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<!DOCTYPE html>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Introduction to Algebra - Multiplication</h1>
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<p class="center"><i>Please read <a href="introduction.html">Introduction to Algebra</a> first</i></p>
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<h2>A Puzzle</h2>
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<p>What is the missing number?</p>
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<table align="center" width="250" border="0">
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<tbody>
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<tr valign="middle">
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<td style="text-align:center; width:50px;">
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<table height="50" width="35" cellspacing="0" cellpadding="0" border="2">
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<tbody>
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<tr>
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<td> </td>
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</tr>
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</tbody></table></td>
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<td class="huge" align="center" width="50"><b>×</b></td>
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<td class="huge" align="center" width="50"><b>4</b></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>8</b></td>
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</tr>
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</tbody></table>
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<p>The answer is 2, right? Because <b>2 × 4 = 8</b>.</p>
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<p>Well, in Algebra we don't use blank boxes, we use
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a <b>letter</b>. So we might write:</p>
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<table align="center" width="250" border="0">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"><b><i>x</i></b></td>
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<td class="huge" align="center" width="50"><b>×</b></td>
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<td class="huge" align="center" width="50"><b>4</b></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>8</b></td>
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</tr>
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</tbody></table>
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<p> </p>
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<p><i class="larger">But the "<font size="4">x</font>" looks like the "<font size="4">×</font>"</i> ... that can be very confusing ... so in Algebra we don't use the multiply symbol (<i>×</i>) between numbers
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and letters:</p>
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<p class="center"><span class="larger">We put the number next to the letter to mean multiply</span>:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"><b>4<i>x</i></b></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>8</b></td>
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</tr>
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</tbody></table>
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<p class="center">In English we say <b><i>"four x equals eight"</i></b>, meaning that 4 x's make 8.</p>
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<p> </p>
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<p>And the answer is written:</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"><i><b>x</b></i></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>2</b></td>
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</tr>
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</tbody></table>
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<h2>How to Solve</h2>
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<p>Instead of saying "<i>obviously</i> x=2", use this neat step-by-step approach:</p>
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<ul>
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<li>Work out <b>what to remove</b> to get "x = ..."</li>
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<li>Remove it by<b> doing the opposite</b></li>
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<li>Do that to<b> both sides</b></li>
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</ul>
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<p>And what is the opposite of multiplying? <b>Dividing!</b></p>
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<p>Have a look at this example:</p>
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<div class="center">
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<div class="boxa bgb1">
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<div class="boxtop">We want<br>
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to
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remove<br>
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the "4"</div>
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<img src="images/4x8a.gif" alt="4x = 8" height="29" width="99">
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</div>
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<div class="boxa bgb1">
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<p class="boxtop">To remove it, <b>do<br>
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the opposite</b>, in<br>
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this case divide by 4</p>
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<img src="images/4x8b.gif" alt="4x=8 divide left by 4" height="95" width="117">
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</div>
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<div class="boxa bgb1">
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<p class="boxtop"><br>
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Do it to<br>
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<b>both sides</b></p>
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<img src="images/4x8c.gif" alt="4x=8 divide left and right by 4" height="95" width="117">
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</div>
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<div class="boxa bgb1">
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<p class="boxtop"><br>
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<br>
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Which is ...</p>
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<img src="images/4x8d.gif" alt="1x = 2" height="28" width="92">
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</div>
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<div class="boxa bgb1">
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<p class="boxtop"><b><i><br>
