Converting between the $y= e+dx$ and $ax + by = c$ forms of a line equation

Question

I can express the equation of a line in two ways

$$y = f(x) = e + dx$$

$$ax + by = c$$

I can easily derive the first equation given $(a, b, c)$ in the second equation. But how can I do it the other way around, if I know $(d, e)$?

Answer 1

If $y = e + dx$, then $-dx + y = e$. So let $a = -d, b=1$ and $c = e$. Then we have $ax + by = c$. When $ax + by = c$, we have $by = c -ax$, so $y = \frac{c}{b} - \frac{a}{b}x$. So let $e = \frac{c}{b}$ and $d = - \frac{a}{b}$ and we have $y = e + dx$. This shows both ways. I hope this helps!