How do we solve this system of equations? a , b ∈ R and a 5 b − b 5 a a − b = 30and a 5 + b 5 = 33I get that a 6 − b 6 = ( a − b ) 63 But I have no idea how to solve after that.

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How do we solve this system of equations?  a  ,  b  ∈  R and                    a        5            b      −              b        5            a              a      −      b        =  30and      a    5    +      b    5    =  33I get that       a    6    −      b    6    =  (  a  −  b  )  63 But I have no idea how to solve after that.

Vitulloh0 · Accepted Answer

a  =  2  ,  b  =  1works. So does  a  =  1  ,  b  =  2.If you draw a graph of       x    5    +      y    5    =  C  &amp;gt;  0 you find that   x  +  y  &amp;gt;  0.. Thus, with  a  b  (  a  +  b  )  (      a    2    +      b    2    )  =  30we find   a  b  &amp;gt;  0. Then, with       a    5    +      b    5    =  33  , we have both positive.Writing  a  =  r  cos  ⁡  θ  ,    b  =  r  sin  ⁡  θand pulling out       r    5    , we get  33  cos  ⁡  θ  sin  ⁡  θ  (  cos  ⁡  θ  +  sin  ⁡  θ  )  =  30  (      cos    5    ⁡  θ  +      sin    5    ⁡  θ  )  .Taking second derivatives, the left hand side has negative second derivative on   0  &amp;lt;  θ  &amp;lt;  π  /  2.. On the same range, the right hand side has negative first derivative until   θ  =  π  /  4  ,, after which it has positive first derivative. As a result, there are at most two points of equality, and we have already found those.

How do we solve this system of equations? a , b ∈ <mi mathvariant=

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