Question

determinati nr. a,b,c stiind ca a+b/5=b+c/7=c+a/8 si 5a +3b+2c=62

Answer 1

   
[tex]\displaystyle\\ \frac{a+b}{5}=\frac{b+c}{7}=\frac{c+a}{8}=k\\\\ a+b=5k\\ b+c=7k\\ a+c = 8k\\\\ 5a +3b+2c=62~~\Big|~5a = 3a + 2a \\ 3a + 2a+3b+2c=62\\ 3a +3b + 2a+2c=62\\ 3(a+b)+2(a+c)=62\\\\ 3\times 5k+2\times 8k = 62\\ 15k+16k=62\\ 31k=62\\\\ k= \frac{62}{31}=\boxed{2} \\\\ a+b=5k=5\times 2=10\\ b+c=7k=7\times 2=14\\ a+c = 8k=8\times 2=16\\\\ [/tex]


[tex]\displaystyle\\ \begin{cases} a+b=10\\ ~~~~~b+c=14\\ a~~~~~+c =16 \end{cases}~~~~~ \Big\}\text{Din ecuatia a 3-a scadem ecuatia a 2-a.}\\ --------\\ \begin{cases} a+b=10\\ a-b =2 \end{cases}~~~~~~~~~\Big\}\text{Adunam ecuatiile}\\ -------\\ 2a~~~~/=12\\\\ a = \frac{12}{2} =\boxed{\bf 6}\\ a+b = 10\Longrightarrow ~b = 10 - a = 10 - 6 = \boxed{\bf 4}\\ b+c=14\Longrightarrow ~c = 14 - b = 14 - 4 = \boxed{\bf 10}\\\\ \text{Solutia:}\\ \boxed{\bf a=6}\\ \boxed{\bf b=4}\\ \boxed{\bf c=10}[/tex]