i am just stuck right now. what was the theorem again about the arbitrary variables in the system of equations? if i have 2 equations with 3 variables, then last variable will be arbitrary, right? and if i have 2 equations with 2 variables, then there will be no arbitrary variables and i can solve for all 2 variables, right?
i am just confused now in solving some systems of equations..
thanks for help
$\endgroup$2 Answers
$\begingroup$Please keep in mind that what really counts is not the number of equations, but the number of independent ones. For instance, $$ \begin{cases} x+y = 1\\ 2x+2y=2 \end{cases} $$ may look like two equations, but it' s really one. (With more equations, it's not that simple.) You should have studied how to reduce to the case of independent equations.
$\endgroup$ 3 $\begingroup$The concept is not quite this simple, but the rough answer to your question is that if you have $m$ equations in $n$ variables, then most of the time you will have 1 solution if $m=n$, no solutions if $m>n$, and $n-m$ arbitrary ("free") variables if $m<n$. If there is a specific named theorem for this, I've forgotten it.
As Andreas points out, the precise definition of "most of the time" is that whenever these solutions are dependent, you should remove some until they aren't, and then use the above idea.
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