\left\{\begin{array}{lll}
3 x+y-3 z+2 t&=&7 \\
x-2 y+z-t&=&-9 \\
2 x-3 y-2 z+t&=&-4 \\
-x+5 z-3 t&=&-11
\end{array}\right.
\left\{\begin{array}{cccc}
x-2 y+z-t&=&-9 & (L_1\longleftrightarrow L_2)\\
3 x+y-3 z+2 t&=&7& \\
2 x-3 y-2 z+t&=&-4& \\
-x+5 z-3 t&=&-11&
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
&7y&-6z&+5t&=&34& (L_2\longleftarrow L_2-3L_1)\\
2 x&-3 y&-2 z&+t&=&-4& \\
-x&&+5 z&-3 t&=&-11&
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
&7y&-6z&+5t&=&18& \\
& y&-4 z&+3t&=&14& (L_3\longleftarrow L_3-2L_1)\\
-x&&+5 z&-3 t&=&-11&
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
&7y&-6z&+5t&=&34& \\
& y&-4 z&+3t&=&14& \\
&-2y&+6z&-4 t&=&-20& (L_4\longleftarrow L_4+L_1)
\end{array}\right.
(S)
- Choix de la 1ère ligne :
(S)\(\iff\)
- Nettoyage de la colonne des "\(x\)":
\(\iff\)
\(\iff\)
\(\iff\)
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
& y&-4 z&+3t&=&14& (L_2\longleftrightarrow L_3)\\
&7y&-6z&+5t&=&34& \\
&-2y&+6z&-4 t&=&-20&
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
& y&-4 z&+3t&=&14& \\
&&22z&-16t&=&-64& (L_3\longleftarrow L_3-7L_2)\\
&-2y&+6z&-4 t&=&-20&
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
& y&-4 z&+3t&=&14& \\
&&22z&-16t&=&-64& \\
&&-2z&+2t&=&8& (L_4\longleftarrow L_4+2L_2)
\end{array}\right.
- Choix de la 2ème ligne :
- Nettoyage de la colonne des "\(y\)":
\(\iff\)
\(\iff\)
\(\iff\)
\(\iff\)
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
& y&-4 z&+3t&=&14& \\
&&-2z&+2t&=&8& (L_3\longleftrightarrow L_4)\\
&&22z&-16t&=&-64& \\
\end{array}\right.
\left\{\begin{array}{ccccccc}
x&-2 y&+z&-t&=&-9 & \\
& y&-4 z&+3t&=&14& \\
&&-2z&+2t&=&8& \\
&&&6t&=&24& (L_4\longleftarrow L_4+11L_3)\\
\end{array}\right.
- Choix de la 3ème ligne :
- Nettoyage de la colonne des "\(z\)":
Le système est échelonné !!!!
- On déduit que:
\left\{\begin{array}{ccc}
x&=& -1 \\
y&=&2 \\
z&=&0 \\
t&=&4 \\
\end{array}\right.
\mathcal{S}=\lbrace (-1;2;0;4)\rbrace
\(\iff\)
\(\iff\)
\(\iff\)
deck
By intermaths.info
deck
- 40
