% systèmes linéaires
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%%%     Feuille d'exercices de mathématiques au format LaTeX     %%%
%%%     créée le Wed, 08 Oct 2008 11:44:05 +0200                 %%%
%%%     sur http://allken-bernard.org/pierre/phpmyexercices      %%%
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% Apparence des exercices
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\begin{document}
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\pagestyle{fancy}
\lhead{\textbf{\today}} 
\chead{\textbf{\thetitle}}
\rhead{\textbf{\thepage/\pageref{fin}}}
\lfoot{\small \textit{Lycée Joachim du Bellay}}
\cfoot{\small \textit{Mathématiques, prépa ECE1}}
\rfoot{\small \textit{http://allken-bernard.org/pierre/ece}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 286
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(A)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&z& 1\\
&&2y&+&2z&1\\
&&&&4z&1
\end{array}\right.\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 287
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(B)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&z& 1\\
3x&-&2y&+&2z&1\\
&&&&4z&-2
\end{array}\right.\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 288
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(C)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&z& 5\\
&&2y&+&2z&1\\
\end{array}\right.\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 289
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(D)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&z& 1\\
&&&&2z&3\\
\end{array}\right.\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 290
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(E)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
4x&+&y&-&3z& 2\\
&&&&0&1\\
\end{array}\right.\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 291
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]On considère le système linéaire~:
\[(F)\quad\begin{array}{rrrrr @{\quad =\quad} r}
x&+&2y&+&2z& 1
\end{array}\]
\begin{enumerate}
\item Est-il échelonné~?
\item Si oui quelles sont les inconnues principales et quelles sont les inconnues secondaires~?
\item Résoudre le système.
\end{enumerate}
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 292
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre le système échelonné suivant~:
\[(S)\quad\left\{\begin{array}{rrrrrrr @{\quad =\quad} r}
3x&+&y&+&z&+&2t& 1\\
&&y&-&2z&+&t&3\\
&&&&4z&-&t&2\\
&&&&&&2t&-4
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 293
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
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\begin{exercice}
\nopagebreak[5]Résoudre le système échelonné suivant~:
\[(S)\quad\left\{\begin{array}{rrrrrrr @{\quad =\quad} r}
x&-&y&+&z&+&2t& -1\\
&&2y&-&2z&+&t&1\\
&&&&2z&-&t&2\\
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 294
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre le système échelonné suivant~:
\[(S)\quad\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&z& 1
\end{array}\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 295
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrr @{\quad =\quad} r}
2x&+&3y& 1\\
5x&+&7y&3
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 296
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrr @{\quad =\quad} r}
2x&+&4y& 10\\
3x&+&6y&15
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 297
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrr @{\quad =\quad} r}
4x&-&2y& 5\\
-6x&+&3y&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 298
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrr @{\quad =\quad} r}
2x&+&3y& 3\\
x&-&2y& 5\\
2x&+&y&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 299
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
2x&+&3y&-&2z& 5\\
2x&+&3y&+&8z&4\\
4x&-&y&+&4z&0
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 300
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&2y&+&2z& 2\\
3x&+&2y&-&z&5\\
2x&-&5y&+&3z&-4\\
x&+&4y&+&6z&0
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 301
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre~:
\[\left\{\begin{array}{rrrrrrrrr @{\quad =\quad} r}
x&-&y&+&2z&+&3t&+&u& 13\\
x&+&y&+&2z&+&7t&+&3u& 25\\
-x&+&4y&-&5z&+&12t&-&4u& 2\\
2x&-&4y&+&5z&+&t&&& 13\\
4x&-&3y&+&4z&+&23t&+&9u& 84\\
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 302
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre le système suivant, $k$ étant un paramètre~:
\[(S)\quad\left\{\begin{array}{rrr @{\quad =\quad} r}
kx&+&y& 1\\
x&+&ky&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 303
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Donner le nombre de solutions du système suivant en fonction du paramètre $k$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&kz& 2\\
3x&+&4y&+&2z&k\\
2x&+&3y&-&z&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 304
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Donner le nombre de solutions du système suivant en fonction du paramètre $a$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&-&z& 1\\
2x&+&3y&+&az&3\\
x&+&ay&+&3z&2
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 305
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Donner le nombre de solutions du système suivant en fonction des paramètres $a$, $b$, $c$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&2y&-&3z& a\\
2x&+&6y&-&11z&b\\
x&-&2y&+&7z&c
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 306
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre en fonction du paramètre $k$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&kz& 2\\
3x&+&4y&+&2z&k\\
2x&+&3y&-&z&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 307
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre en fonction du paramètre $k$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&+&y&+&kz& 1\\
x&+&ky&+&z&1\\
kx&+&y&+&kz&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 308
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre en fonction du paramètre $k$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&&&-&3z& -3\\
2x&+&ky&-&z&-2\\
kx&+&y&+&kz&1
\end{array}\right.\]
\end{exercice}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%     Exercice 309
%%%     Le numéro correspond à la base de données :
%%%     http://allken-bernard.org/pierre/phpmyexercices                  
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{exercice}
\nopagebreak[5]Résoudre en fonction des paramètres $a$, $b$, $c$~:
\[(S)\quad\left\{\begin{array}{rrrrr @{\quad =\quad} r}
x&-&2y&+&4z& a\\
2x&+&3y&-&z&b\\
3x&+&y&+&2z&c
\end{array}\right.\]
\end{exercice}

\label{fin}
\end{document}