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

klipbodok6

klipbodok6

Answered question

2022-07-06

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

Answer & Explanation

Jamiya Costa

Jamiya Costa

Beginner2022-07-07Added 18 answers

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 y
Solving yields
    x + y = 3 ,       x y = 2         { x , y }   =   { 1 , 2 }
nidantasnu

nidantasnu

Beginner2022-07-08Added 7 answers

Looking for linear terms suggests we write the system as
( 1 1 x y x 2 y 2 x 3 y 3 ) ( u v ) = ( 2 1 1 5 ) .
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.

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