When we say that an equation is 'inconsistent' in ordinary speech, what we normally mean is that is that there is some sort of contradiction that obtains. In mathematics, however, an inconsistent equation is an equation for which there is no answer. Take the equation y = y + 1, for example. This is a straightforward example of an inconsistent equation. The reason for this is simple: No matter what number you enter for the variable "y," you will not get an answer that makes any sense. Suppose we enter "0" for y. This means that we will have the equation 0 = 0 + 1, which is obviously false. It is equivalent to saying 0 + 1 = 0, which is obviously incorrect. Suppose we enter 1. Then we will have an equation that is equivalent to saying 1 + 1 = 1, which is obviously incorrect as well. Suppose we enter 2. Then we will have an equation saying that 1 + 2 = 2, which is incorrect. Regardless of what number you enter into the y variable, you will not obtain an answer that makes any sense. It is for this reason that the equation is an inconsistent equation.
March 30, 2014