Caso 1 · Ebanista

Modelo y Gauss–Jordan paso a paso

Variables

$x$: sillas, $y$: mesas de café, $z$: mesas de comedor.

Capacidad y tiempos

• Lijado $16h=960$ min, Pintura $11h=660$ min, Barnizado $18h=1080$ min.
• Tiempos (min): Silla $(10,6,12)$, Café $(12,8,12)$, Comedor $(15,12,18)$.

Modelo (sistema)

\[ \begin{cases} 10x+12y+15z=960\\ 6x+8y+12z=660\\ 12x+12y+18z=1080 \end{cases} \]

Matriz aumentada

\[ \left[ \begin{array}{ccc|c} 10&12&15&960\\ 6&8&12&660\\ 12&12&18&1080 \end{array} \right] \]

Gauss–Jordan

1. $F_2\leftarrow 5F_2-3F_1,\ \ F_3\leftarrow 5F_3-6F_1$

\[ \left[ \begin{array}{ccc|c} 10&12&15&960\\ 0&4&15&420\\ 0&-12&0&-360 \end{array} \right] \]

2. $F_3\leftarrow F_3+3F_2$

\[ \left[ \begin{array}{ccc|c} 10&12&15&960\\ 0&4&15&420\\ 0&0&45&900 \end{array} \right] \]

3. $F_3\leftarrow \tfrac{1}{45}F_3$
4. $F_2\leftarrow F_2-15F_3,\ \ F_1\leftarrow F_1-15F_3$

\[ \left[ \begin{array}{ccc|c} 10&12&0&660\\ 0&4&0&120\\ 0&0&1&20 \end{array} \right] \]

5. $F_2\leftarrow \tfrac{1}{4}F_2=[0\ 1\ 0\ |\ 30]$
6. $F_1\leftarrow F_1-12F_2=[10\ 0\ 0\ |\ 300]$
7. $F_1\leftarrow \tfrac{1}{10}F_1=[1\ 0\ 0\ |\ 30]$