free variable in parametric form

Question

if I have say $x_4$ = free, what value goes in the $x_4$ position of the parametric form, is it $1$ or $0$ or can it be any value since it's free?

Answer 1

"Free" is not a value and "x4=free" is an abuse of the equal sign. You mean $x_4$ is free. Is can denote equality ($=$), but here it denotes membership ($\in$). I had originally put this as a postscript, but your comments on my first drafts indicate this might be a stumbling block for you.

If $x_4$ is a free variable, it should be left as a parameter since it can have, as you say, any value.

You can express the other variables in terms of the free variables, or you can give names like $s$ and $t$ to the parameters and express all $x_i$ variables in terms of $s$ and $t$.

It sounds like your specific general solution is: \begin{align*} x_4 &\text{is free} \\ x_3 &= x_4 \\ x_2 &= 0 \\ x_1 &= 3x_4 \end{align*} "$x_4$ is free" means that you can't say anything more specific than $x_4 = x_4$. So (in vector form) $[x_1,x_2,x_3,x_4] = [3x_4,0,x_4,x_4] = x_4[3,0,1,1]$. Alternatively, you could say $[x_1,x_2,x_3,x_4] = [3x_4,0,x_4,x_4] = t[3,0,1,1]$, where $t\in\mathbb{R}$.

Answer 2

You just say $x_4 = t$, $t\in \Bbb R$

Example: $$\mathbf v = [x_1, x_2, x_3, 0] + t[0, 0, 0, 1],\quad t\in \Bbb R$$ $x_1$, $x_2$, and $x_3$ are set values while $t$ is free.