How many kinds of solutions are there?

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How many kinds of solutions are there?

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Answer 1 · 2015-06-22T22:10:33

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 $n$ $n$ variables, then there are $n + 2$ $n + 2$ kinds of solutions.

Explanation:

If a linear system involves $n$ $n$ variables, $x_{1}, x_{2}, . . x_{n}$ $x_1, x_2,..x_n$ , then the solution set will take one of the following $n + 2$ $n + 2$ forms:

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

$x_{1} = a_{1} \cdot t + b_{1}$ $x_1 = a_1*t + b_1$
$x_{2} = a_{2} \cdot t + b_{2}$ $x_2 = a_2*t + b_2$
...
$x_{n} = a_{n} \cdot t + b_{n}$ $x_n = a_n*t + b_n$

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 $RR^n$

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How many kinds of solutions are there?

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