To my knowledge, this formula can be used to solve the general Diophantine equation
$x=x_0\color{blue}+\frac{b}{d}\cdot t\\y=y_0\color{red}-\frac{a}{d}\cdot t$
or this
$x=x_0\color{red}-\frac{b}{d}\cdot t\\y=y_0\color{blue}+\frac{a}{d}\cdot t$
and both are correct.
therefore, if I can arbitrarily arrange + and -, as long as they are in opposite positions, then this would mean that the following sets of solutions to the equation $999x-49y=5000$ are equal
$\begin{cases}x=-9000+\frac{-49}{1}t=-9000-49t\\y=-1835000-\frac{999}{1}t=-1835000-999t\end{cases}$
is equal to
$\begin{cases}x=-9000-\frac{-49}{1}t=-9000+49t\\y=-1835000+\frac{999}{1}t=-1835000+999t\end{cases}$
Is my line of thinking correct? When I solve the Diophantine equation $ax-by=c$, the signs came out opposite to what the calculator shows me so thats my idea why things are like that.