Let me be more precise. Assume x+3 is an element of R[x] with x an indeterminate and R a ring with characteristic not equal to 3. Then x+3 defines a natural map from R to R. The equation x+3=1 is just a shorthand way of asking for the pre-image of this map.

In two variables we get a map from R^2 to R^2and solutions are ordered pairs. By definition of an element of a polynomial ring over R the variables are ordered.