[jcranberry]

Suppose x is an integer such that x + 3 = 1. Then, x is -2. There's no solutions here, just implications and an alternative way of defining the value of x.

I think variables in equations are not meant to express the existence of variance within an equation, but a sense of context-dependency of the value of x. At least, IMO.

[sykh]

There is a solution. An equation is really a question.

x+3 = 1

is asking the question, “what value for x makes x+3 the number 1?”

The polynomial x+3 is defined for all values in R, the base ring you are working in. We are trying to find the elements of R for which x+3 is the element 1.

[jcranberry]

An equation is an instance of equating.

A solution necessitates a question, and questions associated with equations involving variables aren't restricted to: what is set of possible values which satisfy those equations?

[sykh]

You are incorrect. This is not how mathematicians view a polynomial equation like the one I used as an example. That equation does have a solution.