Answer:
a) If a=20 the system has no solutions.
b) If a≠20 the system has exactly one solution.
Step-by-step explanation:
The augmented matrix of the system is [tex]\left[\begin{array}{cccc}1&2&-3&4\\3&-1&5&2\\4&1&a-14&a+2\end{array}\right][/tex].
Using rows operations we obtain the echelon form of the matrix, ie,
[tex]\left[\begin{array}{cccc}1&2&-3&4\\0&-7&-4&-10\\0&0&a-20&a+4\end{array}\right][/tex]
Since the echelon form of the matrix does not have free variables then the problem has exactly one solution or has no solutions.
a) If a=20 the system has no solutions because in the last equation solving for x3 it has that 0=24. Then the system is inconsistent.
b) If a≠20, the the system has exactly one solution that is obtained solving the system of the echelon form matrix.
