Question

aflati a,b,c stiind ca 0,(3)a+0,25b=0,2c
3ab+5bc-4ac=304

Answer 1

$\it 0,(3) = \dfrac{\ 3^{(3}}{9} = \dfrac{1}{3} \\ \\ \\ 0,25 = \dfrac{\ \ 25^{(25}}{100} = \dfrac{1}{4} \\ \\ \\ 0,2=\dfrac{\ \ 2^{(2}}{10}=\dfrac{1}{5}$

Acum, prima relație din enunț devine:

$\it \dfrac{a}{3} = \dfrac{b}{4} =\dfrac{c}{5} \Rightarrow \begin{cases} \it a=\dfrac{3b}{4} \\ \\ \it c = \dfrac{5b}{4} \end{cases} \ \ \ (*)$

$\it 3ab+5bc-4ac =304 \stackrel{(*)}{\Longrightarrow} \dfrac{9b^2}{4} + \dfrac{25b^2}{4} - \dfrac{\ \ 60b^2^{(4}}{16} =304 \Rightarrow \\ \\ \\ \Rightarrow \dfrac{9b^2}{4} + \dfrac{25b^2}{4} - \dfrac{15b^2}{4} =304 \Rightarrow \dfrac{b^2}{4}(9+25 -15) =304\Rightarrow \\ \\ \\ \Rightarrow \dfrac{b^2}{4} \cdot 19 =304 |_{:19} \Rightarrow \dfrac{b^2}{4} = 16 \Rightarrow b^2=64 \Rightarrow b = \pm8$

$\it b=\pm8 \stackrel{(*)}{\Longrightarrow} a=\pm6,\ \ c = \pm10$