Linear Equations: Determine the values of a in this system of linear equations

Hi,

I have a question regarding linear algebra which I am unsure of how to approach. The question says:

Determine the values of a for which the following system of equations has (a) no solutions, (b) exactly one solution, and (c) infinitely many solutions.

x + y + 7z = -7
2x + 3y + 17z = -16
x + 2y + (a^2+1)z = 3a

So I assume that we would use some form of Gaussian to figure out the solutions because thats what have have been doing in the course so far, but everything I have tried so far has failed!!!

I did get a solution of a=3 for no solutions, but I'm not actually sure if this is correct (what I did was set a^2-6 equal to 3 after using row reduction). Anyway any help I could get would be greatly appreciated or even just any clues as to how I should start this problem.

Thanks in advance!!

Gaussian elimination is the right way to go.
Having done that the third equation is

You have to figure out what this implies for various values of
Specifically those that make the coefficient zero.

Ohhh thank you very much, i think i might be on the right track now

So the solutions I found are:

For no solution: a=3
For infinitely many solutions: a=-3
and
For one solution: a E {R/(-3,3)}

Sorry with that last one I know how I've written it probably isn't the correct way to write it mathematically but what i meant is that a is an element of the Real number system not including -3 and 3 (if you could show me how to write this properly it would be very appreciated as it has been a while since I have done that type of stuff). Are these the correct solutions?

Anyway thank-you very much for the help you have given me so far!