[tex]a)~5x+6y=120 \Rightarrow 6y=120-5x \Leftrightarrow 6y=5(24-x). \\ \\ 5~si~6~sunt~prime~intre~ele \Rightarrow y ~ \vdots ~ 5. \\ \\ Deoarece~x,y \geq 0 ,~rezulta~6y \leq 120 \Rightarrow y \leq 20.[/tex]
[tex]b)~y ~ \vdots ~ 5 ~si~y-prim \Rightarrow y=5. \\ \\ Inlocuim: \\ \\ 5x+6y=120 \\ 5x+6*5=120 ~~~~~~~~~~~|:5 \\ x+6=24 \Rightarrow x=18. \\ \\ (x,y)=(18,5).[/tex]
[tex]5x+6y=120 \Leftrightarrow 5x+5y+y=120 \Leftrightarrow 5(x+y)=120-y \Rightarrow \\ x+y= \frac{120-y}{5} . \\ \\ y \leq 20 \Rightarrow -y \geq 20 \Rightarrow x+y \geq \frac{120-20}{5}=20................(1) \\ \\ y \geq 0 \Rightarrow -y \leq 0. \\ \\ x+y= \frac{120-y}{5} \leq \frac{120+0}{5}=24 ................(2) \\ \\ Din~(1)~si~(2) ~rezulta: \\ 20 \leq x+y \leq 24.[/tex]
