How many kinds of solutions are there?

1 Answer
Jun 22, 2015

Answer:

From the category in which this question is asked, I will assume you mean a finite linear system of equations. If such a system is in #n# variables, then there are #n + 2# kinds of solutions.

Explanation:

If a linear system involves #n# variables, #x_1, x_2,..x_n#, then the solution set will take one of the following #n + 2# forms:

(0) The empty set. The system is inconsistent and has no solutions.
(1) A unique solution in the form of an #n#-tuple
(2) A line of solutions expressible as:

#x_1 = a_1*t + b_1#
#x_2 = a_2*t + b_2#
...
#x_n = a_n*t + b_n#

for all #t in RR#

(3) A plane of solutions expressible as:

#x_1 = a_1*t_1 + b_1*t_2 + c_1#
#x_2 = a_2*t_1 + b_2*t_2 + c_2#
...
#x_n = a_n*t_1 + b_n*t_2 + c_n#

for all #(t_1, t_2) in RR xx RR#

...
(n+1) The whole of #RR^n#