Let $\mathbb{k}$ be an algebraically closed field. We have a system of $d$ linear equations in $d+1$ variables denoted by $u_0,\ldots,u_d$. The coefficients belongs to $\mathbb{K}[x]$ and the system is consistent so there exists a solution.
$u_0 +u_1 h_{1}^{1}+\ldots+u_d h_{d}^{1}=0 $
$u_0 +u_1 h_{1}^{2}+\ldots+u_d h_{d}^{2}=0 $
$\vdots$
$u_0 +u_1 h_{1}^{d}+\ldots+u_d h_{d}^{d}=0 $ where $h_{i}^{j}\in \mathbb{K}[x]$.
So we have $d$ equations.
For each equation we put a degree bound that is $e_j =max\{deg \ h_{i}^{j}\}i\in{1\ldots d}$.
I want a bound on the degree for the solution of the system of linear equations in terms of $e_j, d$. Are this results known in the literature? Can anyone please cite me a reference or a proof?