Are two coincident lines the same line?

[tomwilliam2]

Homework Statement

My son's textbook says In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. He asked me why two lines couldn't be drawn on top of each other...to which I replied that they would have the same algebraic form and would therefore be considered the same line. I also think that if a line is the sum of all points, then there aren't two lines but one.

Later on in his book, though, it describes two forms of parallel lines: coincident lines and separate parallel lines. Is this just loose terminology - aren't coincident lines better described as just one line? It might seem a minor point, but I wasn't really able to explain it satisfactorily to him.

 

[tnich]

Homework Statement

My son's textbook says In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. He asked me why two lines couldn't be drawn on top of each other...to which I replied that they would have the same algebraic form and would therefore be considered the same line. I also think that if a line is the sum of all points, then there aren't two lines but one.

I think your explanation is a good one.

Later on in his book, though, it describes two forms of parallel lines: coincident lines and separate parallel lines. Is this just loose terminology - aren't coincident lines better described as just one line? It might seem a minor point, but I wasn't really able to explain it satisfactorily to him.

I think you are right. I suspect that the context is solving systems of linear equations. When you do this, it is possible to end up with two equations with the same slope. Whether the system has a solution or not depends on whether the two lines are coincident (that is, the same line). For example, given two lines y = x + a and y = x + b, if a=b, you would call them coincident, meaning that they are the same line, and any ordered pair (x,y) that solves one equation would solve the other. If a≠b, then no simultaneous solution exists. So, yes, I think coincident is just a way to say that two representations of a line refer to the same line.

 

[scottdave]

...I think you are right. I suspect that the context is solving systems of linear equations. When you do this, it is possible to end up with two equations with the same slope...

I was thinking the same thing.

 

[mathwonk]

you guys are absolutely right, and this shows you are more perceptive than the book's authors. one might say that "there is only one line passing though a point off the line L and parallel to L, i.e. any two such lines are coincident", to nail it. Then one might say that there are two ways for a line to be parallel to L: either to have no points in common with L, or to be coincident with L; in particular every line is considered to be parallel to itself. The purpose of this odd definition is so that "parallel" will be an equivalence relation, which is convenient.

 

[jack action]

I agree with the textbook, to use the term coincident, you need more than one line.

For example, when using CAD programs, you can draw two cubes, select a ridge from each cube and define the two ridges as coincident. Even though the ridges are defined by the same mathematical equation, you still have two distinct ridges (or lines).

I think that is why the distinction is important.

But it all depends on how you define line. From https://en.wikipedia.org/wiki/Line_(geometry):

In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation, but in a more abstract setting, such as incidence geometry, a line may be an independent object, distinct from the set of points which lie on it.