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246 lines
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<h1 class="center">Equation of a Line from 2 Points</h1>
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<p>First, let's see it in action. Here are two points (you can drag them) and the equation of the line through them. Explanations follow.</p>
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<div class="script" style="height: 360px;">
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../geometry/images/geom-line-equn.js
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</div>
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<h2>The Points</h2>
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<p>We use <a href="../data/cartesian-coordinates.html">Cartesian Coordinates</a> to mark
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a point
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on a graph by <b>how far along</b> and <b>how far up</b> it is:</p>
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<p class="center"><img src="../geometry/images/coordinates-cartesian.svg" alt="graph with point (12,5)" height="232" width="348"><br>
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Example: The point <b>(12,5)</b> is 12 units along, and 5 units up</p>
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<h2>Steps</h2>
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<p>There are 3 steps to find the <a href="../equation_of_line.html">Equation of the Straight Line</a> :</p>
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<ul>
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<li>1. Find the slope of the line</li>
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<li>2. Put the slope and one point into the "Point-Slope Formula"</li>
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<li>3. Simplify</li>
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</ul>
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<h2>Step 1: Find the Slope (or Gradient) from 2 Points</h2>
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<p>What is the <a href="../geometry/slope.html">slope</a> (or gradient) of this line?</p>
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<p class="center"><img src="images/graph-2-points.gif" alt="graph 2 points" height="190" width="220"></p>
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<p>We know two points:</p>
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<ul>
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<li>point "A" is <span class="large">(6,4)</span> (at x is 6, y is 4)</li>
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<li>point "B" is <span class="large">(2,3)</span> (at x is 2, y is 3)</li>
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</ul>
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<p>The slope is the <b>change in height</b> divided by the <b>change in horizontal distance</b>.</p>
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<p>Looking at this diagram ...</p>
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<p class="center"><img src="images/graph-2-points-b.gif" alt="graph 2 points" height="188" width="302"></p>
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<p class="center larger"><a href="../geometry/slope.html">Slope</a> m = <span class="intbl"><em>change in y</em><strong>change in x</strong></span> = <span class="intbl"><em>y<sub>A</sub> − y<sub>B</sub></em><strong>x<sub>A</sub> − x<sub>B</sub></strong></span></p>
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<div style="clear:both"></div>
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<p>In other words, we:</p>
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<ul>
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<li>subtract the Y values,</li>
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<li>subtract the X values</li>
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<li>then divide</li>
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</ul>
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<p>Like this:</p>
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<p class="center larger">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>4−3</em>
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<strong>6−2</strong>
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</span> = <span class="intbl">
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<em>1</em>
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<strong>4</strong>
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</span> = 0.25</p>
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<p>It doesn't matter which point comes first, it still works out the same. Try swapping the points:</p>
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<p class="center larger">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>3−4</em>
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<strong>2−6</strong>
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</span> = <span class="intbl hilite">
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<em>−1</em>
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<strong>−4</strong>
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</span> = 0.25</p>
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<p>Same answer.</p>
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<h2>Step 2: The "Point-Slope Formula"</h2>
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<p>Now put that <b>slope</b> and <b>one point</b> into the "Point-Slope Formula"</p>
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<p class="center larger"><img src="images/graph-2-points.gif" alt="graph 2 points" height="190" width="220"><br></p>
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<p>Start with the <a href="line-equation-point-slope.html">"point-slope" formula</a> (<span class="larger"><b>x<sub>1</sub></b></span> and <span class="larger"><b>y<sub>1</sub></b></span> are the coordinates of a point on the line):</p>
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<p class="center"><span class="larger">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</span></p>
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<p>We can choose <b>any point</b> on the line for <span class="larger"><b>x<sub>1</sub></b></span> and <span class="larger"><b>y<sub>1</sub></b></span>, so let's just use point <span class="larger">(2,3)</span>:</p>
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<p class="center"><span class="larger">y − 3 = m(x − 2)</span></p>
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<p>We already calculated the slope "m":</p>
