Question

Pe multimea numerelor reale se defineste legea de compozitie x*y=1/3 xy+x+y. Demonstrati ca x*y=1/3 (x+3)(y+3)-3, pentru orice numere reale x si y.

Answer 1

x*y = 1/3 xy + x +y

∀ x,y ∈ IR , "x*y" = 1/3 (x+3) (y+3) - 3

x*y = 1/3 (x+3) (y+3) - 3

x*y = 1/3 (xy + 3x + 3y + 9) - 3

*se efectuează înmulțirea:

x*y = 1/3 xy + x + y + 3 - 3

x*y = 1/3 xy + x + y

Answer 2

$\it x*y=\dfrac{1}{3}xy+x+y= \dfrac{1}{3}xy+\dfrac{3x}{3} +\dfrac{3y}{3} +\dfrac{9}{3} -3= \\ \\ \\ = \dfrac{1}{3}(xy+3x+3y+9) -3 =\dfrac{1}{3} [y(x+3)+3(x+3)]-3= \\ \\ \\ = \dfrac{1}{3}(x+3)(y+3) - 3.$