Let-R-fixed-Solve-for-real-numbers-ax-by-2-1-ax-2-by-2-4-1-ax-3-by-3-8-1-ax-4-by-4-16-1-

Question Number 150646 by mathdanisur last updated on 14/Aug/21

L et λ \in R fixed . S olve for real numbers :

{\begin{cases} ax + by = 2 λ + 1 \\ {ax}^{2} + {by}^{2} = 4 λ + 1 \\ {ax}^{3} + {by}^{3} = 8 λ + 1 \\ {ax}^{4} + {by}^{4} = 16 λ + 1 \end{cases}

Answered by MJS_new last updated on 14/Aug/21

looks complicated but indeed is very simple

(1) solve the 1^{st} for b

(2) solve the 2^{nd} for a

(3) solve the 3^{rd} for y ({it}^{'} s linear!)

(4) solve the 4^{th} for x

(x^{2} - 3 x + 2 = 0!)

\Rightarrow

a = b = 1

x = 1 \land y = 2 \lor x = 2 \land y = 1

Commented by mathdanisur last updated on 14/Aug/21

Thank you S er