Question

Rezolvati in R × R sistemul (x^3 + y^3 =9
(x + y = 3

Answer 1

$\it x^3+y^3 = 9\ \ \ \ \ (1)\\ \\ x+y=3\ \ \ \ \ \ (2)\\ \\ (2) \Rightarrow (x+y)^3 = 3^3 \Rightarrow x^3+y^3+3xy(x+y) =27 \ \stackrel{(1),(2)}{\Longrightarrow}\ 9+3xy\cdot3=27\Rightarrow\\ \\ \Rightarrow 9+9xy=27 \Rightarrow 9xy=27-9 \Rightarrow 9xy=18 \Rightarrow xy=2 \ \ \ \ \ \ (3)$

$\it x+y=3 \Rightarrow y=3-x \stackrel{(3)}{\Longrightarrow} x(3-x)=2 \Rightarrow 3x-x^2=2 \Rightarrow 0=x^2-3x+2\Rightarrow\\ \\ \Rightarrow x^2-3x+2=0 \Rightarrow x^2-x-2x+2=0 \Rightarrow x(x-1)-2(x-1)=0 \Rightarrow\\ \\ \Rightarrow (x-1)(x-2)=0\Rightarrow \begin{cases}\it x-1=0 \Rightarrow x_1=1\\ \\ \it x-2=0 \Rightarrow x_2=2\end{cases}$