Maxima: how to reform a equation in a defined format

Question

In Maxima, I want to change the following equation:

ax+b-c-d=0

into the following format

(ax+b)/(c+d)=1

Note:

something like ax+b-c-d+1=1 is not what I want.

Basically I want to have positive elements in one side and negative elements in another side, then divide the positive elements by the negative elements.

Answer 1

Here is a quick attempt. It handles some equations of the form you described, but it's probably easy to find some which it can't handle. Maybe it works well enough, or at least provides some inspiration.

ptermp (e) := symbolp(e) or (numberp(e) and e > 0)
  or ((op(e) = "+" or op(e) = "*") and every (ptermp, args(e)));
matchdeclare (pterm, ptermp);
matchdeclare (otherterm, all);
defrule (r1, pterm + otherterm = 0, ratsimp (pterm/(-otherterm)) = 1);

NOTE: the catch-all otherterm must be precede pterm alphabetically! This is a useful, but obscure, consequence of the simplification of "+" expressions and the pattern-matching process ... sorry for the obscurity.

Examples:

apply1 (a*x - b - c + d = 0, r1);

                              a x + d
                              ------- = 1
                               c + b

apply1 (a*x - (b + g) - 2*c + d*e*f = 0, r1);

                            a x + d e f
                            ----------- = 1
                            g + 2 c + b