Lets not confuse the beginning algebra student.
In x+3 = 4 we have x=1. It is not a line, it is not the point (1,0). It is the solution to an equation.
The line x=3 is a line, it contains many points. In fact it contains each and every point on the x-y plane (just think graph paper) where the x value is 1. Like (1, 3/4), (1, 7), (1, -11), (1, 3sqrt(5)), ...
x=1 never represents the point (1,0). After all, why not (1,3)?
You need to understand where x=1 is coming from. If I ask you if x=1 is a solution to ...., then x is only one. It is not a point it is just a possible solution to an equation.
Same goes for y= some constant.