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374 lines
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<title>Systems of Linear and Quadratic Equations</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 class="center">Solving Systems of Linear <br />and Quadratic Equations
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Graphically
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</h1>
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<p class="center"><i>(also see <a href="systems-linear-quadratic-equations.html">Systems of Linear and Quadratic Equations</a>)</i></p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td bgcolor="#eef"><img src="images/linear-quadratic-a.svg" alt="linear " /></td>
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<td><span class="center">A <a href="linear-equations.html">Linear Equation</a> is an <b>equation</b> of a <b>line</b>.</span></td>
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</tr>
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<tr>
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<td bgcolor="#eef"><img src="images/linear-quadratic-b.svg" alt="quadratic" /></td>
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<td>A <a href="quadratic-equation.html">Quadratic Equation</a> is the equation of a <a href="../geometry/parabola.html">parabola</a> <br />
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and has at least one variable squared
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(such as x<sup>2</sup>)<br /></td>
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</tr>
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<tr>
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<td bgcolor="#eef"><img src="images/linear-quadratic-c.svg" alt="linear and quadratic" /></td>
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<td>And together they form a <b>System</b> <br />
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of a Linear and a Quadratic Equation</td>
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</tr>
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</table>
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</div>
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<p> </p>
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<p>A <b>System</b> of those two equations can be solved (find where they intersect), either:</p>
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<ul>
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<li>Using <a href="systems-linear-quadratic-equations.html">Algebra</a></li>
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<li>Or <b>Graphically</b>, as we will find out!</li>
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</ul>
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<h2>How to Solve Graphically</h2>
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<p>Easy! Plot both equations and see where they cross!</p>
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<h3>Plotting the Equations</h3>
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<p>We can plot them manually, or use a tool like the <a href="../data/function-grapher.html">Function Grapher</a>.</p>
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<p>To plot them manually:</p>
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<ul>
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<li>make sure both equations are in "y=" form</li>
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<li>choose some x-values that will hopefully be near where the two equations cross over</li>
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<li>calculate y-values for those x-values</li>
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<li>plot the points and see! </li>
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</ul>
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<h3>Choosing Where to Plot</h3>
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<p>But what values should we plot? Knowing the <b>center</b> will help! </p>
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<p>Taking the <a href="quadratic-equation.html">quadratic formula</a> and ignoring everything after the <span class="large">±</span> gets us a central x-value:</p>
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<p class="center"><img src="images/system-lin-quad-4.gif" width="268" height="133" alt="x = -b/2a on graph" /></p>
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<p>Then choose some x-values either side and calculate y-values, like this:</p>
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<div class="example">
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<h3>Example: Solve these two equations graphically to 1 decimal place:</h3>
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<ul>
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<li>y = x<sup>2</sup> − 4x + 5</li>
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<li>y = x + 2 </li>
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</ul>
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<p> </p>
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<p class="larger">Find a Central X Value:</p>
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<p>The quadratic equation is <b>y = x<sup>2</sup> − 4x + 5</b>, so a = 1, b = −4 and c = 5</p>
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<table border="0" align="center">
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<tr>
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<td rowspan="2" class="large">central x = </td>
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<td class="large">−b</td>
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<td rowspan="2" class="large"> = </td>
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<td class="large">−(−4)</td>
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<td rowspan="2" class="large"> = </td>
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<td class="large">4</td>
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<td rowspan="2" class="large"> = 2</td>
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</tr>
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<tr>
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<td align="center" class="large" style="border-top: 1px solid black;">2a</td>
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<td align="center" class="large" style="border-top: 1px solid black;">2×1</td>
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<td align="center" class="large" style="border-top: 1px solid black;">2</td>
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</tr>
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</table>
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<p> </p>
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<p class="larger">Now Calculate Values Around x=2</p>
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<table border="0" align="center" cellpadding="5">
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<tr>
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<th width="50" align="center"><br />
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<span class="larger">x</span></th>
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<th align="center">Quadratic<br />
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<span class="larger">x<sup>2</sup> − 4x + 5</span></th>
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<th width="100" align="center">Linear<br />
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<span class="larger">x + 2</span></th>
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</tr>
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<tr>
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<td align="center">0</td>
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<td width="150" align="center">5</td>
