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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Introduction to Linear Systems">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Fri Mar 28 16:06:47 1997}
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\section{Introduction to Systems of Linear Equations\label%
{Introduction to Systems of Linear Equations}}

\subsubsection{\protect\vspace{1pt}Linear Equation\label{linear equation}}

A \textbf{linear equation} in the $n$ variables $x_{1},x_{2},...,x_{n}$ is
an equation that can be written in the form $%
a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b$, where the \textbf{coefficients\label%
{coefficients} }$a_{1},a_{2},...,a_{n}$ and the \textbf{constant term\label%
{constant term}} $b$ are numbers.

\begin{itemize}
\item  A \textbf{solution\label{solution} }of a linear equation is a
collection of values for the variables that make the equation true when
these values are substituted for the variables in the equation.

\item  Variables in the equation are often referred to as \textbf{unknowns%
\label{unknowns}}.
\end{itemize}

\subsubsection{System of Linear Equations\label{system of linear equations}}

The general form of a system of $m$ equations in $n$ unknowns is

\[
\begin{array}{lllllllll}
a_{11}x_{1} & + & a_{12}x_{2} & + & ... & + & a_{1n}x_{n} & = & b_{1} \\ 
a_{21}x_{1} & + & a_{22}x_{2} & + & ... & + & a_{2n}x_{n} & = & b_{2} \\ 
. &  & . &  & ... &  & . &  & . \\ 
. &  & . &  & ... &  & . &  & . \\ 
. &  & . &  & ... &  & . &  & . \\ 
a_{m1}x_{1} & + & a_{m2}x_{2} & + & ... & + & a_{mn}x_{n} & = & b_{m}
\end{array}
\]

This system is called an $m\times n$ ($m$ by $n$) linear system. A system of 
$n$ equations in $n$ unknowns is called an $n\times n$ linear system, or a 
\textbf{square linear system\label{square linear system}}.

\begin{itemize}
\item  A solution of a system of linear equations is a collection of values
for the variables that makes all the equations true.

\item  A linear system that has no solutions is called \textbf{inconsistent}.%
\label{inconsistent}

\item  A linear system that does have solutions is called \textbf{consistent}%
.\label{consistent}
\end{itemize}

\subsection{\ Related Reference Topics}

{\small \hyperref{Solving systems of linear equations by the substitution
method}{}{}{Al06_01.tex#Substitution method for solving systems of equations}%
}

{\small \hyperref{Solving systems of linear equations by the elimination
method}{}{}{Al06_01.tex#Elimination method for solving systems of equations}}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Equations and matrices}{}{}{../../Help/DM8-4.tex#Systems
of equations}}

{\small \hyperref{Systems of equations}{}{}{../../Help/DM3-3.tex#Systems of
equations}}

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