# How many kinds of solutions are there?

Jun 22, 2015

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:

${x}_{1} = {a}_{1} \cdot t + {b}_{1}$
${x}_{2} = {a}_{2} \cdot t + {b}_{2}$
...
${x}_{n} = {a}_{n} \cdot t + {b}_{n}$

for all $t \in \mathbb{R}$

(3) A plane of solutions expressible as:

${x}_{1} = {a}_{1} \cdot {t}_{1} + {b}_{1} \cdot {t}_{2} + {c}_{1}$
${x}_{2} = {a}_{2} \cdot {t}_{1} + {b}_{2} \cdot {t}_{2} + {c}_{2}$
...
${x}_{n} = {a}_{n} \cdot {t}_{1} + {b}_{n} \cdot {t}_{2} + {c}_{n}$

for all $\left({t}_{1} , {t}_{2}\right) \in \mathbb{R} \times \mathbb{R}$

...
(n+1) The whole of ${\mathbb{R}}^{n}$