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 & 1\\
0.08 & 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?

While the eigenvectors and eigenvalues don't play their usual role in this problem 
(as argued in the other answers) 
the eigensystem still has a physical interpretation.
The eigenvector $v_2$ is unphysical since it corresponds to a negative volume.
Let $M$ represent the matrix above,
$$M v_1 = \lambda_1 v_1.$$
Since 
$\lambda_1 y/(\lambda_1 x) = y/x$,
the alcohol content of the final mixture is equal to the ratio of vodka to beer.
In addition
$$x+y = \lambda_1 x,$$ 
so the eigenvalue is the alcohol content of the final mixture plus one. 
This mixture could be used since the alcohol content is about 11 percent. 
To get a 10 liter mixture scale the eigenvector, $v_1 \rightarrow 10v_1/(x+y)$. 
Perhaps the committee would find it a more mathematically interesting mixture. 
