Unsupervised clustering without of Data which is supposed to be on a linear function When I have a dataset where each datum has x and y,
and the (x,y) has a relation of one of y = a_i*x + b_i (i=1,2,...).

Is the process written below available? and which algorithm does it belong to?

The process is.....

I have many points (x, y).
The machine finds 2 linear functions which represents the points.
The machine eliminates points which are far from the 3 lines.

In this case, I think I put parameters the number of linear functions and criterion to judge if a point is on a line or not.. 

The first figure is my dataset.

I want to have a machine which accepts the number of line (3 in this case)
and finds 3 lines(as the second figure (the lines in the figure are just put for idea without computation)), and then finally suggests points which may belong to neither of them. (In this case, for example, (71.6, 22))

Should I, for instance, extend the k-means algorithm to achieve this procedure?///
I would rather look at mixture models. Or, if there is additional noise besides the lines, use computer vision algorithms, for example, some variation of the Hough transform.