Consider four equations in three unknowns:
aix + biy + CiZ = ki (i = 1,..., 4).
Assume that the pair of equations for i = 1, 2 and the pair for i= 3, 4 each represents a line in space as in Problem. Thus the solutions of the four equations represent the points common to two lines in space. Discuss the geometrical alternatives that can occur and relate them to the set of solutions of the given equations.
Problem
Consider two equations in three unknowns:
a1x + b1y + c1z=k1, a2x + b2y + c2z=k2.
a) Show that if Gaussian elimination can be carried out to solve for x and y, then by writing z − t the solutions become parametric equations for a line in space (Section 1.3).
b) Assume that the two equations represent two planes in space. Interpret geometrically the case in which the equations have no solution and the case in which elimination leads to a second equation 0 = 0.
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