I'm attempting to develop an understanding of how equations are developed and wondered whether or not all equations started their development in the quest to document observable phenomenon or are there any 'purely synthetic' equations, which are developed from the basis of a thought in order to produce numeric results which are useful in some way.
Put another way: Do observations of phenomenon drive the development of equations or does a conceptual need for a tool drive their development instead?
The reason I ask, is because as someone who is a novice in mathematics (arithmetic skill, elementary algebra and descriptive stats), the choice of the construction of equations / their form always leads me to wonder why this form was chosen. In particular as I delve further into stats I have seen descriptions where certain equations may be adjusted based 'on ones need' which further deepens the difficulty of just using a tool blindly and wondering at what point or in what way I can develop the skill to discern how to choose how to alter an equation to better fit a desired outcome...
I attempted to articulate this originally HERE, then asked about whether it made sense to start a new question HERE, but for lack of feedback in either, I decided to open this query and post any positive results as link backs to those two questions for the community's later use. If there is a better way to approach asking this please let me know and I'll be happy to adjust accordingly.
Since I can't comment I will summarise my thoughts in this answer. In my experience, and from historical literature that I've read, the answer is "yes" to both of your questions. To clarify: As far as I'm concerned, it starts with the need or want to describe a particular physical phenomena. To this end, we first agree on relevant definitions and then observe the phenomena in order to formulate a "theorem" (e.g an equation). Sometimes it is easy to derive the equation, other times it requires much more technicalities, including mathematical tricks and tools. Whenever that is the case, it means the problem moves to a more theoretical setting, also increasing the risk of a less intuitive final result. But this is where the motivation for research on pure mathematics lies, answering questions such as: can we develop a theory which improves the equation in terms of computational complexity, intuitiveness, simplicity etc? This research does not (necessarily) have any particular practical importance but rather focuses on the theoretical parts of an equation/model, and has in turn introduced pure mathematics as an independent field of study.
You also ask how to decipher when to adjust an equation based on your needs. The short answer is simply to read proofs and practice on problem solving. Make sure you have seen all relevant equations and definitions for your course and how they depend on one another - there is always a reason for an adjustment to an equation and you should be able to find it from there. Once you have seen enough you will get a feel for it.
:)