Question

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

Answer 1

a) If a=20 the system has no solutions.

b) If a≠20 the system has exactly one solution.

Explanation:

The augmented matrix of the system is
`\left[\begin{array}{cccc}1&2&-3&4\\3&-1&5&2\\4&1&a-14&a+2\end{array}\right]`.

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

`\left[\begin{array}{cccc}1&2&-3&4\\0&-7&-4&-10\\0&0&a-20&a+4\end{array}\right]`

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.