# Consistent and Inconsistent Linear Systems

## Key Questions

• Two curves are consistent if it is possible for some point to be on both. (Being on one curve is consistent with being on the other.) There is an intersection. (Possibly many intersections.)

Two curves are inconsistent is it is impossible for any point to be on both. (Being on one curve is inconsistent with being on the other -- it contradicts, being on the other.) There is no intersection.

Statements are consistent if it is possible for both to be true, statements are inconsistent if it is not possible for both to be true. (The truth of one is consistent or inconsistent with the truth of the other.)

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 ${\mathbb{R}}^{n}$

• A system of equations is said to be consistent if it has at least one solution; otherwise, it is inconsistent.

I hope that this was helpful.