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<title>Equation of a Straight Line</title>
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<!-- #BeginEditable "Body" --><a name="Top" id="Top"></a>
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<center>
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<h1>Equation of a Straight Line</h1>
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</center>
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<p>The equation of a straight line is usually written this way:</p>
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<div class="center80">
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<div class="center"><span class="largest">y = mx + b</span> </div>
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</div>
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<p class="center"><span class="larger"> (or "y = mx + c" in the UK <a href="#Countrynote">see
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below</a>) </span></p>
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<p> </p>
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<h3>What does it stand for?</h3>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="algebra/images/y-mxpb-graph.svg" alt="y=mx+b graph" height="241" width="183" ></p>
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<table class="large" align="center" width="300" border="0">
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<tbody>
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<tr>
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<td colspan="2"><img src="algebra/images/y-mxpb.svg" alt="Equation of a Straight Line y=mx+b" height="74" width="247" ></td>
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</tr>
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<tr>
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<td style="text-align:center;"><a href="geometry/slope.html">Slope</a> or<br>
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<a href="gradient.html">Gradient</a></td>
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<td style="text-align:right;"><b>y</b> value when <b>x=0</b><br>
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<i>(see <a href="y_intercept.html">Y Intercept</a>)</i></td>
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</tr>
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</tbody></table><br>
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<div style="float:right; margin: 0 0 5px 10px;">
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<p class="large"><b>y</b> = how far up</p>
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<p class="large"><b>x</b> = how far along</p>
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<p class="large"><b>m</b> = Slope or Gradient (how steep the line is)</p>
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<p class="large"><b>b</b> = value of <b>y</b> when <b>x=0</b></p>
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</div>
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<div style="clear:both"></div>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="algebra/images/y-mxpb-intercept.svg" alt="y=mx+b intercept" height="83" width="148" ></p>
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<h3>How do you find "m" and "b"?</h3>
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<ul>
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<li><span class="large">b</span> is easy: just see where the line crosses the Y axis.</li>
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<li><span class="large">m</span> (the Slope) needs some calculation:</li>
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</ul>
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<div class="flexy">
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<div class="col">
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<b>m</b> = <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>
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</div>
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<div class="col">
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<img src="algebra/images/slope.svg" alt="gradient" height="173" width="273" >
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</div>
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</div>
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<h3>Knowing this we can work out the equation of a straight line:</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/2xp1-graph.gif" alt="y=2x+1 graph" height="247" width="215" ></p>
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<p class="large center">m = <span class="intbl">
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<em>2</em>
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<strong>1</strong>
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</span> = 2</p>
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<p class="center"><span class="large">b = 1 </span><i>(value of <b>y</b> when x=0)</i><br></p>
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<p>Putting that into <b>y = mx + b</b> gets us:</p>
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<p class="center larger">y = 2x + 1</p>
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</div>
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<div class="center80">
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<p>With that equation we can now ...</p>
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<p class="center">... choose any value for <span class="large">x</span> and find the matching value for <span class="large">y</span></p>
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<p>For example, when <span class="large">x</span> is 1:</p>
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<p class="center"><span class="larger">y = 2×<b>1</b> + 1 = 3</span></p>
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<p>Check for yourself that <span class="large">x=1</span> and <span class="large">y=3</span> is actually on the line.</p>
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<p><br></p>
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<p>Or we could choose another value for <span class="large">x</span>, such as 7:</p>
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<p class="center"><span class="larger">y = 2×<b>7</b> + 1 = 15</span></p>
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<p>And so when <span class="large">x=7</span> you will have <span class="large">y=15</span></p>
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</div>
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<h2>Positive or Negative Slope?</h2>
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<p>Going from left-to-right, the cyclist has to <b>P</b>ush on a <b>P</b>ositive Slope:</p>
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<p class="center"><img src="geometry/images/slope-neg.svg" alt="negative slope" height="80" width="130" > <img src="geometry/images/slope-0.svg" alt="zero slope" height="80" width="130" > <img src="geometry/images/slope-pos.svg" alt="positive slope" height="80" width="130" ></p>
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<div class="example">
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<h3>Example 2</h3>
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<center>
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<img src="algebra/images/m3x-graph2.svg" alt="y=-3x graph" height="248" width="180" >
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</center>
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<p class="large center">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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<p class="center large">b = 0</p>
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<p>This gives us:</p>
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<p class="center large">y = −3x + 0</p>
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<p>We do not need the zero! So:</p>
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<p class="larger center">y = −3x</p>
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</div>
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<h3> </h3>
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<div class="example">
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<h3>Example 3: Vertical Line</h3>
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<div class="center"><img src="data/images/graph-x-2.gif" alt="graph x=2" height="233" width="198" > </div>
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<p class="center"><span class="larger">What is the equation for a vertical line?</span><br>
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The slope is <b>undefined</b> ... and where does it cross the Y-Axis?</p>
