# How does equation of a line change as scale of axes changes?

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.

• $\frac a\alpha x + \frac b\beta y + c = 0$. – fleablood Jul 27 '18 at 15:22

By the new coordinates $X$ and $Y$, we have
• $X=\alpha x$
• $Y=\beta y$
$$ax+by+c=0 \implies \frac a {\alpha}X+\frac b {\beta}Y+c=0 \implies a\beta X+b\alpha Y+\alpha \beta c=0$$