MHB Solving a system of linear equations with one unknown value

[exqiron]

Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values.

3x1+6x2 = −6
3x1+9x2−6x3 = −12
x1+x2+ax3 = 1No Solutions: ?
Unique Solution: ?
Infinitely Many Solutions: ?

all i could conclude from this was
1 1 a 1
0 1 -2 -2
0 0 (2-a) -1
don't know what to do next i have tried different questions, don't understand :/

 

[Prove It]

Determine the values of a for which the following system of linear equations has no solutions, a unique solution, or infinitely many solutions. You can select 'always', 'never', 'a = ', or 'a ≠', then specify a value or comma-separated list of values.

3x1+6x2 = −6
3x1+9x2−6x3 = −12
x1+x2+ax3 = 1No Solutions: ?
Unique Solution: ?
Infinitely Many Solutions: ?

all i could conclude from this was
1 1 a 1
0 1 -2 -2
0 0 (2-a) -1
don't know what to do next i have tried different questions, don't understand :/

To start with, you can find where you are going to have unique solutions by evaluating the determinant of your coefficient matrix. For all values of a that give a nonzero determinant, you will have unique solutions.

 

[exqiron]

To start with, you can find where you are going to have unique solutions by evaluating the determinant of your coefficient matrix. For all values of a that give a nonzero determinant, you will have unique solutions.

so in other words if i write this is this correct?
No solutions: when a = 2
Unique solution: when a not equal to 2
Infinite Many solutions: Never