Need help with these equations

Mario23 · 2017-01-17 02:47:25

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?

bobbym · 2017-01-17 03:23:49

Hi;

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

Mario23 · 2017-01-17 04:06:13

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

bobbym · 2017-01-17 04:27:13

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.

Mario23 · 2017-01-17 05:03:19

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)

bobbym · 2017-01-17 08:38:17

I can solve in terms of x:

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

So, there are infinite number of solutions.

Mario23 · 2017-01-17 09:24:48

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
Just wondering,how did you find all those numbers above?

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

bobbym · 2017-01-17 09:46:34

By computer and using a CAS on that computer.

Math Is Fun Forum

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

Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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

Re: Need help with these equations

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