Find the general solutions of the systems whose augmented matrices are given as

12. \(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\{ - 1}&7&{ - 4}&2&7\end{array}} \right]\).

A basic principle states that row operations do not affect the solution set of a linear system. Perform an elementary row operation to produce the first augmented matrix.

Replace row \(3\) by adding row 1 to row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\0&0&{ - 4}&8&{12}\end{array}} \right]\)

Perform an elementary row operationto produce a second augmented matrix.

Replace row 3 by adding\(4\)times row 2 to row 3.

\(\left[ {\begin{array}{*{20}{c}}1&{ - 7}&0&6&5\\0&0&1&{ - 2}&{ - 3}\\0&0&0&0&0\end{array}} \right]\)

To obtain the general solution of the system, you have to convert the augmented matrix into the system of equations.

Write the obtained matrix into the equation notation.

\(\begin{array}{l}{x_1} - 7{x_2} + 6{x_4} = 5\\{x_3} - 2{x_4} = - 3\end{array}\)

Thus, the general solution of the system is

\({x_1} = 5 + 7{x_2} - 6{x_4}\)

\({x_2}\)is free.

\({x_3} = - 3 + 2{x_4}\)

\({x_4}\) is free.