An equation is meant to be solved, that is, there are some unknowns. A formula is meant to be evaluated, that is, you replace all variables in it with values and get the value of the formula. Your example is a formula for mpg. But it can become an equation if mpg and one of the other value is given and the remaining value is sought.
I love your answer for a line equation in the form of z = f (x, y)... Unfortunately calculating square roots can be impractical from the calculational standpoint and hence I really doubt any plotting software will be able to graph it correctly.
The equation of an object is a way of telling whether a point is part of an object -- if you substitute the coordinates of the point into the equation and the equation is true, then the point is on the object; if the equation is not true for that point, then the point is not on the object.
(Equations which can be easily transformed to Cauchy functional equation or can be solved by using similar methods.) Is there some overview of basic facts about Cauchy equation and related functional equations - preferably available online?
Equation means equality. They are both related to the word equal. If such an equality is true for all values of the variable, it is called an identity, e.g., $\sin^2x+\cos^2x=1$ is true for all x. If however the equation in question only holds for some values, which one is supposed to determine, then it's called conditional, and its variable is termed an unknown.
Note that there are 4 unknowns, so there is no single solution for a,b, and c since you can just multiply the equation by some constant and it stays the same, so you just solve for ratio of a to b to c. Also if there is no single solution, then there is a solution set (multiple planes can satisfy the conditions).
A Bernoulli differential equation is a non-linear differential equation of the form $$ \\frac{dy}{dx} + P(x)y = Q(x)y^n. $$ I understand this is special; Because its exact solution is known though ...