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<td align="center">2</td>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">2</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center"><b>2</b></td>
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<td align="center"><b>1 </b></td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">2</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">5</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">5</td>
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<td align="center">10</td>
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<td align="center">7</td>
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</tr>
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</table>
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<p>(We only calculate first and last of the linear equation as that is all we need for the plot.)</p>
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<p> </p>
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<p class="larger">Now Plot Them:</p>
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<p class="center"><img src="images/system-lin-quad-1.gif" width="181" height="293" alt="system linear and quadratic points" /></p>
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<p class="larger">We can see they cross at <b>about x = 0.7</b> and <b>about x = 4.3</b></p>
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<p>Let us do the calculations for those values:</p>
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<table border="0" align="center" cellpadding="5">
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<tr>
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<th width="50" align="center"><br />
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<span class="larger">x</span></th>
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<th align="center">Quadratic<br />
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<span class="larger">x<sup>2</sup> − 4x + 5</span></th>
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<th width="100" align="center">Linear<br />
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<span class="larger">x + 2</span></th>
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</tr>
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<tr>
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<td align="center">0.7</td>
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<td width="150" align="center">2.69</td>
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<td align="center">2.8</td>
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</tr>
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<tr>
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<td align="center">4.3</td>
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<td align="center">6.29</td>
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<td align="center">6.2</td>
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</tr>
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</table>
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<p>Yes they are close.</p>
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<p class="center">To 1 decimal place the two points are <b>(0.7, 2.8)</b> and <b>(4.3, 6.2)</b></p>
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</div>
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<h2>There Might Not Be 2 Solutions!</h2>
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<p>There are three possible cases:</p>
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<ul>
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<li><b>No</b> real solution (happens when they never intersect)</li>
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<li><b>One</b> real solution (when the straight line just touches the quadratic)</li>
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<li><b>Two</b> real solutions (like the example above)</li>
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</ul>
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<p class="center"><img src="images/linear-quadratic-2.svg" width="80%" alt="linear and quadratic different intersections" /></p>
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<p>Time for another example:</p>
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<div class="example">
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<h3>Example: Solve these two equations graphically:</h3>
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<ul>
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<li>4y − 8x = −40</li>
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<li>y − x<sup>2</sup> = −9x + 21</li>
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</ul>
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<p>How do we plot these? They are not in "y=" format!</p>
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<p class="larger">First make both equations into "y=" format:</p>
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<p>Linear equation is: 4y − 8x = −40</p>
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<div class="so">Add 8x to both sides: 4y = 8x − 40</div>
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<div class="so">Divide all by 4: <b>y = 2x − 10</b></div>
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<p>Quadratic equation is: y − x<sup>2</sup> = −9x + 21</p>
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<div class="so">Add x<sup>2</sup> to both sides: <b>y = x<sup>2</sup> − 9x + 21</b></div>
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<p class="larger"> </p>
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<p class="larger">Now Find a Central X Value:</p>
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<p>The quadratic equation is <b>y = x<sup>2</sup> − 9x + 21</b>, so a = 1, b = −9 and c = 21</p>
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<table border="0" align="center">
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<tr>
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<td rowspan="2" class="large">central x = </td>
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<td class="large">−b</td>
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<td rowspan="2" class="large"> = </td>
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<td class="large">−(−9)</td>
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<td rowspan="2" class="large"> = </td>
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<td class="large">9</td>
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<td rowspan="2" class="large"> = 4.5</td>
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</tr>
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<tr>
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<td align="center" class="large" style="border-top: 1px solid black;">2a</td>
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<td align="center" class="large" style="border-top: 1px solid black;">2×1</td>
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<td align="center" class="large" style="border-top: 1px solid black;">2</td>
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</tr>
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</table>
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<p> </p>
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<p class="larger">Now Calculate Values Around x=4.5</p>
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<table border="0" align="center" cellpadding="5">
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<tr>
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<th width="50" align="center"><br />
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<span class="larger">x</span></th>
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<th align="center">Quadratic<br />
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<b>x<sup>2</sup> − 9x + 21</b></th>
