Question

xy +2x-3y=16, Aflaţi x si y numere întregi

Answer 1

$xy+2x-3y=16 \\ x(y+2)-3y-6=10 \\ x(y+2)-3(y+2)=10 \\ (y+2)(x-3)=10 \Rightarrow (y+2) \in D_{10}~si~(x-3) \in D_{10}. \\ \\ (y+2) \in \{ \pm 1 ; \pm 2 ; \pm 5 ; \pm10 \}.$

$y+2=-10 \Rightarrow y=-12 \Rightarrow x-3=-1 \Rightarrow x=2. \\ y+2=-5 \Rightarrow y=-7 \Rightarrow x-3=-2 \Rightarrow x=1. \\ y+2=-2 \Rightarrow y=-4 \Rightarrow x-3=-5 \Rightarrow x=-2. \\ y+2=-1 \Rightarrow y=-3 \Rightarrow x-3=-10 \Rightarrow y=-7. \\ y+2=1 \Rightarrow y=-1 \Rightarrow x-3=10 \Rightarrow x=13. \\ y+2=2 \Rightarrow y=0 \Rightarrow x-3=5 \Rightarrow x=8. \\ y+2=5 \Rightarrow y=3 \Rightarrow x-3=2 \Rightarrow x=5. \\ y+2=10 \Rightarrow y=8 \Rightarrow x-3=1 \Rightarrow x=4.$

$\underline{Solutie}: (x,y) \in \{(2,-12);(1,-7);(-2,-4);(-7,-3);(13,-1); \\ ~~~~~~~~~~~~~~~~~~~~~~~~(8,0); (5,3);(4,8)\}.$