Answer:
[tex]x=\frac{45}{4}, y=-\frac{201}{4}, z=-\frac{61}{2}[/tex]
Step-by-step explanation:
We start by putting our equation in a matricial form:
[tex]\left[\begin{array}{cccc}12&2&1&4\\3&3&-4&5\\2&-2&4&1\end{array}\right][/tex]
Then, we multiply the second row by 4 and substract the first row:
[tex]\left[\begin{array}{cccc}12&2&1&4\\0&10&-17&16\\2&-2&4&1\end{array}\right][/tex]
Now, multiply the third row by 6 and substract the first row:
[tex]\left[\begin{array}{cccc}12&2&1&4\\0&10&-17&16\\0&-14&23&2\end{array}\right][/tex]
Next, we will add [tex]\frac{7}{5}[/tex] times the second row to the third row:
[tex]\left[\begin{array}{cccc}12&2&1&4\\0&10&-17&16\\0&0&\frac{-4}{5}&\frac{122}{5}\end{array}\right][/tex]
Now we can solve [tex]\frac{-4}{5} z=\frac{122}{5}[/tex] to obtain
[tex]z=-\frac{61}{2}[/tex]
Then [tex]10y-17\frac{-61}{2}=16[/tex] wich implies that
[tex]y=\frac{16-\frac{17*61}{2}}{10} =\frac{\frac{32-17*61}{2}}{10}=\frac{-1005}{20}=\frac{-201}{4}[/tex]
Finally
[tex]x=\frac{4-2*\frac{-201}{4}+\frac{61}{2}}{12} =\frac{\frac{8+201+61}{2}}{12}=\frac{270}{24}=\frac{135}{12}=\frac{45}{4}[/tex].
[tex]z=-\frac{61}{2}\\ y=-\frac{201}{4} \\x=\frac{45}{4}[/tex]
