# Help solving linear system

#### amcgl064

##### New member
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

##### Well-known member
MHB Math Helper
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.