a)
[tex] \it a+b+c = 130\ \ \ \ \ \ (1)\\ \\ a = 2b\ \ \ \ \ (2)\\ \\ a = c-15 \stackrel{(2)}{\Longrightarrow} 2b=c-15 \Rightarrow c = 2b+15 \ \ \ \ \ \ (3)\\ \\ (1),\ (2),\ (3) \Rightarrow 2b+b+2b+15=130 \Rightarrow 5b+15=130 \Rightarrow \\ \\ \Rightarrow 5b = 130-15 \Rightarrow 5b = 115 \Rightarrow b= 115:5 \Rightarrow b = 23 [/tex]
[tex] \it b=23\stackrel{(2)}{\Longrightarrow} a=2\cdot23 \Rightarrow a = 46\\ \\ b=23\stackrel{(3)}{\Longrightarrow} c=2\cdot23 +15 \Rightarrow c = 61\\ \\ Numerele\ \ cerute\ \ sunt: a=46, \ \ b=23,\ \ c = 61. [/tex]
b)
[tex] \it a+b+c=130 \ \ \ \ \ (1)\\ \\ a<b\ de\ 3\ ori \Rightarrow b>a\ de\ 3\ ori \Rightarrow b =3a\ \ \ \ (2)\\ \\ c=2b \stackrel{(2)}{\Longrightarrow} c=2\cdot3a\Rightarrow c = 6a\ \ \ \ (3)\\ \\ (1),\ (2),\ (3) \Rightarrow a+3a+6a= 130 \Rightarrow 10a=130 \Rightarrow a = 13 [/tex]
[tex] \it a=13\stackrel{(2)}{\Longrightarrow} b = 3\cdot13 \Rightarrow b = 39\\ \\ a=13\stackrel{(2)}{\Longrightarrow} c = 6\cdot13 \Rightarrow c = 78\\ \\ Numerele \ \ cerute\ \ sunt: a = 13,\ \ b=39,\ \ c=78. [/tex]
c)
[tex] \it \ a+b+c = 130\ \ \ \ \ \ (1)\\ \\ \dfrac{a}{2} =2b \Rightarrow a= 2\cdot2b \Rightarrow a=4b\ \ \ \ \ \ (2)\\ \\ b<c\ cu\ 4 \ \Rightarrow c>b\ cu\ 4 \Rightarrow c=b+4\ \ \ \ \ \ (3)\\ \\ (1),\ (2),\ (3) \Rightarrow 4b+b+b+4=130 \Rightarrow 6b+4=130|_{-4} \Rightarrow\\ \\ \Rightarrow 6b=126|_{:6} \Rightarrow b = 21 [/tex]
[tex] \it b=21\stackrel{(2)}{\Longrightarrow} a=4\cdot21 \Rightarrow a = 84
\\ \\
b=21\stackrel{(2)}{\Longrightarrow} c=21 +4 \Rightarrow c = 25
\\ \\
Numerele\ \ cerute\ \ sunt: a=84,\ \ b=21,\ \ c=25.
[/tex]
