I know a way to find new equation of line but it's a bit lengthy. I can first get any two different points on line, then scale those points according to the scale change of the axes, and finally go back to line.
But I was wondering how can I directly get a new line equation from the original one. For instance, if I have a line of form $ax+by+c=0$, how I can get new parameters $[a, b, c]$ for this line when the x and y axes have changed by scales $\alpha$ and $\beta$. Thanks.