Question

How many kinds of solutions are there?

Answer 1

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`