Characteristic equation is $|4-x \ \ 3|$ $|2 \ \ 3-x|=0$ It give $x_n=a+3b6^{n-1},y_n=-a+2b6^{n-1}$. From $x_1=y_1=5$ we have $a=-1,b=2$. Therefore $x_n=6^n-1,y_n=4*6^{n-1}+1,\lim_{n\to \infty} \frac{x_n}{y_n}=3/2,$

$3y_n=2x_n+5\Longrightarrow \frac 32=\frac{x_n}{y_n}+\frac{5}{2y_n},\ y_n\rightarrow\infty\Longrightarrow \frac{x_n}{y_n}\rightarrow \frac 32\ .$

Could You Rust or anyone write something about finding the general method for finding $ (x_n) (y_n)$ Because You wrote about characteristic equation. Could You explain more clearly what are the properties and other interesting concludes if we will compute that?

This is just linear algebra; what we have is $ \left[ \begin{array}{c} x_n \\ y_n \end{array} \right] = \left[ \begin{array}{cc} 4 & 3 \\ 2 & 3 \end{array} \right] \left[ \begin{array}{c} x_{n - 1} \\ y_{n - 1} \end{array} \right]$ hence, letting the above matrix be $ \mathbf{M}$ and $ \mathbf{x}_n$ denote the column vector whose entries are $ x_n, y_n$, we have $ \mathbf{x}_n = \mathbf{M}^n \mathbf{x}_0$ which we can compute by diagonalizing $ \mathbf{M}$.