Determine the values of a for which the system has no solutions, exactly one solution, or infinitely many solutions. x + 2y – 3z = 4 3x – y + 5z = 2 4x + y +(a– 14)z = a +2

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 .

Using rows operations we obtain the echelon form of the matrix, ie,

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.