SOLUTION: X^x+y^y=31 X+y=5. Find x and y

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Question 1058114: X^x+y^y=31
X+y=5. Find x and y
Found 3 solutions by stanbon, Boreal, ikleyn:
Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
X^x+y^y=31
X+y=5. Find x and y
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X = 5-y
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Substitute for "X" and solve for "y"::
(5-y)^x + y^y = 31
Answer by Boreal(15235)   (Show Source): You can put this solution on YOUR website!
x^2+y^2=31
x+y=5
y=(5-x)
x^2+(5-x)^2=31
x^2+25-10x+x^2=31
2x^2-10x-6=0
x^2-5x-3=0
x=(1/2)(5+/- sqrt(25+12)); sqrt (37)=6.083)
x=(1/2)(11.083)=5.54
x=(1/2)(-1.083)=-0.54
(5.54,-0.54) and (-0.54,5.54), correct within rounding error.
It is a line intersecting a circle centered on the origin with radius sqrt (31)
Answer by ikleyn(52776)   (Show Source): You can put this solution on YOUR website!
.
The system of two equations 

 +  = 31,
x + y = 5

Has two solutions:

     1)  x = 2,  y = 3

and/or

     2)  x = 3,  y = 2.

I don't know if it has other solutions.



Plots y = , y = 31

Looking in the plot above, it seems that there are no other solutions.


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