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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Linear Equations">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 23 15:53:27 1997}
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\section{Linear Equations\label{Linear Equations}}

A \textbf{linear equation} is an equation in which each term is either a
constant or a nonzero multiple of the variable.

\begin{itemize}
\item[Example]  These are linear equations: 
\begin{eqnarray*}
4x-5 &=&3 \\
2x &=&\frac{1}{2}x-5
\end{eqnarray*}
These are nonlinear equations: 
\begin{eqnarray*}
x^{2}+2x &=&8 \\
\sqrt{x}-\frac{3}{x} &=&6x-1
\end{eqnarray*}
\bigskip 
\end{itemize}

\subsection{Solving a Linear Equation\label{Solving Linear Equations}}

You solve a linear equation by using the properties of equality to transform
it into an equivalent statement of the form 
\[
x=? 
\]
That is, you perform steps in which you add the \emph{same number} to both
sides or multiply the \emph{same number} times both sides of the equation
until you isolate the variable on one side of the equation.

\begin{itemize}
\item[Example]  Solve the equation: 
\[
7x-4=3x+8
\]
\textbf{Solution}: Add $4$ to each side: 
\[
\left( 7x-4\right) +4=\left( 3x+8\right) +4
\]
Simplify: 
\[
7x=3x+12
\]
Subtract $3x$ from each side: 
\[
7x-3x=3x+12-3x
\]
Simplify: 
\[
4x=12\text{ }
\]
Multiply each side by $\frac{1}{4}:$%
\[
\frac{1}{4}\cdot 4x=\frac{1}{4}\cdot 12\text{ }
\]
Simplify: 
\[
x=3
\]
\bigskip 

\item[Example]  Solve the equation: 
\[
\frac{x}{6}+\frac{2}{3}=\frac{3}{4}x
\]
\textbf{Solution}: The Least Common Denominator (LCD) of $6$, $3$, and $4$
is $12$. Rewrite the fractions with the common denominator: 
\[
\frac{2x}{12}+\frac{8}{12}=\frac{9}{12}x
\]
Multiply each side by $12$: 
\[
12\left( \frac{2x}{12}+\frac{8}{12}\right) =12\left( \frac{9}{12}x\right) 
\]
Simplify: 
\[
2x+8=9x
\]
Subtract $2x$ from each side: 
\[
\left( 2x+8\right) -2x=9x-2x
\]
Simplify: 
\[
8=7x
\]
Multiply each side by $\frac{1}{7}$: 
\[
\frac{8}{7}=x
\]
\bigskip 
\end{itemize}

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