Homework 4, Math 152-205, additional problem: Consider a linear system of some number of equations in 4 variables, x1, x2, x3, x4, where the coefficients of the x's are fixed real numbers, but the constants of the equations involve a parameter c as well as fixed real numbers. An example of this is Problem 6 of this assignment. Another example would be: 5 x1 - 4 x2 + 3 x3 + 7 x4 = 2 + 5 c - 12 sin(c) 3 x1 - 9 x2 - 2 x3 + 2 x4 = 4 + 9 c - 97 tan(c) For any such system, is it possible that (for the same system) there exists (1) a value of c for which the system has no solutions, AND (2) a value of c for which the system has a unique solution, AND (3) a value of c for which the system has infinitely many solutions? More generally, give all combinations of (1), (2), and (3) that can occur. Hint: you can ask the same question about a linear system involving one variable, x1, and the answer will be the same. But you should be able to argue the same no matter how many variables there are. Second hint: you might first consider the homogeneous form of the system.