Κάτω το σύστημα , αλλά με δουλειά !

KARKAR · #1

Να λυθεί το σύστημα : $\left\{\begin{matrix} x^{2}+y^{2}+x+y & =112\\ xy+x+y& =21 \end{matrix}\right.$

#2

$\displaystyle{ \left\{ \begin{gathered} x^2 + y^2 + x + y = 112 \\ xy + x + y = 21 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \left( {x + y} \right)^2 - 2xy + x + y = 112 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \left( {x + y} \right)^2 - 2\left[ {21 - \left( {x + y} \right)} \right] + x + y = 112 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \Leftrightarrow }$

$\displaystyle{ \left\{ \begin{gathered} \left( {x + y} \right)^2 - 42 + 2\left( {x + y} \right) + \left( {x + y} \right) = 112 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \left( {x + y} \right)^2 + 3\left( {x + y} \right) - 154 = 0 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right.\mathop \Leftrightarrow \limits^{x + y = \omega } \left\{ \begin{gathered} \omega ^2 + 3\omega - 154 = 0 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \Leftrightarrow }$

$\displaystyle{ \left\{ \begin{gathered} \omega ^2 + 3\omega - 154 = 0 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right.\mathop \Leftrightarrow \limits^{\Delta = 9 + 4 \cdot 154 = 625 \Rightarrow \sqrt \Delta = 25} \left\{ \begin{gathered} \left\{ \begin{gathered} \omega = 11 \\ v \\ \omega = - 14 \\ \end{gathered} \right. \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \left\{ \begin{gathered} \omega = 11 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \\ v \\ \left\{ \begin{gathered} \omega = - 14 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \\ \end{gathered} \right.\mathop \Leftrightarrow \limits^{x + y = \omega } \left\{ \begin{gathered} \left\{ \begin{gathered} x + y = 11 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \\ v \\ \left\{ \begin{gathered} x + y = - 14 \\ xy = 21 - \left( {x + y} \right) \\ \end{gathered} \right. \\ \end{gathered} \right. }$ $\displaystyle{ \Leftrightarrow \left\{ \begin{gathered} \left\{ \begin{gathered} x + y = 11 \\ xy = 21 - 11 \\ \end{gathered} \right. \\ v \\ \left\{ \begin{gathered} x + y = - 14 \\ xy = 21 - \left( { - 14} \right) \\ \end{gathered} \right. \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \left( {\Sigma _1 } \right):\left\{ \begin{gathered} x + y = 11 \\ xy = 10 \\ \end{gathered} \right. \\ v \\ \left( {\Sigma _2 } \right):\left\{ \begin{gathered} x + y = - 14 \\ xy = 35 \\ \end{gathered} \right. \\ \end{gathered} \right. }$

Για το $\displaystyle{ \left( {\Sigma _1 } \right) }$ τα $\displaystyle{ x,y }$ είναι ρίζες τις εξίσωσης: $\displaystyle{ t^2 - St + p = 0\mathop \Leftrightarrow \limits^{S = x + y = 11,p = xy = 10} t^2 - 11t + 10 = 0 \Leftrightarrow \ldots \left\{ \begin{gathered} t = 1 \\ t = 10 \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \boxed{\left\{ \begin{gathered} x = 1 \hfill \\ y = 10 \hfill \\ \end{gathered} \right.} \\ v \\ \boxed{\left\{ \begin{gathered} x = 10 \hfill \\ y = 1 \hfill \\ \end{gathered} \right.} \\ \end{gathered} \right. }$

Για το $\displaystyle{ \left( {\Sigma _2 } \right) }$ τα $\displaystyle{ x,y }$ είναι ρίζες τις εξίσωσης: $\displaystyle{ t^2 - St + p = 0\mathop \Leftrightarrow \limits^{S = x + y = - 14,p = xy = 35} t^2 + 14t + 35 = 0 \Leftrightarrow \ldots \left\{ \begin{gathered} t_1 = \frac{{ - 14 - 2\sqrt {14} }} {2} \\ t_2 = \frac{{ - 14 + 2\sqrt {14} }} {2} \\ \end{gathered} \right. \Leftrightarrow \left\{ \begin{gathered} \boxed{\left\{ \begin{gathered} x = - 7 - \sqrt {14} \hfill \\ y = - 7 + \sqrt {14} \hfill \\ \end{gathered} \right.} \\ v \\ \boxed{\left\{ \begin{gathered} x = - 7 + \sqrt {14} \hfill \\ y = - 7 - \sqrt {14} \hfill \\ \end{gathered} \right.} \\ \end{gathered} \right. }$

Στάθης

Λάμπρος Ευσταθίου · #3

Στάθη στο Σ2, xy=35 αλλά οκ..

Επίσης, αν έθετες και το γινομενο xy=κάτι.. θα γινόταν πιο ευέλικτη η πορεία προς τη λύση.

Κάτω το σύστημα , αλλά με δουλειά !

Κάτω το σύστημα , αλλά με δουλειά !

Re: Κάτω το σύστημα , αλλά με δουλειά !

Re: Κάτω το σύστημα , αλλά με δουλειά !

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