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Compare them (Posted on 2018-09-08) Difficulty: 3 of 5
Two positive distinct numbers fulfill the equation ba- ab=1

Which is larger:a or b?

For what (x,y) couple are both terms i.e xy and yx equal?

No Solution Yet Submitted by Ady TZIDON    
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Graphical solution | Comment 2 of 7 |
https://www.desmos.com/calculator/ly4es9gpq2

The first equation, in red, shows two distinct branches.  One with x<y and one with x>y.  This implies either a or b could be larger.

It is also interesting to note that for any value of a>1 there appear to be two values of b.  For example, if a=3, b can be either 2 or about 3.221.  

The same is true for b, except it cannot be between about 2.018 and 3.196.  Also if b is in the lower range both a's will be larger and visa versa.

The second equation, in green, shows all solutions for x^y=y^x with x=1 and y=1 as asymptotes.  It can also be that x=y, but this is partially obscured by the third equation.

The third equation is x=y, a blue dotted line separate the regions for the first question.


Edited on September 8, 2018, 5:14 pm
  Posted by Jer on 2018-09-08 17:06:13

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