When 2 or more straight lines meet...the point where they meet is called the point of intersection
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A vertex is defined as the place where two or more straight lines meet. A cube has eight vertices, which are the corners of the cube.
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No. A linear equation represents a straight line and the solution to a set of linear equations is where the lines intersect; two straight lines can only intersect at most at a single point - two straight lines may be parallel in which case they will not intersect and there will be no solution. With more than two linear equations, it may be that they do not all intersect at the same point, in which case there is no solution that satisfies all the equations together, but different solutions may exist for different subsets of the lines.
Yes. Any two distinct lines of longitude, for example, meet at two points - the poles. On a plane, though, two points define a unique line. So if two lines intersect at more than one point they must be coincident.
It is called The Origin. You can go to Wikipedia.com and get more information on The Origin.