# How to make $x_3 x_4 x_5$ independent variables in this linear system of equation?

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.

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.