This problem requires string processing and the ability to implement conditions.

A simple way to process the string is to take only the expression with x without y = and to strip the string of surrounding whitespace (just to be safe - the test data shouldn't actually contain whitespace after the expressions). We can use built-in functions such as find(), substr(), and atoi() in C++ instead of using many loops to get the values of ~a~ and ~b~ easily.

Afterward, we simply have to implement the simplifying conditions given in the problem statement. It is recommended to test the conditions in order of decreasing priority. For example, first check if ~ab = 0~, then if ~b-1 = 0~, etc.

Time Complexity: ~\mathcal{O}(T)~