Performing the indicated operations (Definition CVA, Definition CVSM), we obtain the vector equations
\begin{align*}
\colvector{5\\0}
&= \alpha\colvector{2\\1} + \beta\colvector{1\\3}
= \colvector{2\alpha + \beta \\ \alpha + 3\beta}
\end{align*}
Since the entries of the vectors must be equal by Definition CVE, we obtain the system of equations
\begin{align*}
2\alpha + \beta &= 5\\
\alpha + 3\beta &= 0.
\end{align*}
which we can solve by row-reducing the augmented matrix of the system,
\begin{align*}
\begin{bmatrix}
2 & 1 & 5\\
1 & 3 & 0
\end{bmatrix}
&\rref
\begin{bmatrix}
\leading{1} & 0 & 3\\
0 & \leading{1} & -1
\end{bmatrix}
\end{align*}
Thus, the only solution is $\alpha = 3$, $\beta = -1$.