Question

III. Utiliza el método de eliminación de Gauss-Jordan para encontrar, si existen, todas las soluciones de los sistemas dados. a) -2x1 x2 6x3 = 18 5x1 8x3 = -16 3x1 2x2 - 10x3 = -3

Answer 1

Utilizando el método de eliminación de Gauss Jordan se obtiene:

x₁ = -4

x₂ = 7

x₃ = 1/2

Explicación:

Datos;

-2x₁+ x₂+ 6x₃ = 18

5x₁ +8x₃ = -16

3x₁+ 2x₂ - 10x₃ = -3

El método de Gauss Jordan para la resolución de sistemas de ecuaciones plantea, hallar una matriz Mx = I, siendo I la matriz identidad.  

$\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33} \end{array}\right] .\left[\begin{array}{c}x&y&z\end{array}\right] = \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$  

Sustituir;

$\left[\begin{array}{ccc}-2&1&6\\5&0&8\\3&2&-10\end{array}\right] .\left[\begin{array}{c}18&-16&-3\end{array}\right]$

-1/2f₁

$=\left[\begin{array}{ccc}1&-1/2&-3\\5&0&8\\3&2&-10\end{array}\right] .\left[\begin{array}{c}-9&-16&-3\end{array}\right]$

f₂-5f₁

f₃-3f₁

$=\left[\begin{array}{ccc}1&-1/2&-3\\0&5/2&23\\0&7/2&-1\end{array}\right] .\left[\begin{array}{c}-9&29&24\end{array}\right]$

2/5f₂

$=\left[\begin{array}{ccc}1&-1/2&-3\\0&1&46/5\\0&7/2&-1\end{array}\right] .\left[\begin{array}{c}-9&58/5&24\end{array}\right]$

f₃-7/2f₂

$=\left[\begin{array} {ccc}1&-1/2&-3\\0&1&46/5\\0&0&-166/5\end{array}\right] .\left[\begin{array}{c}-9&58/5&-83/3\end{array}\right]$

-5/166f₃

$=\left[\begin{array} {ccc}1&-1/2&-3\\0&1&46/5\\0&0&1\end{array}\right] .\left[\begin{array}{c}-9&58/5&1/2\end{array}\right]$

f₁+3f₃

f₂-46/5f₃

$=\left[\begin{array} {ccc}1&-1/2&0\\0&1&0\\0&0&1\end{array}\right] .\left[\begin{array}{c}-15/2&7&1/2\end{array}\right]$

f₁+1/2f₂

$=\left[\begin{array} {ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] .\left[\begin{array}{c}-4&7&1/2\end{array}\right]$