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

• Mar 6th 2013, 12:38 AM
Cotty
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.

• Mar 6th 2013, 01:04 AM
BobP
Re: Linear Equations: Determine the values of a in this system of linear equations
Gaussian elimination is the right way to go.
Having done that the third equation is
\$\displaystyle (a^{2}-9)z=3a+9.\$
You have to figure out what this implies for various values of \$\displaystyle a.\$
Specifically those that make the \$\displaystyle z\$ coefficient zero.
• Mar 6th 2013, 01:26 AM
Cotty
Re: Linear Equations: Determine the values of a in this system of linear equations
Ohhh thank you very much, i think i might be on the right track now
• Mar 6th 2013, 04:42 AM
Cotty
Re: Linear Equations: Determine the values of a in this system of linear equations
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!