Answer:
[tex]x=\frac{-363}{70}[/tex] ≈ -5.1857
[tex]y=\frac{-192}{35}[/tex] ≈ -5.4857
[tex]z=\frac{313}{84}[/tex] ≈ 3.7262
Step-by-step explanation:
Rewrite the equation system as:
[tex]6x+8y=-75[/tex]
[tex]-3x+6y+6z=5[/tex]
[tex]2x-9y=39[/tex]
Now, write the system in its augmented matrix form:
[tex]\left[\begin{array}{cccc}6&8&0&-75\\-3&6&6&5\\2&-9&0&39\end{array}\right][/tex]
applying row reduction process to its associated augmented matrix:
Swap R1 and R3, and then Swap R1 and R2:
[tex]\left[\begin{array}{cccc}-3&6&6&5\\2&-9&0&39\\6&8&0&-75\end{array}\right][/tex]
R3+2R1
[tex]\left[\begin{array}{cccc}-3&6&6&5\\2&-9&0&39\\0&20&12&-65\end{array}\right][/tex]
3R2+2R1
[tex]\left[\begin{array}{cccc}-3&6&6&5\\0&-15&12&127\\0&20&12&-65\end{array}\right][/tex]
15R3+20R2
[tex]\left[\begin{array}{cccc}-3&6&6&5\\0&-15&12&127\\0&0&420&1565\end{array}\right][/tex]
Now we have a simplified system:
[tex]-3x+6y+6z=5\\0-15y+12z=127\\0+0+420z=1565[/tex]
[tex]-3x+6y+6z=5\hspace{5 mm}(1)\\0-15y+12z=127\hspace{3 mm}(2)\\0+0+420z=1565\hspace{3 mm}(3)[/tex]
From (3):
[tex]z=\frac{313}{84}[/tex] (4)
Replacing (4) in (2)
[tex]y=\frac{-192}{35}[/tex] (5)
Finally replacing (5) and (4) in (1)
[tex]x=\frac{-363}{70}[/tex]
