01 Systems of Equations

A system of linear equations is a collection of linear equations with some variables in common, such as

$$ \begin{aligned} &a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n} &= b_{1} \\ &a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n} &= b_{2} \\ &a_{31}x_{1}+a_{32}x_{2}+\cdots+a_{3n}x_{n} &= b_{3} \\ &\vdots \\ &a_{m1}x_{1}+a_{m2}x_{2}+\cdots+a_{mn}x_{n} &= b_{m} \\ \end{aligned} $$

where all $a$'s and $b$'s are known, and $x$'s are unknown.

Solution
All $x$'s that solve every equation simultaneously
Solution Set
The set of all possible solutions to the system.

Two systems are equivalent if the solution sets are equivalent.

Three possibilities for the solution set

No solutions
Same slope, parallel lines
One unique solution
Lines cross once
Infinitely many solutions
Equivalent Systems