Ensino Médio ⇒ Sistema de Equações Não-Lineares Tópico resolvido

[mvrasseli]

Qual o valor de [tex3]a\,+\,b\,+\,c,[/tex3] se [tex3]a\,+\,ab\,=\,1,\, b\,+\,bc\,=\,1[/tex3] e [tex3]c\,+\,ac\,=\,1[/tex3].

[Alexandre_SC]

• [tex3]\left\{\begin{aligned}
  &b(1+c)=1\Longrightarrow b=\frac{1}{1+c}\\
  &a(1+b)=1\Longrightarrow a=\frac{1}{1+b}=\frac{1+c}{2+c}\\
  &c(1+a)=1\Longrightarrow c=\frac{1}{1+a}=\frac{2+c}{3+2c}\Longrightarrow c^2+c-1=0\Longrightarrow c_1=\frac{\sqrt{5}-1}{2}\text{ e }c_2=\frac{-\sqrt{5}-1}{2}
  \end{aligned}\right.[/tex3]

• [tex3]a+b+c= \frac{1+c}{2+c} + \frac{1}{1+c} + c = \frac{5}{2+c}[/tex3]

• [tex3]\left\{\begin{array}{l}
  c_1=\frac{\sqrt{5}-1}{2}\Longrightarrow a+b+c=\frac{5\cdot (3-\sqrt{5})}{2}\\
  c_2 = \frac{-\sqrt{5}-1}{2}\Longrightarrow a+b+c=\frac{5\cdot (3+\sqrt{5})}{2}
  \end{array}\right.[/tex3]