Determine the values of a for which the system has no solutions, exactly one solution, or infinitely many solutions.

Question

Determine the values of a for which the system has no solutions, exactly one solution, or infinitely many solutions.

x + 2y - 2z = 3

2x - y + 2z = 3

4x + 3y + (a² - 11)z = a + 6

Answer 1

[ 1 2 -2 3]

[ 2 -1 2 3]

[4 3 a²-11 a+6]

Doing a few row operations

[1 2 -2 3]

[0 -5 6 -3]

[0 0 a²-9 a-3]

(a²-9)z = a-3

z = (a-3)/((a+3)(a-3)) = 1/(a+3)

If a = -3, then no solution (1/0)

if a = 3, then the last row becomes zero, so infinitely many solutions because z is free

"a" can be anything else, each time giving one solution, but that would be an infinite number of solutions