Do the equations x = 4y + 1 and x = 4y – 1 have the same solution? How can I explain this to someone that does not know algebra?

No. The solution of a linear function in two variables is an infinite set of points - so we typically represent it as a graph, or we may talk about particular solutions. For example what is x when y=0?

Graphing each of these equations we will get two different lines - that's one sure-fire way to know they have different solutions.

:)