Finding the general solution of a system of equations

Find the general solution to the following system of equations and indicate which variables are free and which are basic.

Putting it in augmented matrix form to start we have:

1 -1 -1 4 | -3

1 0 -1/2 3 | -1

1 1 0 2 | 1

Now performing the following fundamental row operations:

R1<-->R2

R2+R3-->R2

-2R3+R2-->R2

-R3+R1-->R3

R2/-2

R2+R3-->R2

-3R3+R1-->R1

And finally I end with the augmented matrix:

1 0 -2 0 | 5

0 1 0 0 | 0

0 0 -1/2 1 |-2

Can someone please tell me if I got the correct matrix at the end and if so how do I determine which variables are free and which are basic?

Thank you.

Re: Finding the general solution of a system of equations

Hints:

For this this linear system to have a solution:

x3 = 5-x1-3x2

x4 = (1-x1-x2)/2

Also try to solve using least squares when number of unknowns and equations are not equal...to obtain:

{x1, x2, x3, x4} = {17/50, 69/50, 13/25, -9/25}

Re: Finding the general solution of a system of equations

I am not familiar with least squares. I presented this method because this is what I was taught and I need to know where I went wrong with the method I presented.