(1/17),
  z -> -(17/16)}, {x -> 1036, y -> -(1/1037),
  z -> -(1037/1036)}, {x -> -342, y -> 1/341,
  z -> -(341/342)}, {x -> 7533, y -> -(1/7534),
  z -> -(7534/7533)}, {x -> -8781, y -> 1/8780,
  z -> -(8780/8781)}, {x -> 684, y -> -(1/685),
  z -> -(685/684)}, {x -> -(2661/3334), y -> -(3334/673),
  z -> 673/2661}, {x -> -1960, y -> 1/1959,
  z -> -(1959/1960)}, {x -> -(1019/5001), y -> -(5001/3982),
  z -> 3982/1019}, {x -> 7750, y -> -(1/7751),
  z -> -(7751/7750)}, {x -> 8643, y -> -(1/8644),
  z -> -(8644/8643)}, {x -> -(26/1667), y -> -(1667/1641),
  z -> 1641/26}, {x -> 9659, y -> -(1/9660),
  z -> -(9660/9659)}, {x -> 2765, y -> -(1/2766),
  z -> -(2766/2765)}, {x -> 7307, y -> -(1/7308),
  z -> -(7308/7307)}, {x -> -5606, y -> 1/5605,
  z -> -(5605/5606)}, {x -> -3266, y -> 1/3265,
  z -> -(3265/3266)}, {x -> -(1037/5001), y -> -(5001/3964),
  z -> 3964/1037}, {x -> 7906, y -> -(1/7907),
  z -> -(7907/7906)}, {x -> -(1640/1667), y -> -(1667/27),
  z -> 27/1640}, {x -> 7264, y -> -(1/7265),
  z -> -(7265/7264)}, {x -> -9128, y -> 1/9127,
  z -> -(9127/9128)}, {x -> -4750, y -> 1/4749,
  z -> -(4749/4750)}, {x -> -847, y -> 1/846,
  z -> -(846/847)}, {x -> -(8881/10002), y -> -(10002/1121),
  z -> 1121/8881}, {x -> 5682, y -> -(1/5683),
  z -> -(5683/5682)}, {x -> -8386, y -> 1/8385,
  z -> -(8385/8386)}, {x -> -4467, y -> 1/4466,
  z -> -(4466/4467)}, {x -> -7428, y -> 1/7427,
  z -> -(7427/7428)}, {x -> -7210, y -> 1/7209,
  z -> -(7209/7210)}, {x -> -6297, y -> 1/6296,
  z -> -(6296/6297)}, {x -> -(497/5001), y -> -(5001/4504),
  z -> 4504/497}, {x -> -(3401/10002), y -> -(10002/6601),
  z -> 6601/3401}, {x -> -1157, y -> 1/1156,
  z -> -(1156/1157)}, {x -> -(8083/10002), y -> -(10002/1919),
  z -> 1919/8083}, {x -> 3292, y -> -(1/3293),
  z -> -(3293/3292)}, {x -> -(1300/1667), y -> -(1667/367),
  z -> 367/1300}, {x -> 1605, y -> -(1/1606),
  z -> -(1606/1605)}, {x -> 8865, y -> -(1/8866),
  z -> -(8866/8865)}, {x -> -(1457/1667), y -> -(1667/210),
  z -> 210/1457}, {x -> -(3991/5001), y -> -(5001/1010),
  z -> 1010/3991}, {x -> -(2422/5001), y -> -(5001/2579),
  z -> 2579/2422}, {x -> -5372, y -> 1/5371,
  z -> -(5371/5372)}, {x -> -(113/5001), y -> -(5001/4888),
  z -> 4888/113}, {x -> -(7949/10002), y -> -(10002/2053),
  z -> 2053/7949}, {x -> -(893/5001), y -> -(5001/4108),
  z -> 4108/893}, {x -> -9981, y -> 1/9980,
  z -> -(9980/9981)}, {x -> -1136, y -> 1/1135,
  z -> -(1135/1136)}, {x -> -4269, y -> 1/4268,
  z -> -(4268/4269)}, {x -> 2015, y -> -(1/2016),
  z -> -(2016/2015)}, {x -> -3587, y -> 1/3586,
  z -> -(3586/3587)}, {x -> -(1423/5001), y -> -(5001/3578),
  z -> 3578/1423}, {x -> -(2135/10002), y -> -(10002/7867),
  z -> 7867/2135}, {x -> -4795, y -> 1/4794,
  z -> -(4794/4795)}, {x -> -(3946/5001), y -> -(5001/1055),
  z -> 1055/3946}, {x -> -(2761/5001), y -> -(5001/2240),
  z -> 2240/2761}, {x -> -(1901/5001), y -> -(5001/3100),
  z -> 3100/1901}, {x -> 7228, y -> -(1/7229),
  z -> -(7229/7228)}, {x -> -2431, y -> 1/2430,
  z -> -(2430/2431)}, {x -> -(489/1667), y -> -(1667/1178),
  z -> 1178/489}, {x -> -(7021/10002), y -> -(10002/2981),
