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353 lines
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HTML
353 lines
13 KiB
HTML
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<!-- Mirrored from www.mathsisfun.com/algebra/linear-equations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:00:20 GMT -->
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<head>
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<title>Linear Equations</title>
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<body id="bodybg">
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<div id="logo"><a href="../index.html"><img src="../images/style/logo.svg" alt="Math is Fun" /></a></div>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<a name="Top" id="Top"></a>
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<center>
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<h1>Linear Equations</h1> </center>
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<p align="center" class="larger">A <b>linear</b> equation is an <a href="definitions.html">equation</a> for a straight <b>line</b></p>
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<div class="example">
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<h3>These are all linear equations:</h3>
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<table border="0" align="center">
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td>y = 2x + 1</td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td>5x = 6 + 3y</td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td>y/2 = 3 − x</td>
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</tr>
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</table>
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</div>
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<p>Let us look more closely at one example:</p>
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<div class="example">
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<h3>Example: <b>y = 2x + 1</b> is a linear equation:</h3>
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<p align="center"><img src="images/graph-line.svg" alt="line on a graph" style="background-color:white;" /></p>
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<p align="center">The graph of <b>y = 2x+1</b> is a straight line</p>
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<p align="center"> </p>
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<ul>
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<li>When x increases, y increases <b>twice as fast</b>, so we need <span class="large">2x</span></li>
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<li>When x is 0, y is already 1. So <span class="large">+1</span> is also needed</li>
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<li>And so: <span class="large">y = 2x + 1</span></li>
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</ul>
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<p>Here are some example values:</p>
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<table border="0" align="center">
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<tr align="center">
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<th width="50">x</th>
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<th width="200">y = 2x + 1</th>
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</tr>
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<tr align="center">
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<td width="50"><b>-1</b></td>
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<td width="200">y = 2 × (-1) + 1 = <b>-1</b></td>
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</tr>
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<tr align="center">
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<td width="50"><b>0</b></td>
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<td width="200">y = 2 × 0 + 1 = <b>1</b></td>
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</tr>
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<tr align="center">
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<td width="50"><b>1</b></td>
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<td width="200">y = 2 × 1 + 1 = <b>3</b></td>
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</tr>
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<tr align="center">
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<td width="50"><b>2</b></td>
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<td width="200">y = 2 × 2 + 1 = <b>5</b></td>
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</tr>
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</table>
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<p>Check for yourself that those points are part of the line above!</p>
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</div>
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<h2>Different Forms</h2>
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<p>There are many ways of writing linear equations, but they usually have <b>constants</b> (like "2" or "c") and must have simple <b>variables</b> (like "x" or "y").</p>
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<div class="example">
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<h3>Examples: These are linear equations:</h3>
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<table border="0" align="center">
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td><span class="larger">y = 3x − 6</span></td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td><span class="larger">y − 2 = 3(x + 1)</span></td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td><span class="larger">y + 2x − 2 = 0</span></td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td><span class="larger">5x = 6</span></td>
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</tr>
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<tr>
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<td><img src="../images/style/yes.svg" alt="yes" /></td>
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<td> </td>
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<td><span class="larger">y/2 = 3</span></td>
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</tr>
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</table>
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</div>
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<p><b>But</b> the variables (like "x" or "y") in Linear Equations do <b>NOT</b> have:</p>
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<ul>
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<li><a href="../exponent.html">Exponents</a> (like the 2 in x<sup>2</sup>)</li>
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<li><a href="../square-root.html">Square roots</a>, <a href="../numbers/cube-root.html">cube roots</a>, etc</li>
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</ul>
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<div class="example">
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<h3>Examples: These are <b>NOT</b> linear equations:</h3>
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<table border="0" align="center">
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<tr>
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<td><img src="../images/style/no.svg" alt="not" /></td>
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<td> </td>
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<td><span class="larger">y<sup>2</sup> − 2 = 0</span></td>
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</tr>
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<tr>
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<td><img src="../images/style/no.svg" alt="not" /></td>
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<td> </td>
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<td class="larger">3√x − y = 6</td>
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</tr>
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<tr>
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<td><img src="../images/style/no.svg" alt="not" /></td>
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<td> </td>
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<td class="larger">x<sup>3</sup>/2 = 16</td>
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</tr>
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</table>
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</div>
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<h2>Slope-Intercept Form</h2>
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<p>The most common form is the <a href="../equation_of_line.html">slope-intercept equation of a straight line</a>: </p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/y-mxpb-graph.svg" alt="y=mx+b graph" /></p>
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<table width="300" border="0" class="larger" align="center">
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<tr>
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<td colspan="2"><img src="images/y-mxpb.svg" alt="Equation of a Straight Line y=mx+b" /></td>
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</tr>
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<tr>
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<td align="center"><a href="../geometry/slope.html" class="larger">Slope</a> (or <a href="../gradient.html" class="larger">Gradient</a>)</td>
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<td align="right"><a href="../y_intercept.html" class="larger">Y
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Intercept</a></td>
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</tr>
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<tr>
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<td colspan="2" align="center"> </td>
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</tr>
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</table>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: y = 2x + 1</h3>
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<ul>
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<li>Slope: <b>m = 2</b></li>
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<li>Intercept: <b>b = 1</b></li>
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</ul>
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</div>
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<table width="80%" border="0" align="center">
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<tr>
