TITLE: Why can any type be realized?
QUESTION [6 upvotes]: I couldn't find this question asked previously, which means it's probably an especially daft question.
Given an $\mathcal{L}$-structure $\mathcal{M}$, my textbook defines an $n$-type over $A\subseteq M$ to be a set $p$ of sentences all in the same $n$ free variables such that $p\cup Th_A(\mathcal{M})$ is satisfiable. ($Th_A(\mathcal{M})$ is here the complete theory of $\mathcal{M}$ considered as a structure in the language $\mathcal{L}\cup\{c_a: a\in A\}$.)
The proof in the book proceeds by showing that since the union of $p$ and the elementary diagram of $\mathcal{M}$ is satisfiable, there's a model $\mathcal{N}$ into which $\mathcal{M}$ is elementarily embedded and which obviously satisfies $p\cup Th_A(\mathcal{M})$. This much I understand perfectly. 
The (imo, important) step of showing that there's an $\overline{a}\in N^n$ such that it satisfies every formula in $p$ is sort of brushed over. "Now let $c_i\in N$ be the interpretations of $v_i$. Then $(c_1,\ldots,c_n)$ is a realization of $p$." (This is David Marker; Chang & Keisler are even less helpful.)
I have clearly misunderstood something important; I know what the interpretation of $v_i$ with respect to a sequence $\overline{a}\in N^m$ for $m > i$ is, but I don't see a warrant in the proof or in the definitions surrouding interpretation for such a thing as "the" intepretation of a free variable. Without such a thing,  though, I'm not sure what actually guarantees realization of $p$.
So, what's the step I'm missing here?

REPLY [3 votes]: It all just hinges on the definition of satisfiable formula. Note that you misquoted Marker (you wrote sentence instead).
There is of course only one sensible possible definition (which I didn't manage to locate in Marker, unfortunately):

A set of formulae $\Phi$ in the free variables $(x_n)_n$ is satisfiable if there exist a model $\mathcal M$ and elements $(m_n)_n \in M$ such that for each $\phi \in \Phi$: $$\mathcal M \models \phi(m_1,\ldots,m_n)$$ (where $n$ is the highest index of a variable occurring in $\phi$).

Now the desired existence of $\bar a$ is an immediate consequence of the definition.