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
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