Full d'exercicis I

Exercici 1

Resol el següents sistemes:

1. \( \left\lbrace\begin{array}{rrrrrrr} 2x&-&3y&+&9z&=&8 \\ 3x&-&4y&-&3z&=&8 \\ 5x&-&7y&+&9z&=&16 \end{array}\right. \)

   Solució:

   Sistema Compatible Determinat

   \( x=-8 \quad \land \quad y=-8 \quad \land \quad z=0 \)

2. \( \left\lbrace\begin{array}{rrrrrrr} 2x&+&7y&-&9z&=&4 \\ x&-&3y&+&3z&=&1 \\ x&-&7y&+&2z&=&6 \end{array}\right. \)

   Solució:

   Sistema Compatible Determinat

   \( x=1 \quad \land \quad y=-1 \quad \land \quad z=-1 \)

3. \( \left\lbrace\begin{array}{rrrrrrr} 3x&+&4y&+&2z&=&0 \\ x&+&y&+&2z&=&-11 \\ x&+&2y&+&2z&=&-6 \end{array}\right. \)

   Solució:

   Sistema Compatible Determinat

   \( x=-2 \quad \land \quad y=5 \quad \land \quad z=7 \)

4. \( \left\lbrace\begin{array}{rrrrrrr} 4x&+&2y&+&z&=&10 \\ 4x&+&3y&+&4z&=&14 \\ 3x&+&y&+&4z&=&15 \end{array}\right. \)

   Solució:

   Sistema Compatible Determinat

   \( x=3 \quad \land \quad y=-2 \quad \land \quad z=2 \)

5. \( \left\lbrace\begin{array}{rrrrrrr} x&+&2y&+&3z&=&0 \\ 2x&+&2y&+&2z&=&3 \\ 3x&+&2y&+&z&=&6 \end{array}\right. \)

   Solució:

   Sistema Compatible Indeterminat

   \( \displaystyle x=\lambda +3 \quad \land \quad y=-\frac{4\lambda +3}{2} \quad \land \quad z=\lambda \)

6. \( \left\lbrace\begin{array}{rrrrrrr} 2x&+&4y&+&6z&=&2 \\ 3x&+&6y&+&9z&=&3 \\ 4x&+&8y&+&12z&=&4 \end{array}\right. \)

   Solució:

   Sistema Compatible Indeterminat

   \( \displaystyle x=1-2\lambda - 3\mu \quad \land \quad y=\lambda \quad \land \quad z=\mu \)

7. \( \left\lbrace\begin{array}{rrrrrrr} 4x&+&3y&+&z&=&17 \\ x&+&y&-&z&=&9 \\ 7x&+&7y&-&6z&=&60 \end{array}\right. \)

   Solució:

   Sistema Compatible Determinat

   \( \displaystyle x=2 \quad \land \quad y=4 \quad \land \quad z=-3 \)

8. \( \left\lbrace\begin{array}{rrrrrrr} 2x&+&3y&+&z&=&5 \\ x&+&y&+&z&=&1 \\ 4x&+&5y&+&3z&=&0 \end{array}\right. \)

   Solució:

   Sistema Incompatible