Equations

If you have seen an equation, which you probably have, it was given to you by an educator along with an explicit task... solve for x or get everything on one side of the equals sign. But equations don't give themselves to you in daily life. So where do they come from and what are they?

They are "models" - someone takes raw statistics or measurements and figures out "trends" and then finds a "common form" of an equation that comes as close as possible to explaining the data. A simple example is: width times height equals area. Someone back in ancient times probably measured a couple hundred pastures and looked at the data long enough to figure out a way to generalize it for any pasture as a time saver. The most common equations are so simple that we use them without realizing that we are using them.

I am studying probabilities, which use the same models. By definition, an equation is a probability if the area is 1. The model is usually some scale and solving the equation for a part of the area gives you the corresponding probability of that part of the model. Here is an example. A common form of an equation is a exponential. Most commonly seen as: y equals "e" to the power of negative x. It starts out very small and then gets large very fast. In probabilities is is often used to model the lifetime of a part. The part is more likely (y, the probability, is a progressively bigger number) to die later (as x becomes a larger number) in the timescale as opposed to earlier. This is where math becomes so much easier when the equation is looked at as a picture.

Some key points:

Setting the equation equal to zero and solving for x (using any method such as completing the square, quadratic equation, etc.) just tells you what value of x has a corresponding y=0. In probabilities this means, when there is zero/no chance of x occuring. (Meaning the part can never die under those circumstances.) This is just as important in many other applications. In physics, when y=0 might mean the tank will not explode or the project is finished or earliest point that the shuttle will escape Earth's atmosphere successfully. Anyway, you get the idea. Solving for the x that corresponds to a y=0 is important. They just never tell you why when you are learning the 20 different methods to do it. (Nor do they tell you that the 20 things they are teaching you are different methods to get at the same result, so you never know what the thing you are learning is for... they tell you the good stuff well after everyone else has lost interest.)