Taking the example of projective space of dimension three, there will be homogeneous coordinates (x:y:z:w). The plane at infinity is usually identified with the set of points with w = 0. Away from this plane we can use (x/w, y/w, z/w) as an ordinary Cartesian system; therefore the affine space complementary to the plane at infinity is coordinatised in a familiar way, with a basis corresponding to (1:0:0:1), (0:1:0:1), (0:0:1:1).
If we try to intersect the two planes defined by equations x = w and x = 2w then we clearly will derive first w = 0 and then x = 0. That tells us that the intersection is contained in the plane at infinity, and consists of all points with coordinates (0:y:z;0). It is a line, and in fact the line joining (0:1:0:0) and (0:0:1:0). It cannot be given by a single equation in the coordinates. In fact a line in threedimensional projective space corresponds to a twodimensional subspace of the underlying fourdimensional vector space, therefore given by two linear conditions.
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