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<br>
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Solved!</i></b></p>
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<img src="images/4x8e.gif" alt="x = 2" height="31" width="75">
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</div>
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</div>
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<h2>Why did we divide by 4 on both sides?</h2>
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<h3>Because of the need for balance ...</h3>
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<div class="simple center">
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<div class="boxb">
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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> </td>
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</tr>
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<tr>
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<td><img src="images/balance-m-1.gif" alt="balance 4x vs 8" height="104" width="204"></td>
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</tr>
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<tr>
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<td style="text-align:center;">In Balance</td>
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</tr>
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</tbody></table>
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</div>
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<div class="boxb">
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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 style="text-align:left;">Divide Left by 4</td>
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</tr>
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<tr>
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<td><img src="images/balance-m-2.gif" alt="unbalanced 1x vs 8" height="104" width="198"></td>
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</tr>
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<tr>
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<td><b>Out of Balance!</b></td>
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</tr>
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</tbody></table>
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</div>
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<div class="boxb">
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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 style="text-align:right;">Divide Right by 4 Also</td>
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</tr>
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<tr>
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<td><img src="images/balance-m-3.gif" alt="balance x vs 2" height="103" width="203"></td>
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</tr>
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<tr>
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<td style="text-align:center;">In Balance Again</td>
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</tr>
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</tbody></table>
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</div>
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<div style="clear:both"></div>
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<p> </p>
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<h3>Just remember ...</h3>
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</div>
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<div class="beach">
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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 class="larger" align="center"> To keep the balance, what we do to <b>one side</b> of the "="<br>
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we should also do to the <b>other side</b>!</td>
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</tr>
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</tbody></table>
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</div>
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<h2>Another Puzzle</h2>
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Solve this one:
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<table align="center" width="250" border="0">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"><b><i>x</i></b></td>
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<td class="huge" align="center"><b>/</b></td>
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<td class="huge" align="center" width="50"><b>3</b></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>5</b></td>
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</tr>
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</tbody></table>
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<p>What we want is an answer like "x = ...", but the <i>divide by 3</i> is in the way of that!</p>
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<p>If we <i>multiply by 3</i> we can cancel out the <i>divide by 3</i> (because 3/3=1)</p>
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<div class="tbl">
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<div class="row"><span class="left">So, let us try multiplying by 3 on <b>both sides</b>:</span><span class="right" style="text-align: center;"><span class="intbl"><em>x</em><strong>3</strong></span> <b>×3</b> = 5 <b>×3</b></span></div>
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<div class="row"><span class="left">A little arithmetic (<span class="intbl"><em>1</em><strong>3</strong></span><b>×</b>3 = 1 and 5<b>×</b>3 = 15) becomes:</span><span class="right" style="text-align: center;">1x = 15</span></div>
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<div class="row"><span class="left">Which is just:</span><span class="right" style="text-align: center;"><b>x = 15</b></span></div>
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<div class="row"><span class="left"> </span><span class="right" style="text-align: center;"><b><i>Solved!</i></b></span></div>
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<div class="row"><span class="left">(Quick Check: 15/3 = 5)</span><span class="right"> </span></div>
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</div>
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<!--
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<h2>Have a Try Yourself</h2>
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<p>Now practice on this <a href="../worksheets/print.php?w=11191&ID=3262">Algebra Multiplication Worksheet</a> and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!</p>
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-->
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<h2>More Complicated Example</h2>
