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208 lines
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<title>Point-Slope Equation of a Line</title>
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<center>
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<h1>Point-Slope Equation of a Line</h1>
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</center>
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<h3 align="center"><br>
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The "point-slope" form of the equation of a straight line is:<br>
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<br></h3>
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<div class="center80">
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<p align="center" class="largest">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</p>
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</div>
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<p><center>
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</center></p>
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<p>The equation is useful when we know:</p>
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<ul>
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<li>one <a href="../data/cartesian-coordinates.html">point</a> on the line: <b>(x<sub>1</sub>,y<sub>1</sub>)</b></li>
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<li>and the <a href="../geometry/slope.html">slope</a> of the line: <b>m</b>,</li>
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</ul>
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<p>and want to find other points on the line.</p>
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<p>Have a play with it first (move the point, try different slopes):</p>
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<script src="../geometry/images/geom-line-equn.js"></script>
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<script>geomlineequnMain('pt');</script>
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<p> </p>
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<p>Now let's discover more.</p>
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<h2>What does it stand for?</h2>
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<p style="float:left; margin: 0 30px 5px 0;"><img src="../data/images/graph-point-slope-a.gif" alt="graph with slope m" width="221" height="184" /></p>
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<p><span class="large">(x<sub>1</sub>, y<sub>1</sub>)</span> is a <b>known</b> point</p>
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<p><span class="large">m</span> is the <b>slope</b> of the line</p>
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<p><span class="large">(x, y)</span> is any other point on the line</p>
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<div style="clear:both"></div>
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<h2>Making sense of it</h2>
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<p>It is based on the slope:</p>
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<p class="center"><img src="../data/images/graph-point-slope-b.gif" alt="graph" width="303" height="186" /></p>
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<p class="center large"><a href="../geometry/slope.html">Slope</a> m = <span class="intbl">
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<em>change in y</em>
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<strong>change in x</strong>
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</span> = <span class="intbl">
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<em>y − y<sub>1</sub></em>
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<strong>x − x<sub>1</sub></strong>
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</span></p>
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<p> </p>
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<table style="border: 0; margin:auto;">
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<tr>
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<td align="right"><p>Starting with the slope:</p>
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<p>we rearrange it like this:</p>
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<p> </p>
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<p>to get this:</p></td>
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<td width="20" align="right"> </td>
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<td><img src="../data/images/equn-line-slope-rearrange.gif" width="211" height="186" alt="equation of line slope rearrange" /></td>
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</tr>
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</table>
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<p>So, it is just the slope formula in a different way!</p>
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<h3>Now let us see how to use it.</h3>
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<div class="example">
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<h3>Example 1:</h3>
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<p class="center"><img src="../data/images/graph-point-slope-d.gif" alt="graph with slope m=3" width="219" height="187" /></p>
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<p class="center large">slope "m" = <span class="intbl"><em>3</em><strong>1</strong></span> = 3</p>
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<p class="center large">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</p>
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<p>We know <span class="large">m</span>, and also know that <span class="large">(x<sub>1</sub>, y<sub>1</sub>) </span>=<span class="large"> (3,2)</span>, and so we have:</p>
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<div class="center80">
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<p class="center large">y − 2 = 3(x − 3)</p>
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</div>
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<p>That is a perfectly good answer, but we can simplify it a little:</p>
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<p class="center"><span class="large">y − 2 = 3x − 9</span></p>
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<p class="center"><span class="large">y = 3x − 9 + 2</span><br></p>
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<p class="center"><span class="large">y = 3x − 7</span><br></p>
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</div>
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<div class="example">
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<h3>Example 2:</h3>
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<p class="center"><img src="images/m3x-graph2.svg" alt="y=-3x graph" /></p>
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<center>
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<p class="large">m = <span class="intbl">
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<em>−3</em>
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<strong>1</strong>
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</span> = −3</p>
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</center>
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<p class="center large">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</p>
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<p>We can pick any point for <span class="large">(x<sub>1</sub>, y<sub>1</sub>)</span>, so let's choose <span class="large"> (0,0)</span>, and we have:</p>
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<p class="center large">y − 0 = −3(x − 0)</p>
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<p>Which can be simplified to:</p>
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<div class="center80">
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<p class="center large">y = −3x</p>
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</div>
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</div>
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<div class="example">
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<h3>Example 3: Vertical Line</h3>
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<table style="border: 0; margin:auto;">
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<tr>
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<td align="center"><img src="../data/images/graph-x-2.gif" alt="graph x=2" width="198" height="233" /></td>
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</tr>
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</table>
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<p class="center">What is the equation for a vertical line?<br>
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The slope is undefined!</p>
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<p>In fact, this is a <b>special case</b>, and we use a different equation, like this:</p>
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<div class="center80">
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<p class="center large">x = 1.5</p>
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</div>
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<p class="center">Every point on the line has <b>x</b> coordinate <b>1.5</b>,<br>
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that’s why its equation is <b>x = 1.5</b></p>
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</div>
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<h2>What About y = mx + b ?</h2>
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<p>You may already be familiar with the "<a href="../equation_of_line.html">y=mx+b</a>" form (called the slope-intercept form of the equation of a line).</p>
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<p class="center larger">It is the same equation, in a different form!</p>
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<p>The "b" value (called the <a href="../y_intercept.html">y-intercept</a>) is where the line crosses the y-axis.</p>
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<p class="center">So point <span class="large">(x<sub>1</sub>, y<sub>1</sub>)</span> is actually at <span class="large">(0, b)</span></p>
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<p>and the equation becomes:</p><div class="tbl">
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<div class="row"><span class="left">Start with</span><span class="right"><span class="large">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</span></span></div>
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<div class="row"><span class="left"><span class="large">(x<sub>1</sub>, y<sub>1</sub>)</span> is actually <span class="large">(0, b)</span>:</span><span class="right"><span class="large">y − b = m(x − 0)</span></span></div>
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<div class="row"><span class="left">Which is:</span><span class="right"><span class="large">y − b = mx</span></span></div>
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<div class="row"><span class="left">Put b on other side:</span><span class="right"><span class="large"><b>y = mx + b</b></span></span></div>
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</div>
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<p> </p>
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<div class="questions">
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<script>getQ(519, 521, 522, 1160, 1161, 1162, 2074, 2075, 9027, 9028, 9029);</script> </div>
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<div class="related">
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<a href="../gradient.html">Slope (Gradient) of a Straight Line</a>
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<a href="../y_intercept.html">Y Intercept of a Straight Line</a>
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<a href="../quiz/linear_equation_test.html">Test Yourself</a>
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<a href="../data/straight_line_graph.html">Explore the Straight Line Graph</a>
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<a href="../straight-line-graph-calculate.html">Straight Line Graph Calculator</a>
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