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      <text>7.1.2 (Example of solving two lin. sim. eqns. graphically)</text>
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      <text><![CDATA[<h3>Example</h3>

<p>By considering their graphs solve the simultaneous equations \[ \begin{align*} &amp;y=3x-1 \\ &amp;x+y=3 \end{align*} \]</p>

<h3>Solution</h3>

<p>If we plot these functions on the same axes we obtain the following.</p>
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<p><img style="display: block; margin-left: auto; margin-right: auto;" src="@@PLUGINFILE@@/Simultaneous%20eqns%20lin%201.png?time=1602587286374" alt="graph of two linear simultaneous equations" width="402" height="358" /></p>
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<p>We can see that the graphs meet at the point \((1,2)\) and therefore the solution to the simultaneous equations is \(x = 1, y = 2\).</p>
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<p>As with the other methods of solving simultaneous equations, we can check the solution by substituting the values we found into the initial equations and checking that they hold. In this case note that 
\[
y = 3(1) -1 = 2  \qquad \mbox{and} \qquad 1 + 2 = 3,
\]
both of these equations are true, therefore the solutions we found are correct.</p>
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<p>We can see that the graphical method appears to be the quickest way to solve simultaneous equations. This, in general, is probably true, however the main difficulty is to be able to accurately plot the functions. This can become more difficult with more complicated equations. Not that this graphical method explains why there is only one solution to two linear simultaneous equations. It is because two straight lines can intersect each other at a maximum of one point. </p>]]></text>
<file name="Simultaneous eqns lin 1.png" path="/" encoding="base64">[encoded data, 75592 characters]</file>
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