Determining Basic and Free Variables in a Linear System

[amcgl064]

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.

 

[Sudharaka]

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.

Hi amcgl064, :)

The answer you have obtained for the row echelon form is incorrect. The correct answer is,

\[\left(\begin{matrix}1&-1&-1&4\\0&1&\frac{1}{2}&-1\\0&0&0&0\end{matrix}\right)\]

Please refer >>this<< for a basic introduction about basic variables and free variables. I hope you can do the rest. :)

Kind Regards,
Sudharaka.