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I'm given the following linear system of equations:

$$x_1 + x_2 + 4x_3 + x_4 = 10$$

$$x_2 + 2x_3 + 1/2x_4 + 1/2x_5 = 10$$

I'm asked to solve the system using gauss-jordan elimination to express $x_3 x_4 x_5$ as independent variables.

$$\begin{pmatrix} 1 & 1 & 4 & 1 & 0 & 10\\ -1 & 1 & 0 & 0 & 1 & 10 \end{pmatrix}$$

What I do is to add to the second line the first line. I get :

$$\begin{pmatrix}1 & 1 & 4 & 1 & 0 & 10 \\ 0 & 2 & 4 & 1 & 1 & 20 \end{pmatrix}$$

I then multiply the second line by 2. I get : $\begin{pmatrix}1 & 1 & 4 & 1 & 0 & 10 \\ 0 & 1 & 2 & 1/2 & 1/2 & 10 \end{pmatrix}$

There doesn't seem to be much more that I can do. My problem with this question is that I do not understand what to do to make the variables $x_3 x_4 x_5 $ independent variables in the resulting system of equations. I would really like to have an explanation on this part of the problem.

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1 Answer 1

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You can subtract the second line from the first. At that point, ANY assignment of values to $x_3, x_4, x_5$ will lead to forced-choices for $x_1, x_2$, i.e., $x_{3,4,5}$ are independent, and $x_{1,2}$ are dependent.

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