Question

Se da sistemul :
{x}+2{y}=2/3
3{y} -2{x} =13/6
Aflati x si y.

Answer 1

$\left\{ \begin{array}{c} \{x\}+2\{y\} = \dfrac{2}{3}\Big|\cdot2 \\ \\ 3\{y\} -2\{x\} = \dfrac{13}{6} \end{array} \right | (adunam): \Rightarrow 4\{y\}+3\{y\} + 0 = \dfrac{4}{3} + \dfrac{13}{6} \Rightarrow \\ \\ \Rightarrow 7\{y\} = \dfrac{21}{6} \Rightarrow 7\{y\} = \dfrac{7}{2} \Rightarrow \{y\} = \dfrac{\dfrac{7}{2}}{7} \Rightarrow \{y\} = \dfrac{1}{2}, (\frac{1}{2} = 0,5) \\ \\ \Rightarrow \boxed{y = \dfrac{1}{2}}$

$\\ Inlocuim in prima relatie fiindca e mai usor de calculat:\\ \\ \{x\} +2\cdot \dfrac{1}{2} = \dfrac{2}{3} \Rightarrow \{x\} +1 = \dfrac{2}{3} \Rightarrow \{x\} = \dfrac{2}{3} - 1 \Rightarrow \{x\} = -\dfrac{1}{3} \ \textless \ 0\Rightarrow \\ \\ \Rightarrow x\in \O \\ \\ \Rightarrow S = \O$

Answer 2

fie{x}=u si {y}=v
atunci
u+2v=2/3
-2u+3v=13/6

rezolvam prin metoda reducerii, inmultind prima relatie cu 2

2u+4v=4/3
-2u+3v=13/6


7v=4/3+13/6=8/6+13/6=21/6
v=3/6=1/2

u=2/3-2v=2/3-1=-1/3
ptca a dat u negativ, verrificam daca (-1/3;1/2) este solutia sistemului
-1/3+2*1/2=-1/3+1=2/3 adevarat

3*(1/2) -2*(-1/3)=3/2+2/3=9/6+4/6=13/6 adevarat
adevarate ambele, deci solutia siostemului  este (u;v)=(-1/3;1/2)

cum {x}∈[0;1) si -1/3<0, sistemul cu {x} si {y} NU are solutie
 (x;y)∈∅