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<p>How do we solve this?</p>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr valign="middle">
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<td class="huge" align="center" width="50"><b><i>x</i></b></td>
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<td class="huge" align="center"><b>/</b></td>
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<td class="huge" align="center" width="50"><b>3</b></td>
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<td class="huge" align="center" width="50"><b>+</b></td>
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<td class="huge" align="center" width="50"><b>2</b></td>
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<td class="huge" align="center" width="50"><b>=</b></td>
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<td class="huge" align="center" width="50"><b>5</b></td>
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</tr>
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</tbody></table>
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<p>It might look hard, but not if we solve it <b>in stages</b>.<br>
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<br>
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<b>First let us get rid of the "+2":</b></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">x/3 + 2 = 5</span></div>
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<div class="row"><span class="left"> To remove the <i>plus 2</i> use <i>minus 2</i> (because 2−2=0) </span><span class="right">x/3 + 2 <b>−2</b> = 5 <b>−2</b></span></div>
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<div class="row"><span class="left">A little arithmetic (2−2 = 0 and 5−2 = 3) becomes:</span><span class="right">x/3 + 0 = 3</span></div>
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<div class="row"><span class="left">Which is just:</span><span class="right">x/3 = 3</span></div>
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</div>
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<p><b>Now, get rid of the "/3":</b></p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">x/3 = 3</span></div>
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<div class="row"><span class="left"><i>multiply by 3</i> to cancel out the <i>divide by 3:</i> </span><span class="right">x/3 <b>×3</b> = 3 <b>×3</b></span></div>
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<div class="row"><span class="left">A little arithmetic (3/3 = 1 and 3<b>×</b>3 = 9) becomes:</span><span class="right">1x = 9</span></div>
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<div class="row"><span class="left">Which is just:</span><span class="right"><b>x = 9</b></span></div>
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<div class="row"><span class="left"> </span><span class="right"><b><i>Solved!</i></b></span></div>
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<div class="row"><span class="left">(Quick Check: 9/3 + 2 = 3+2 = 5)</span><span class="right"> </span></div>
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</div>
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<h2>When you get more experienced:</h2>
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<p>When you get more experienced, you can solve it like this:</p>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">x/3 + 2 = 5</span></div>
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<div class="row"><span class="left"> Subtract 2 from both sides: </span><span class="right">x/3 + 2 <b>−2</b> = 5 <b>−2</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right">x/3 = 3</span></div>
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<div class="row"><span class="left"> Multiply by 3 on both sides: </span><span class="right">x/3 <b>×3</b> = 3 <b>×3</b></span></div>
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<div class="row"><span class="left">Simplify:</span><span class="right"><b>x = 9</b></span></div>
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</div>
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Or <i>faster</i> like this:
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<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right">x/3 + 2 = 5</span></div>
|
||
<div class="row"><span class="left">Subtract 2 from both sides:</span><span class="right">x/3 = 3</span></div>
|
||
<div class="row"><span class="left"> Multiply by 3: </span><span class="right"><b>x = 9</b></span></div>
|
||
</div>
|
||
|
||
|
||
<h2>Real World Example</h2>
|
||
|
||
<div class="example">
|
||
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/chocolates.jpg" alt="chocolates" height="73" width="120"></p>
|
||
|
||
<h3>Example: Sam bought 3 boxes of chocolates online.<br>
|
||
Postage was $9 and the total cost was $45.<br>
|
||
How much was each box?</h3>
|
||
<p> </p>
|
||
<p>Let's use <b>x</b> for the price of each box.</p>
|
||
<p class="center">3 times <b>x</b> plus $9 is $45:</p>
|
||
<p class="center larger">3x + 9 = 45</p>
|
||
<p>Let's solve!</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right">3x + 9 = 45</span></div>
|
||
<div class="row"><span class="left">Subtract 9 from both sides:</span><span class="right">3x + 9 <b>− 9</b> = 45 <b>− 9</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right">3x = 36</span></div>
|
||
<div class="row"><span class="left">Divide by 3:</span><span class="right">3x <b>/3</b> = 36 <b>/3</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b>x = 12</b></span></div>
|
||
</div>
|
||
<p>So each box was <b>$12</b></p>
|
||
</div>
|
||
<p><b>Advanced:</b> we can also do the "divide by 3" first (but we must do it to all terms):</p>
|
||
<div class="tbl">
|
||
<div class="row"><span class="left">Start with:</span><span class="right">3x + 9 = 45</span></div>
|
||
<div class="row"><span class="left">Divide by 3:</span><span class="right">3x<b>/3</b> + 9<b>/3</b> = 45<b>/3</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right">x + 3 = 15</span></div>
|
||
<div class="row"><span class="left">Subtract 3 from both sides:</span><span class="right">x + 3 <b>− 3</b> = 15 <b>− 3</b></span></div>
|
||
<div class="row"><span class="left">Simplify:</span><span class="right"><b>x = 12</b></span></div>
|
||
</div>
|
||
<p class="center">Same answer!</p>
|
||
|
||
|
||
<h2>Have a Try Yourself</h2>
|
||
|
||
<p>Now practice on this <a href="../worksheets/ws-algebra8283.html?steps=2&n=10&amin=2&amax=9&ops=am&seed=7772">Algebra (Two Steps to Solve) Worksheet</a> and then check your answers on the page after. Try to use the steps we have shown you here, rather than just guessing!</p>
|
||
<p> </p>
|
||
<div class="questions">1729, 1730, 1731, 1732, 1733, 1734, 3139, 3140, 3141, 3142</div>
|
||
|
||
<div class="related">
|
||
<a href="introduction.html">Introduction to Algebra</a>
|
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<a href="index.html">Algebra Menu</a>
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