In classical or Euclidean plane geometry two points defines exactly one line. On a sphere two points can define infinitely many lines only one of which will represent the shortest distance between the points. On other curved surfaces, or in non-Euclidean geometries, the number of lines determined by two points can vary.
Even in the Euclidean plane, two points determine infinitely many lines that are not straight!
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If two lines have different slopes, then they intersect at exactly one point. It makes no difference what their y-intercepts are.
one
1 straight line. An infinite number of curved lines.
Just one.