When does a linear system have infinitely many solutions, yet some of them don't depend on the others?

Question

Consider this system: $$ \begin{cases} w + x + y + z = 1 \\ w + x + y + 2z = 2 \end{cases} $$ Its solution set is $\{z = 1,\, y= -w - x \;|\; w,\,x \in \mathbb{R}\}$. So, $z$ is "fixed," in a sense. Is there a general case where this happens? (In other words, what are the general conditions where the system is consistent and dependent but the general solution leaves one of the coordinates fixed?)

The dimension of the solution of the associated homogenous system is $2$, but I don't think it tells us anything.

Answer 1

It is too difficult to answer in general, which is why we use row operations to reduce it to 'reduced row echelon form' :$$w+x+y=0\\z=1$$
Then $x$ and $y$ are free variables, and can take any value. Since they don't appear in the equation where $z$ is the pivot variable, $z$ will be fixed.
Conversely, start with a reduced row echelon form where $z$ is fixed, and apply row operations; $z$ will still be fixed.

Answer 2

I am new to this, but here's what I think is the answer. Let's consider an easier case with three variables and two equations. A solution would correspond to two planes intersecting, to form a line. If z were fixed it would then mean that the z-coordinate of this line was constant. This in turn would mean that the line was perpendicular to the z-axis (i.e. has a constant value for z).