Question

Soluçao da equação vetorial : $5~ \vec{x}~+ 2~ \vec{a}~-~ \vec{b}~ = ~ 2~\vec{x}~-4~ \vec{x}~+2~ \vec{b}~, na~ variavel ~ \vec{x}~.$

resposta final é $\vec{x}~=~-2~\vec{a}~+\vec{b}~$

gostaria de ver a conta completa passo a passo.

Answer 1

$\mathtt{5\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=2\overrightarrow{\mathtt{x}}-4\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{5\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=(2-4)\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{5\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=-2\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}$


Somando o vetor $\mathtt{2\overrightarrow{\mathtt{x}}}$ aos dois lados da igualdade acima,

$\mathtt{2\overrightarrow{\mathtt{x}}+5\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=2\overrightarrow{\mathtt{x}}-2\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{(2+5)\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=(2-2)\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=0\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=\overrightarrow{\mathtt{0}}+2\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}=2\overrightarrow{\mathtt{b}}}$


Somando $\mathtt{\overrightarrow{\mathtt{b}}}$ aos dois lados da igualdade acima,

$\mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-\overrightarrow{\mathtt{b}}+\overrightarrow{\mathtt{b}}=2\overrightarrow{\mathtt{b}}+\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}+(-1+1)\overrightarrow{\mathtt{b}}=(2+1)\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}+0\overrightarrow{\mathtt{b}}=3\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}+\overrightarrow{\mathtt{0}}=3\overrightarrow{\mathtt{b}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}=3\overrightarrow{\mathtt{b}}}$


Somando o vetor $\mathtt{-2\overrightarrow{\mathtt{a}}}$ aos dois lados da equação,

$\mathtt{7\overrightarrow{\mathtt{x}}+2\overrightarrow{\mathtt{a}}-2\overrightarrow{\mathtt{a}}=3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+(2-2)\overrightarrow{\mathtt{a}}=3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+0\overrightarrow{\mathtt{a}}=3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}+\overrightarrow{\mathtt{0}}=3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}}\\\\ \mathtt{7\overrightarrow{\mathtt{x}}=3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}}$


Multiplicando os dois lados por $\mathtt{\dfrac{1}{7}},$

$\mathtt{\dfrac{1}{7}\cdot 7\overrightarrow{\mathtt{x}}=\dfrac{1}{7}\cdot \left(3\overrightarrow{\mathtt{b}}-2\overrightarrow{\mathtt{a}}\right)}\\\\\\ \mathtt{\left(\dfrac{1}{7}\cdot 7 \right )\!\!\overrightarrow{\mathtt{x}}=\dfrac{1}{7}\cdot 3\overrightarrow{\mathtt{b}}-\dfrac{1}{7}\cdot 2\overrightarrow{\mathtt{a}}}\\\\\\ \mathtt{1\overrightarrow{\mathtt{x}}=\dfrac{3}{7}\overrightarrow{\mathtt{b}}-\dfrac{2}{7}\overrightarrow{\mathtt{a}}}\\\\\\\\ \boxed{\begin{array}{c} \mathtt{\overrightarrow{\mathtt{x}}=\dfrac{3}{7}\overrightarrow{\mathtt{b}}-\dfrac{2}{7}\overrightarrow{\mathtt{a}}} \end{array}}$


Dúvidas? Comente.


Bons estudos! :-)