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\begin_layout Title
Solving equations
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\begin_layout Author
N
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'a
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ndor Sieben
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Northern Arizona University
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\begin_layout Section*

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Necessary or sufficient
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\noindent
Consider the equation 
\begin_inset Formula 
\[
x+1=x.
\]

\end_inset

To solve it we first square both sides and get
\begin_inset Formula 
\[
x^{2}+2x+1=x^{2}.
\]

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Subtracting 
\begin_inset Formula $x^{2}+1$
\end_inset

 from both sides gives
\begin_inset Formula 
\[
2x=-1.
\]

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Finally dividing by 
\begin_inset Formula $2$
\end_inset

 results in the incorrect
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\[
x=-\frac{1}{2}.
\]

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Style
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\noindent

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Example of what not to do:
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We solve the equation 
\begin_inset Formula $2(x+2)=x$
\end_inset

.
 
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\noindent
\begin_inset Formula 
\[
\begin{gathered}2(x+2)=x\\
2x+4=x\\
x+4=0\\
x=-4
\end{gathered}
\]

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Correct way:
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We solve the equation 
\begin_inset Formula $2(x+2)=x$
\end_inset

.
 
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\noindent
Expanding the parentheses gives
\begin_inset Formula 
\[
2x+4=x.
\]

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Subtracting 
\begin_inset Formula $x$
\end_inset

 from both sides results in
\begin_inset Formula 
\[
x+4=0.
\]

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Finally subtracting 
\begin_inset Formula $4$
\end_inset

 from both sides implies that the only possibility for the solution is 
\begin_inset Formula 
\[
x=-4.
\]

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Substituting 
\begin_inset Formula $-4$
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 for 
\begin_inset Formula $x$
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 in the original equation verifies that this is in fact a solution.
 So the only solution is 
\begin_inset Formula $x=-4$
\end_inset

.
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