# Finding the general solution of a system of equations

• Sep 24th 2012, 06:10 AM
dumbledore
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.

$\displaystyle x_1 + 4x_4 + 3 = x_2 + x_3$
$\displaystyle x_1 + 3x_4 + 1 = \frac{1}{2}x_3$
$\displaystyle x_1 + x_2 + 2x_4 = 1$

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.
• Sep 24th 2012, 10:10 AM
MaxJasper
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}
• Sep 24th 2012, 10:13 AM
dumbledore
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.