Explaining the physical meaning of an eigenvalue in a real world problem Contextual Problem

A PhD student in Applied Mathematics is defending his dissertation and needs to make 10 gallon keg consisting of vodka and beer to placate his thesis committee.  Suppose that all committee members, being stubborn people, refuse to sign his dissertation paperwork until the next day.   Since all committee members will be driving home immediately after his defense, he wants to make sure that they all drive home safely.  To do so, he must ensure that his mixture doesn't contain too much alcohol in it!  

Therefore, his goal is to make a 10 gallon mixture of vodka and beer such that the total alcohol content of the mixture is only $12$ percent.  Suppose that beer has $8\%$ alcohol while vodka has $40\%$.  If $x$ is the volume of beer and $y$ is the volume of vodka needed, then clearly the system of equations is 

\begin{equation}
x+y=10 \\ 0.08 x +0.4 y = 0.12\times 10 
\end{equation}

My Question

The eigenvalues and eigenvectors of the corresponding matrix

\begin{equation}
\left[
\begin{array}{cc}
1 &amp; 1\\
0.08 &amp; 0.4
\end{array}
\right]
\end{equation} 

are

\begin{align}
\lambda_1\approx 1.1123 \\
\lambda_2\approx 0.2877 \\
v_1\approx\left[\begin{array}{c}
0.9938 \\ 0.1116
\end{array}
\right] \\
v_2\approx\left[\begin{array}{c}
-0.8145 \\ 0.5802
\end{array}
\right]
\end{align}

How do I interpret their physical meaning in the context of this particular problem?
An interpretation of eigenvalues and eigenvectors of this matrix makes little sense because it is not in a natural fashion an endomorphism of a vector space: On the "input" side you have (liters of vodka, liters of beer) and on the putput (liters of liquid, liters of alcohol). For example, nothing speaks against switching the order of beer and vodka (or of liquid and alcohol), which would result in totally different eigenvalues.
You don't really need eigenvectors to solve this problem.  Just treat it as a linear programming problem and let the Simplex Algorithm (or your algorithm of choice) come up with a feasible solution.  Since you need an objective function, you can choose one arbitrarily; for example, you might choose to minimize the total cost.  If you don't know the costs of beer and  vodka, just make some up.