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.