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<th width="100" align="center">Linear<br />
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<b>2x − 10</b></th>
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</tr>
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<tr>
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<td align="center">3</td>
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<td width="150" align="center">3</td>
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<td align="center">-4</td>
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</tr>
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<tr>
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<td align="center">4</td>
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<td align="center">1</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center"><b>4.5</b></td>
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<td align="center"><b>0.75</b></td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">5</td>
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<td align="center">1 </td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">6</td>
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<td align="center">3</td>
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<td align="center"> </td>
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</tr>
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<tr>
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<td align="center">7</td>
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<td align="center">7</td>
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<td align="center">4</td>
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</tr>
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</table>
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<p> </p>
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<p class="larger">Now Plot Them:</p>
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<p class="center"></p>
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<p class="center"><img src="images/system-lin-quad-2.gif" width="237" height="311" alt="system linear and quadratic points" /></p>
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<p class="center">They never cross! There is <b>no solution</b>.</p>
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</div>
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<p> </p>
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<div class="example">
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<h3>Real World Example</h3>
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<p><span class="larger">Kaboom!</span></p>
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<p>The cannon ball flies through the air, following a <a href="../geometry/parabola.html">parabola</a>: <span class="larger">y = 2 + 0.12x - 0.002x<sup>2</sup></span></p>
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<p>The land slopes upward: <span class="larger">y = 0.15x </span></p>
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<p>Where does the cannon ball land?</p>
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<p class="center"><img src="images/linear-quadratic-5.gif" width="471" height="235" alt="linear quadratic cannon" /></p>
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<p>Let's fire up the <a href="../data/function-grapher42f4.html?func1=2+0.12x-0.002x^2&func2=0.15x&xmin=-48&xmax=48&ymin=-32&ymax=32">Function Grapher</a>!</p>
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<p>Enter <span class="larger">2 + 0.12x - 0.002x^2</span> for one function and <span class="larger">0.15x</span> for the other.</p>
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<p>Zoom out, then zoom in where they cross. You should get something like this:</p>
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<p class="center"><img src="images/linear-quadratic-6.gif" width="230" height="101" alt="linear quadratic " /></p>
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<p class="center">By zooming in far enough we can find they cross at <b>(25, 3.75)</b></p>
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</div>
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<h2>Circle and Line</h2>
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<div class="example">
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<h3>Example: Find the points of intersection to 1 decimal place of </h3>
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<ul>
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<li>The circle <b>x<sup>2</sup> + y<sup>2</sup> = 25 </b></li>
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<li>And the straight line <b>3y - 2x = 6</b></li>
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</ul>
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<p> </p>
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<h3>The Circle</h3>
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<p>The "Standard Form" for <a href="circle-equations.html">the equation of a circle</a> is <span class="large">(x-a)<sup>2</sup> + (y-b)<sup>2</sup> = r<sup>2</sup></span>
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</p>
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<p>Where <span class="large">(a, b)</span> is the center of the circle and <span class="large">r</span> is the radius.</p>
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<p>For <b>x<sup>2</sup> + y<sup>2</sup> = 25 </b>we can see that</p>
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<ul>
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<li>a=0 and b=0 so the center is at <b>(0, 0)</b>,</li>
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<li>and for the radius <b>r<sup>2</sup> = 25 </b>, so <b>r = √25 = 5</b></li>
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</ul>
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<p>We don't need to make the circle equation in "y=" form, as we have enough information to plot the circle now.</p>
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<p> </p>
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<h3>The Line</h3>
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<p>First put the line in "y=" format: </p>
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<div class="so">Move 2x to right hand side: 3y = 2x + 6</div>
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<div class="so">Divide by 3: y = 2x/3 + 2</div>
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<p>To plot the line, let's choose two points either side of the circle:</p>
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<ul>
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<li>at <b>x = −6</b>, <b>y =</b> (2/3)(<b>−</b>6) + 2 = <b>−2</b></li>
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<li>at <b>x = 6</b>, <b>y =</b> (2/3)(6) + 2 = <b>6</b></li>
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</ul>
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<p>Now plot them!</p>
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<p class="center"><img src="images/system-lin-quad-3.gif" width="332" height="325" alt="line vs circle" /></p>
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<p>We can now see that they cross at <b>about (-4.8, -1.2)</b> and <b> (3.0, 4.0)</b></p>
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<p>For an exact solution see <a href="systems-linear-quadratic-equations.html">Systems of Linear and Quadratic Equations </a></p></div>
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<p> </p>
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<div class="related">
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<a href="systems-linear-quadratic-equations.html">Systems of Linear and Quadratic Equations</a>
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