Understanding initial example in "Mathematical Explanation for the Repertoire Method" I am working my way through this excellent answer and the first section "Repertoire method (some basics)" (and associated concrete example) I am having trouble following. The crux is that an equation with the additive term
$$
\alpha a_n+\beta b_n
$$
will have the solution
$$
\alpha x_n + \beta y_n
$$
We are conveniently provided with two recurrences in the repertoire
$$
\begin{align*}
x_0&=3&y_0&=1\\
x_n&=3+x_{n-1},\quad n>0&y_n&=5n^2+1+y_{n-1},\quad n>0
\end{align*}
$$
to which, via linearity, the solution of the recurrence
$$
\begin{align*}
z_0&=7\\
z_n&=2n^2+7+z_{n-1}
\end{align*}
$$
is going to be
$$
\begin{align*}
z_n=\frac{11}{5}x_n+\frac{2}{5}y_n
\end{align*}
$$
What steps did were taken to get to that solution? I know it'll be plugging in the provided recurrences somehow and then doing algebraic manipulations but I'm not quite getting it.
PS: I appreciate how friendly, welcoming, and patient the Math SE community has been with my very basic questions so far.

My confusion was caused by the fact that I had it backwards as to which closed form term is associated with which recurrence term. I see now that it is as follows

Which made it much clearer - I can see now how the "scaling factors" to apply to the repertoire were arrived at. For completeness I will outline the steps I took here in the hopes it'll be useful for someone else.
$n^2$ Term
We're looking for our mysterious "factor" that'll linearly combine with our repertoire to give the $2$ target recurrence coefficient. So, representing our mysterious factor as $\alpha$
$$
5\alpha=2
$$
$$
\alpha=\frac 25
$$
Constant Term
Again, we're looking for something to linearly combine with our repertoire to give the $7$ from the recurrence. We can represent that mysterious factor as $\beta$ that'll scale the repertoire's "$3$" term; so $3\beta$. Thus the equation to find $\beta$ will be $\beta + \alpha = 7$. We already know $\alpha$ is $\frac 25$ so
$$
\begin{aligned}
3\beta + \frac 25&=7\\
15\beta+2&=35\\
15\beta&=33\\
\beta&=\frac {33}{15}\\ 
\beta&=\frac {11}{5}
\end{aligned}
$$
Closed form
Then the closed form is constructed using these "factors":
$$
\begin{align*}
z_n=\frac{11}{5}x_n+\frac{2}{5}y_n
\end{align*}
$$
