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<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-1"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">1.1</span> <span class="title">Review of two equations in two unknowns</span>
</h2>
<section class="subsection" id="TwoEquationsTwoUnknownsSample"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">1.1.1</span> <span class="title">The standard method for finding the solution</span>
</h3>
<p id="p-1">Suppose we want to find <em class="emphasis">all</em> solutions to the equations</p>
<div class="displaymath" id="p-2">
\begin{align*}
2x+3y \amp = 5\\
3x+2y \amp= 7
\end{align*}
</div>
<p id="p-3">The <em class="emphasis">standard</em> technique is to manipulate one or both of the equations until either \(x\) has the same coefficient in both equations or \(y\) has the same coefficient, and then subtract to eliminate one variable. Since there is only one variable left, its value can be found; with this information, the value for the other variable can be found.</p>
<p id="p-4">In the case above, we can multiply both sides of the second equation by \(\frac23\text{,}\) to get</p>
<div class="displaymath">
\begin{equation*}
2x+\tfrac43 y= \tfrac{14}3,
\end{equation*}
</div>
<p class="continuation">and subtracting from the first equation gives</p>
<div class="displaymath">
\begin{equation*}
\frac53y = \frac13,
\end{equation*}
</div>
<p class="continuation">or</p>
<div class="displaymath">
\begin{equation*}
y=\frac15.
\end{equation*}
</div>
<p id="p-5">Using the first equation and the (now) known value of \(y\text{,}\) we find that</p>
<div class="displaymath">
\begin{equation*}
x=\frac{11}5.
\end{equation*}
</div>
<p id="p-6">Hence we see that there is exactly one pair of values for \(x\) and \(y\) that simultaneously satisfy both equations: \(x=\frac{11}5\) and \(y=\tfrac15.\) We can also write this as \((x,y)=(\frac{11}5,\frac15).\) In this case we say that there is a <em class="emphasis">unique</em> solution (in mathematics, the term unique means <em class="emphasis">exactly one</em>).</p></section><section class="subsection" id="subsection-2"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">1.1.2</span> <span class="title">The geometric method of finding the solution</span>
</h3>
<p id="p-7">The set of equations solved in <a href="section-1.html#TwoEquationsTwoUnknownsSample" class="internal" title="Subsection 1.1.1: The standard method for finding the solution">Subsection 1.1.1</a> can also be viewed geometrically. The points in the \(x\)-\(y\) plane satisfying one of the equations lie on a straight line, and any point satisfying both of the equations must lie on both lines. The plot of the two lines looks like this:</p>
<figure class="figure figure-like" id="figure-1"><div class="image-box" style="width: 40%; margin-left: 30%; margin-right: 30%;">
<div class="asymptote-box" style="padding-top: 141.585618493235%"><iframe src="images/TwoLinesInPlane.html" class="asymptote"></iframe></div>
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</div>
</div>
<figcaption><span class="type">Figure</span><span class="space"> </span><span class="codenumber">1.1.1<span class="period">.</span></span><span class="space"> </span>Intersection of two lines in the plane</figcaption></figure><p id="p-8">The point of intersection is \((x,y)=(\frac{11}5,\frac15)\text{,}\) so this is the only solution, just as before.</p>
<article class="exercise exercise-like" id="exercise-1"><a data-knowl="" class="id-ref exercise-knowl original" data-refid="hk-exercise-1"><h6 class="heading">
<span class="type">Checkpoint</span><span class="space"> </span><span class="codenumber">1.1.2</span><span class="period">.</span>
</h6></a></article><div class="hidden-content tex2jax_ignore" id="hk-exercise-1"><article class="exercise exercise-like"><p id="p-9">Are the points satisfying the equation \(2x+3y = 5\) on the red line or the green line?</p>
<div class="solutions">
<a data-knowl="" class="id-ref solution-knowl original" data-refid="hk-solution-1" id="solution-1"><span class="type">Solution.</span> </a><div class="hidden-content tex2jax_ignore" id="hk-solution-1"><div class="solution solution-like"><p id="p-10">Setting \(x=0\) gives \(y=\frac53\) as the point where the line intersects the \(y\)-axis. Hence it is the green line.</p></div></div>
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