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<p>In fact, this is a <b>special case</b>, and we use a different equation, not "<span class="larger"><b>y</b>=...</span>", but instead we use "<span class="larger"><b>x</b>=...</span>".</p>
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<p class="center">Like this:</p>
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<p class="center largest">x = 1.5</p>
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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 is why its equation is <b>x = 1.5</b></p>
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</div>
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<h2>Rise and Run</h2>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="algebra/images/graph-rise-run.svg" alt="rise and run" height="197" width="238" ></p>
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<p>Sometimes the words "rise" and "run" are used.</p>
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<ul>
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<li>Rise is how far up</li>
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<li>Run is how far along</li>
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</ul>
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<p>And so the slope "m" is:</p>
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<p class="center large">m = <span class="intbl"> <em>rise</em> <strong>run</strong> </span></p>
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<p>You might find that easier to remember.</p>
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<table align="center" width="90%" border="0">
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<tbody>
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<tr>
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<td><a href="data/straight_line_graph.html"><img src="data/images/equn-line-anim.gif" alt="Animation" height="88" width="150" ></a></td>
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<td> </td>
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<td>
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<h2>Now Play With The Graph !</h2>
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<p>You can see the effect of different values of <b>m</b> (the slope) and <b>b</b> (the y intercept) at <a href="data/straight_line_graph.html">Explore the Straight Line Graph</a></p></td>
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</tr>
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</tbody></table>
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<p> </p>
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<h2>Other Forms</h2>
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<p>We have been looking at the "slope-intercept" form. The equation of a straight line can be written in <b>many other ways</b>.</p>
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<p>Another popular form is the <a href="algebra/line-equation-point-slope.html">Point-Slope Equation of a Straight Line</a>.</p>
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<p> </p>
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<div class="questions">358,359,517,518, 1156, 1157, 3204, 3205, 3206, 3207</div>
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<p> </p>
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<h2>Footnote</h2>
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<center>
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</center>
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<div class="def">
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<h3><a name="Countrynote" id="Countrynote"></a> Country
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Note:</h3>
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<p>Different Countries teach different "notation" (as sent to me by kind readers):</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>In the <b>US, Australia, Canada, Eritrea, Iran, Mexico, Portugal, Philippines and Saudi Arabia</b> the notation is: </td>
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<td style="text-align:center; width:110px;">y = mx + b</td>
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</tr>
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<tr>
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<td>In the <b>UK, Australia (also), Bahamas, Bangladesh, Belgium, Brunei, Bulgaria, Cyprus, Egypt, Germany, Ghana, India, Indonesia, Ireland, Jamaica, Kenya, Kuwait, Malaysia, Malawi, Malta, Nepal, New Zealand, Nigeria, Oman, Pakistan, Peru, Singapore, Solomon Islands, South Africa, Sri Lanka, Turkey, UAE, Zambia</b> and <b>Zimbabwe</b></td>
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<td style="text-align:center;">y = mx + c</td>
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</tr>
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<tr>
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<td>In <b>Afghanistan, Albania, Algeria, Brazil, China, Czech Republic, Denmark, Ethiopia, France, Lebanon, Netherlands, Kosovo, Kyrgyzstan, Norway, Poland, Romania, South Korea, Surinam, Spain, Tunisia</b> and <b>Viet Nam:</b></td>
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<td style="text-align:center;">y = ax + b</td>
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</tr>
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<tr>
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<td>In <b>Azerbaijan, China, Finland, Russia and Ukraine</b>:</td>
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<td style="text-align:center;">y = kx + b</td>
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</tr>
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<tr>
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<td>In <b>Greece</b>:</td>
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<td style="text-align:center;">ψ = αχ + β</td>
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</tr>
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<tr>
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<td>In <b>Italy</b>:</td>
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<td style="text-align:center;">y = mx + q</td>
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</tr>
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<tr>
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<td>In <b>Japan</b>:</td>
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<td style="text-align:center;">y = mx + d</td>
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</tr>
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<tr>
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<td>In <b>Cuba</b> and <b>Israel</b>:</td>
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<td style="text-align:center;">y = mx + n</td>
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</tr>
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<tr>
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<td>In <b>Romania</b>:</td>
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<td style="text-align:center;">y = gA + C</td>
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</tr>
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<tr>
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<td>In <b>Latvia and Sweden</b>:</td>
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<td style="text-align:center;">y = kx + m</td>
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</tr>
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<tr>
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<td>In <b>Serbia</b> and <b>Slovenia</b>:</td>
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<td style="text-align:center;">y = kx + n</td>
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</tr>
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<tr>
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<td> </td>
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<td style="text-align:center;"> </td>
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</tr>
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<tr>
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<td>In your country: </td>
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<td style="text-align:center;"><a href="contact.html">let us know!</a></td>
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</tr>
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</tbody></table>
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<p>... but it all means the same thing, just different letters.</p>
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</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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<a href="data/graphs-index.html">Graph Index</a>
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</div>
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