Exercicios de Matematica 9 Ano - Sistema de Equações - Exercicio 1

Exercicios de Matematica 9 Ano - Sistema de Equações

Resolva e classifique cada um dos sistemas.

$\left\{ \begin{gathered} 6x + 2y + 1 = 0 \hfill \\ x + \frac{y}{3} = \frac{{23}}{6} \hfill \\ \end{gathered} \right.$

$\left\{ \begin{gathered} 2x + 3y = - 7 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right.$

$\left\{ \begin{gathered} \frac{x}{3} + \frac{y}{2} = 1 \hfill \\ 3y = 6 - 2x \hfill \\ \end{gathered} \right.$

Resolução do Exercício de Matemática:

$\left\{ \begin{gathered} 6x + 2y + 1 = 0 \hfill \\ x + \frac{y}{3} = \frac{{23}}{6} \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 6x + 2y + 1 = 0 \hfill \\ x = \frac{{23}}{6} - \frac{y}{3} \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 6x + 2y + 1 = 0 \hfill \\ x = \frac{{23 - 2y}}{6} \hfill \\ \end{gathered} \right. \Leftrightarrow$

$\Leftrightarrow \left\{ \begin{gathered} 6\left( {\frac{{23 - 2y}}{6}} \right) + 2y + 1 = 0 \hfill \\ x = \frac{{23 - 2y}}{6} \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 23 - 2y + 2y + 1 = 0 \hfill \\ x = \frac{{23 - 2y}}{6} \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 24 = 0 \hfill \\ x = \frac{{23 - 2y}}{6} \hfill \\ \end{gathered} \right.$

Sistema impossível $S = \emptyset$

$\left\{ \begin{gathered} 2x + 3y = - 7 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 2x + 3y = - 7 \hfill \\ x = - y - 2 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 2\left( { - y - 2} \right) + 3y = - 7 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow$

$\Leftrightarrow \left\{ \begin{gathered} - 2y - 4 + 3y = - 7 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} - 2y + 3y = - 7 + 4 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = - 3 \hfill \\ x + y + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow$

$\Leftrightarrow \left\{ \begin{gathered} y = - 3 \hfill \\ x + \left( { - 3} \right) + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = - 3 \hfill \\ x - 3 + 2 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = - 3 \hfill \\ x - 1 = 0 \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = - 3 \hfill \\ x = 1 \hfill \\ \end{gathered} \right.$

Sistema possível e determinado $S = \left\{ {\left( {1, - 3} \right)} \right\}$

$\left\{ \begin{gathered} \frac{x}{3} + \frac{y}{2} = 1 \hfill \\ 3y = 6 - 2x \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \frac{x}{3} + \frac{y}{2} = 1 \hfill \\ y = \frac{{6 - 2x}}{3} \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \frac{x}{3} + \frac{{\frac{{6 - 2x}}{3}}}{2} = 1 \hfill \\ y = 2 - \frac{{2x}}{3} \hfill \\ \end{gathered} \right. \Leftrightarrow$

$\Leftrightarrow \left\{ \begin{gathered} \frac{x}{3} + \frac{{6 - 2x}}{6} = 1 \hfill \\ y = 2 - \frac{2}{3}x \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \frac{x}{3} - \frac{{2x}}{6} + \frac{6}{6} = 1 \hfill \\ y = 2 - \frac{2}{3}x \hfill \\ \end{gathered} \right. \Leftrightarrow$

$\Leftrightarrow \left\{ \begin{gathered} \frac{x}{3} - \frac{x}{3} + 1 = 1 \hfill \\ y = 2 - \frac{2}{3}x \hfill \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 0 = 0{\text{ }} \to {\text{condi\c{c}\~a o universal}} \hfill \\ y = 2 - \frac{2}{3}x \hfill \\ \end{gathered} \right.$