Mathematical Equations exist because problems exist. In other words, we have equations because humans have problems to solve. Even the simple statement A = B implies that at a prior stage we were interested in the relationship between A and B.
Consider a quick trip to the supermarket. Say, you get 1 loaf of bread, 1 gallon of milk and 2lbs of banana.
The problem of interest to both you and the owner of the supermarket is the sum of the price of each of the items you have brought to the check-out counter.
The solution of this problem requires the following equation:
price of 1 loaf of bread + price of 1gallon of milk + price of 2lbs of banana = Total price.
This simple equation contains the key concepts associated with mathematical equations:
(1) Understanding the problem to be solved.
(2) Formulation of the relationships between the variables involved in the problem.
(3) Determination of the appropriate method for solving the problem.
The so called "word problems" are usually used by examiners to test students' knowledge of concepts (1) and (2).
Equations can be grouped into two broad categories: univarite equations which consists of just one variable and multivariate equations which consists of more than one variable. Each group is further categorized according to the highest power (exponent) of the variables; for example a univariate equation in which the exponent of the variable is 1 is called a linear equation. Some other commonly encountered univariate equations are:
Quadratic equations (highest exponent of variable is 2)
Cubic equations (highest exponent of variable is 3)
Quartic equations (highest exponent of variable is 4).
A significant number of research mathematicians devote much of their time to finding appropriate methods for arriving at solutions to many types of equations. Efficient methods have been established for the solution of linear equations , quadratic equations, cubic equations and quartic equations.
Mathematical equality is different from the everyday conception of equality.
The mathematical assertion that A = B without qualification implies that A = B in total. This total equality is non-existent in nature. It does exist however, in the legal, philosophical
and political space.
The philosophical notion of equality with respect to an entity's right to a happy existence in the Universal Space made possible by the Creator of all creatures, makes good sense. However, it does not imply that A is as tall as B, or that B is as intelligent as A, etc.
The notion of total equality in the mathematical sense, nullifies the notion of intrinsic variability. A rare scenario in mathematics and in nature.