u + v = 2 u x + v y = 1 u x 2 + v y 2 = − 1 u x 3 + v y 3 = − 5Can anyone give me a hint for solving this ?

Question

u  +  v  =  2  u  x  +  v  y  =  1  u      x    2    +  v      y    2    =  −  1  u      x    3    +  v      y    3    =  −  5Can anyone give me a hint for solving this ?

Jamiya Costa · Accepted Answer

Exploit the innate symmetry:          u      x    2    +  v      y    2       =     (  x  +  y  )     (  u  x     +     v  y  )     −     x  y     (  u  +  v  )          u      x    3    +  v      y    3       =     (  x  +  y  )     (  u      x    2    +  v      y    2    )  −    x  y     (  u  x  +  v  y  )yields                   −  1     =        x  +  y     −     2     x  y                       5     =        x  +  y          +          x  ySolving yields            x  +  y  =  3  ,           x  y  =  2        ⇒        {  x  ,    y  }     =     {  1  ,    2  }

nidantasnu · Accepted Answer

Looking for linear terms suggests we write the system as

(\begin{matrix} 1 & 1 \\ x & y \\ x^{2} & y^{2} \\ x^{3} & y^{3} \end{matrix}) (\begin{matrix} u \\ v \end{matrix}) = (\begin{matrix} 2 \\ 1 \\ - 1 \\ - 5 \end{matrix}) .

Row-reduction has to give two entirely zero rows of coefficients,yielding two equations for

x

and

y

, one of which is linear in

y

. (It will help to recognize that

x - y

is a factor of

x^{2} - y^{2}

and of

x^{3} - y^{3}

.) That makes it straightforward to find

x

, from which you obtain y. With

x

and

y

in hand the coefficients are completely determined, allowing you to solve for

(u, v)

in the usual ways.

u + v = 2 u x + v y = 1 u x 2 </msu

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