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<td>
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<a href="../data/straight_line_graph.html"><img src="../data/images/equn-line-anim.gif" alt="Animation" width="150" height="88" /></a>
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</td>
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<td> </td>
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<td>
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<h3>Play With It !</h3>
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<p>You can see the effect of different values of <b>m</b> and <b>b</b> at <a href="../data/straight_line_graph.html">Explore the Straight Line Graph</a></p>
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</td>
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</tr>
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</table>
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<h2>Point-Slope Form</h2>
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<p>Another common one is the <a href="line-equation-point-slope.html">Point-Slope Form</a> of the equation of a straight line:</p>
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<table border="0" align="center">
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<tr>
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<td class="larger">
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<p align="center"><span class="largest">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</span></p>
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</td>
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<td class="larger"> </td>
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<td><img src="../data/images/graph-point-slope-a-sm.gif" alt="Point-Slope Form" width="120" height="98" /></td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: y − 3 = (¼)(x − 2)</h3>
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<p>It is in the form <b>y − y<sub>1</sub> = m(x − x<sub>1</sub>)</b> where:</p>
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<ul>
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<li>y<sub>1</sub> = 3</li>
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<li>m = ¼</li>
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<li>x<sub>1</sub> = 2</li>
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</ul>
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</div>
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<h2>General Form</h2>
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<p>And there is also the <a href="line-equation-general-form.html">General Form</a> of the equation of a straight line:</p>
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<table border="0" align="center">
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<tr>
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<td>
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<p align="center" class="largest">Ax + By + C = 0</p>
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</td>
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</tr>
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<tr>
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<td>(A and B cannot both be 0)</td>
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</tr>
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</table>
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<div class="example">
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<h3>Example: 3x + 2y − 4 = 0</h3>
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<p>It is in the form <b>Ax + By + C = 0</b> where: </p>
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<ul>
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<li>A = 3</li>
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<li>B = 2</li>
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<li>C = −4</li>
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</ul>
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</div>
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<p>There are other, less common forms as well.</p>
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<h2>As a Function</h2>
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<p>Sometimes a linear equation is written as a <a href="../sets/function.html">function</a>, with <span class="large">f(x)</span> instead of <span class="large">y</span>:</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 width="300" align="center" class="larger"><span class="large">y = 2x − 3</span></td>
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</tr>
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<tr>
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<td align="center" class="larger"><span class="large">f(x) = 2x − 3</span></td>
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</tr>
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<tr>
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<td align="center" class="larger">These are the same!</td>
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</tr>
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</table>
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</div>
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<p>And functions are not always written using f(x):</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 width="300" align="center" class="larger"><span class="large">y = 2x − 3</span></td>
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</tr>
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<tr>
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<td align="center" class="larger"><span class="large">w(u) = 2u − 3</span></td>
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</tr>
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<tr>
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<td align="center" class="larger"><span class="large">h(z) = 2z − 3</span></td>
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</tr>
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<tr>
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<td align="center" class="larger">These are also the same!</td>
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</tr>
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</table>
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</div>
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<h2>The Identity Function</h2>
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<p>There is a special linear function called the "Identity Function":</p>
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<p align="center"> <span class="large">f(x) = x</span></p>
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<p>And here is its graph:</p>
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<p align="center"><img src="images/graph-line-x.gif" alt="Identity Function" width="220" height="222" /><span class="large"><br />
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It makes a 45° (its slope is 1)</span></p>
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<p>It is called "Identity" because what comes out is <b>identical</b> to what goes in:</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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<th width="60" align="center">In</th>
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<th width="60" align="center">Out</th>
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</tr>
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<tr>
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<td width="60" align="center">0</td>
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<td width="60" align="center">0</td>
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</tr>
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<tr>
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<td width="60" align="center">5</td>
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<td width="60" align="center">5</td>
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</tr>
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<tr>
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<td width="60" align="center">−2</td>
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<td width="60" align="center">−2</td>
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</tr>
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<tr>
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<td width="60" align="center">...etc</td>
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<td width="60" align="center">...etc</td>
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</tr>
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</table>
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</div>
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<h2>Constant Functions</h2>
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<p>Another special type of linear function is the Constant Function ... it is a horizontal line: </p>
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<p align="center"><img src="images/function-constant.gif" alt="Constant Function" width="218" height="196" /></p>
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<p align="center" class="large">f(x) = C</p>
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<p>No matter what value of "x", f(x) is always equal to some constant value. </p>
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<h2>Using Linear Equations</h2>
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<p>You may like to read some of the things you can do with lines:</p>
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<div class="bigul">
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<ul>
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<li><a href="line-midpoint.html">Finding the Midpoint of a Line Segment</a></li>
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<li><a href="line-parallel-perpendicular.html">Finding Parallel and Perpendicular Lines</a></li>
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<li><a href="line-equation-2points.html">Finding the Equation of a Line from 2 Points</a>
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<br />
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<br /> </li>
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</ul>
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</div>
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<div class="questions">
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