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Infinite.
many points, even curved line as long as the line is not broken it could go on for infinity!
In general, a linear equation CANNOT be made to go through three points. That will only happen if the three points are collinear and in that case, the equation of the line will only require two points.
Infinitely many, though in a plane, only one straightline.However, an arbitrarily large number of curved linescould pass through two points.[As an aside there some non-Euclidian geometries in which the above do not apply, but I'm sure they lie outside your scope of inquiry.]
planes are used for cargo shipping, passenger shipping, or just sight seeing.