What is the relationship between spans that contain some of the same vectors? I have been given the following problem:

Let x, y, z be non-zero vectors and suppose w = 4x + y -3z.
a) If z = 4x + y, then w = _x + _y.
b) Using the calculation in (a), mark the statements below that must be true.

Then I'm given a number of statements of the same type. I will quote only the one for purposes of this discussion and hope to be able to do the rest myself by the end of it!

(i) Span (w, y, z) = Span (w, x)

For reference, I got w = -8x - 2y and this has been marked as correct (whew!)
I'm confused by this question on a couple of fronts. Firstly, I understood a span to be the set of all linear combinations of all vectors in a given space. However, the equation given for w looks like a specific linear combination (i.e. 4 of x and 1 of y etc). Is my understanding of the definition incorrect? Is it actually any linear combination? Am I wrong about w and it's not actually a space (I see the question doesn't say it is) Have I lost the plot completely and this has nothing to do with anything?!!!
I'm also confused about how to engage with this question. Should I be taking the equations for w, y and z and summing them together to see if I get the same as the RHS? (i.e. the sum of w and x?)
I would really appreciate any light you guys can throw on this. I have a ream of questions to answer on spans and this is one of the intro ones. It's clear to me that there's something I'm not getting.
 A: Ok so the span of a bunch of vectors is intuitively "the vectors that you can make with these ones". By make I mean by using only the operations of addition/subtraction/scalar multiplication. The span makes a vector space.
So in your question the fact that $w = 4x + y - 3z$ is telling you that "$w$ can be made from $x,y,z$". Formally this means $w\in\text{span}\{x,y,z\}$.
But the next fact, that $z = 4x + y$ is telling you that actually "$z$ can be made from $x,y$", i.e. $z\in\text{span}\{x,y\}$. 
So really from this you can conclude that "$w$ can be made from $x$ and $y$ only". Indeed, replace $z$ in the definition of $w$ and you get $w = 4x + y - 3(4x+y) = -8x - 2y$, so that $w\in\text{span}\{x,y\}$.
Anyway, for your second point, lets have a think about it. The statement that $\text{span}\{w,y,z\} = \text{span}\{w,x\}$ is saying "can anything made from $w,y,x$ be made only from $w,x$ and conversely, can everything made from $w,x$ be made from $w,y$ and $z$?"
Let's answer the first of these. Well clearly we can make $w$ from $w$ and $x$. Also we have just seen that $w$ can be made from $x$ and $y$, so rearranging, we can make $y$ from $w$ and $x$. Also $z$ is made from $x$ and $y$ and $y$ is made from $w$ and $x$ so that $z$ can be made from $w$ and $x$. Thus anything made from $w,y,z$ CAN be made from $w$ and $x$.
The opposite inclusion is similar. We see that $w$ can obviously be made from $w,y$ and $z$. Also $x$ can be made from $w,y$ and $z$. So anything made from $w$ and $y$ CAN be made from $w,y$ and $z$.
Hope this helps.
