Take some matrix $A$ such that
$$ A = \begin{bmatrix} \alpha_{11} & \alpha_{12} & \dots & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & \dots & \alpha_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \alpha_{n1} & \alpha_{n2} & \dots & \alpha_{nn} \end{bmatrix}, $$
where each $\alpha_{ij} \in \mathbb{Z}$, and assume $\det(A)$ is nonzero. Also take column vectors $\mathbf{x} = (x_1, \dots, x_n)^T, \mathbf{b}= (\beta_1, \dots, \beta_n)^T$, where $\beta_i \in \mathbb{Z}$ and the $x_i$'s are unknown. Consider the matrix equation
$$ A\mathbf{x} = \mathbf{b} $$
which is equivalent to the system of linear equations
$$ \sum_{j=1}^{n} \alpha_{ij} x_j = \beta_i, $$
for $1 \leq i \leq n$. I understand that the equation can be solved via Gaussian reduction of $A$, however, I was wondering if there was an explicit way to calculate each $x_i$ in terms of the $\alpha_i, \beta_i$ by repeated substitution?