  z -> 2981/7021}, {x -> -1413, y -> 1/1412,
  z -> -(1412/1413)}, {x -> -(2090/5001), y -> -(5001/2911),
  z -> 2911/2090}, {x -> 1041, y -> -(1/1042),
  z -> -(1042/1041)}, {x -> -(2671/3334), y -> -(3334/663),
  z -> 663/2671}, {x -> 9561, y -> -(1/9562),
  z -> -(9562/9561)}, {x -> 9458, y -> -(1/9459),
  z -> -(9459/9458)}, {x -> -188, y -> 1/187,
  z -> -(187/188)}, {x -> -2658, y -> 1/2657,
  z -> -(2657/2658)}, {x -> 2014, y -> -(1/2015),
  z -> -(2015/2014)}, {x -> 8477, y -> -(1/8478),
  z -> -(8478/8477)}, {x -> -6491, y -> 1/6490,
  z -> -(6490/6491)}, {x -> -2633, y -> 1/2632,
  z -> -(2632/2633)}, {x -> -5836, y -> 1/5835,
  z -> -(5835/5836)}, {x -> -5822, y -> 1/5821,
  z -> -(5821/5822)}, {x -> -(6427/10002), y -> -(10002/3575),
  z -> 3575/6427}, {x -> -671, y -> 1/670,
  z -> -(670/671)}, {x -> 1761, y -> -(1/1762),
  z -> -(1762/1761)}, {x -> 6228, y -> -(1/6229),
  z -> -(6229/6228)}, {x -> -5397, y -> 1/5396,
  z -> -(5396/5397)}, {x -> -191, y -> 1/190,
  z -> -(190/191)}, {x -> -(4235/10002), y -> -(10002/5767),
  z -> 5767/4235}, {x -> 7467, y -> -(1/7468),
  z -> -(7468/7467)}, {x -> -7228, y -> 1/7227,
  z -> -(7227/7228)}, {x -> -5515, y -> 1/5514,
  z -> -(5514/5515)}, {x -> 3703, y -> -(1/3704),
  z -> -(3704/3703)}, {x -> 4663, y -> -(1/4664),
  z -> -(4664/4663)}, {x -> -(9941/10002), y -> -(10002/61),
  z -> 61/9941}, {x -> -4319, y -> 1/4318,
  z -> -(4318/4319)}, {x -> -9316, y -> 1/9315,
  z -> -(9315/9316)}, {x -> 7160, y -> -(1/7161),
  z -> -(7161/7160)}, {x -> 3059, y -> -(1/3060),
  z -> -(3060/3059)}, {x -> 2269, y -> -(1/2270),
  z -> -(2270/2269)}, {x -> -3399, y -> 1/3398,
  z -> -(3398/3399)}, {x -> 6726, y -> -(1/6727), z -> -(6727/6726)}}

So

Find all the triplets of distinct real numbers (x,y,z) for which:

will be difficult to do in a post. I could write the x,y relationships that would.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#5 2017-01-17 05:03:19

Mario23
Member
Registered: 2017-01-03
Posts: 27

Re: Need help with these equations

Oh my God that's a lot of numbers.Which are the relationships you are thinking at?

Last edited by Mario23 (2017-01-17 05:07:53)

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#6 2017-01-17 08:38:17

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Need help with these equations

I can solve in terms of x:

y=1/(-x-1) and z->(-x-1)/x

So, there are infinite number of solutions.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2017-01-17 09:24:48

Mario23
Member
Registered: 2017-01-03
Posts: 27

Re: Need help with these equations

Yeah that's what I tried to do too.It seems that the equations have an infinity of solutions.I thought I was missing something big_smile
Just wondering,how did you find all those numbers above?

Last edited by Mario23 (2017-01-17 09:25:26)

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#8 2017-01-17 09:46:34

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Need help with these equations

By computer and using a CAS on that computer.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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