We are given the task to find the numerical values of $a, b, c, d$ for the following equation $ax+by+cz+d=0$. We are given that this plane should intersect with the points $M=(4,4,4), N=(6,0,8), L=(5,5,-5)$.
We know the following: $$ \begin{align*} 4a+4b+4c+d &= 0 \\ 6a+8c+d &= 0 \\ 5a+5b-5c+d &= 0 \end{align*} $$
We solve this by creating the linear system of equations as such: $$ \begin{bmatrix} 4 & 4 & 4 & 1 & 0\\ 6 & 0 & 8 & 1 & 0\\ 5 & 5 & -5 & 1 & 0 \end{bmatrix} $$
and end up finding the following
$$ \begin{bmatrix} 1 & 0 & 0 & \frac{2}{15} & 0\\ 0 & 1 & 0 & \frac{11}{120} & 0\\ 0 & 0 & 1 & \frac{1}{40} & 0 \end{bmatrix} $$
which gives $$ \begin{align*} a &= -\frac{2}{15}d \\ b &= -\frac{11}{120}d \\ c &= -\frac{1}{40}d \end{align*} $$ where we can plot this back into our equation for the plane.
Now, according to exercise text $d$ is a called a free variable. However, if we use $d=0$ we get $0=0$ for the equation $ax+by+cz+d=0 \to 0x+0y+0z+0=0$ thus leaving us without an equation for the plane intersecting the three points.
Now, the way I understod free variables, was that you could choose whatever you felt like and they would satisfy the equations provided. However, why is $d=0$ an invalid value here is there some assumption being done that restricts my free variable? Am I missing something? Or is it wrong to imply that $d$ is a free variable in this case?
Your understanding of free variables is exactly right when it comes to solving linear systems. But having $a,b,c,d$ which solve your linear system is not the same thing as having $a,b,c,d$ which determine a plane. The only difference is that $a=b=c=0$ does not determine a plane (in general, hyperplanes have the form $\vec{a}\cdot \vec{x}+d=0$ where $\vec{a}$ is a fixed, non-zero vector and $d$ is a constant. (In your case, $\vec{a}=(a,b,c)$).
I would think about this problem in two steps: 1. Find one or more solutions to the system, and 2. make sure the solution you choose from 1. determines a plane.
The choice $d=0$ is perfectly valid to find a solution for 1. It is part 2. which invalidates it as a response to this particular question.