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<p class="center large"><b>m</b> = <span class="intbl"><em>change in y</em><strong>change in x</strong></span> = <span class="intbl"><em>4−3</em><strong>6−2</strong></span> = <span class="intbl"><em>1</em><strong>4</strong></span></p>
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<p>And we have:</p>
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<div class="center80">
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<p class="center larger">y − 3 = <span class="intbl"><em>1</em><strong>4</strong></span>(x − 2)</p>
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</div>
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<p><b>That is an answer</b>, but we can simplify it further.</p>
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<h2>Step 3: Simplify</h2>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right"><span class="larger">y − 3 = <span class="intbl"><em>1</em><strong>4</strong></span>(x − 2)</span></span></div>
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<div class="row"><span class="left">Multiply <span class="intbl"><em>1</em><strong>4</strong></span> by (x−2):</span><span class="right"><span class="larger">y − 3 = <span class="intbl"><em>x</em><strong>4</strong></span> − <span class="intbl"><em>2</em><strong>4</strong></span></span></span></div>
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<div class="row"><span class="left">Add 3 to both sides:</span><span class="right"><span class="larger">y = <span class="intbl"><em>x</em><strong>4</strong></span> − <span class="intbl"><em>2</em><strong>4</strong></span> + 3</span></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><span class="larger">y = <span class="intbl"><em>x</em><strong>4</strong></span> + <span class="intbl"><em>5</em><strong>2</strong></span></span></span></div>
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</div>
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<p>And we get:</p>
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<div class="center80">
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<p class="center larger">y = <span class="intbl"><em>x</em><strong>4</strong></span> + <span class="intbl"><em>5</em><strong>2</strong></span></p>
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</div>
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<p>Which is now in the <a href="../equation_of_line.html">Slope-Intercept (<b>y = mx + b</b>)</a> form.</p>
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<p> </p>
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<div class="center80">
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<h3>Check It!</h3>
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<p>Let us confirm by testing with the second point <span class="larger">(6,4)</span>:</p>
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<p class="center larger"><b>y</b> = <b>x</b>/4 + 5/2 = <b>6</b>/4 + 2.5 = 1.5 + 2.5 = <b>4</b></p>
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<p>Yes, when x=6 then y=4, so it works!</p>
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</div>
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<h2>Another Example</h2>
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<div class="example">
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<h3>Example: What is the equation of this line?</h3>
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<p class="center"><img src="images/graph-2-points-c.gif" alt="graph 2 points" height="200" width="190"></p>
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<p>Start with the <a href="line-equation-point-slope.html">"point-slope" formula</a>:</p>
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<p class="center"><span class="larger">y − y<sub>1</sub> = m(x − x<sub>1</sub>)</span></p>
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<p>Put in these values:</p>
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<ul>
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<li>x<sub>1</sub> = 1</li>
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<li>y<sub>1</sub> = 6</li>
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<li>m = (2−6)/(3−1) = −4/2 = −2</li>
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</ul>
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<p>And we get:</p>
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<p class="center"><span class="larger">y − 6 = −2(x − 1)</span></p>
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<p>Simplify to <a href="../equation_of_line.html">Slope-Intercept (<b>y = mx + b</b>)</a> form:</p>
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<p class="center larger">y − 6 = −2x + 2</p>
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<p class="center larger">y = −2x + 8</p>
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<p>DONE!</p>
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</div>
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<h2>The Big Exception</h2>
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<p>The previous method works nicely except for one particular case: a <b>vertical line</b>:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/graph-2-points-vertical.gif" alt="graph vertical line" height="189" width="148"></p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>
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<p>A vertical line's gradient is undefined (because <a href="../numbers/dividing-by-zero.html">we cannot divide by 0</a>):</p>
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<p class="center large">m = <span class="intbl"><em>y<sub>A</sub> − y<sub>B</sub></em><strong>x<sub>A</sub> − x<sub>B</sub></strong></span> = <span class="intbl"><em>4 − 1</em><strong>2 − 2</strong></span> = <span class="intbl"><em>3</em><strong>0</strong></span> = undefined</p>
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<p>But there is still a way of writing the equation: use <b>x=</b> instead of <b>y=</b>, like this:</p>
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<div class="center80">
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<p class="center large">x = 2</p>
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</div></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p> </p>
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<div class="questions">7270, 525, 526, 1165, 1166, 7291, 7292, 7300, 7301, 7302</div>
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<div class="related">
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<a href="../equation_of_line.html">Equation of a Straight Line</a>
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<a href="../straight-line-graph-calculate.html">Straight Line Graph Calculator</a>
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<a href="index.html">Algebra Index</a>
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