Συστηματική προσοχή

KARKAR · #1

Λύστε το σύστημα : $\begin{cases} x+\frac{2}{y} & \text= 3 \\ y+\frac{2}{x}& \text =\displaystyle \frac{8}{3} \end{cases}$

ΝΟΤΗΣ ΚΟΥΤΣΙΚΑΣ · #2

KARKAR έγραψε:Λύστε το σύστημα : $\begin{cases} x+\frac{2}{y} & \text= 3 \\ y+\frac{2}{x}& \text =\displaystyle \frac{8}{3} \end{cases}$

Καλημέρα σας κύριε Θανάση

$\displaystyle{\left\{ \begin{gathered} x + \frac{2}{y} = 3 \\ y + \frac{2}{x} = \frac{8}{3} \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \frac{{xy + 2}}{y} = 3 \\ \frac{{xy + 2}}{x} = \frac{8}{3} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} xy + 2 = 3y \\ \frac{{xy + 2}}{x} = \frac{8}{3} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} xy + 2 = 3y \\ \frac{{3y}}{x} = \frac{8}{3} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} xy + 2 = 3y \\ 8x = 9y \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} xy + 2 = 3y \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow }$

$\displaystyle{\left\{ \begin{gathered} \frac{{9y}}{8} \cdot y + 2 = 3y \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 9{y^2} + 16 = 24y \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow }$ $\displaystyle{\left\{ \begin{gathered} 9{y^2} - 24y + 16 = 0 \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} {\left( {3y - 4} \right)^2} = 0 \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} 3y - 4 = 0 \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow }$

$\displaystyle{\left\{ \begin{gathered} 3y = 4 \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = \frac{4}{3} \ne 0 \\ x = \frac{{9y}}{8} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = \frac{4}{3} \\ x = \frac{9}{8} \cdot \frac{4}{3} \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} y = \frac{4}{3} \\ x = \frac{3}{2} \ne 0 \\ xy \ne 0 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \boxed{y = \frac{4}{3}} \\ \boxed{x = \frac{3}{2}} \\ \end{gathered} \right.}$

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