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318 lines
12 KiB
HTML
318 lines
12 KiB
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</h1>
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<p class="center"><i>Algebra is great fun - you get to solve puzzles!</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>
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</td>
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<td class="huge" align="center"><b>−</b></td>
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<td class="huge" align="center"><b>2</b></td>
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<td class="huge" align="center"><b>=</b></td>
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<td class="huge" align="center"><b>4</b></td>
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</tr>
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</tbody></table>
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<p>OK, the answer is 6, right? Because <b>6 − 2 = 4</b>. Easy stuff.</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> (usually an x or y, but any letter is fine). So we write:</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" 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>4</b></td>
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</tr>
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</tbody></table>
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<p>It is really that simple. The letter (in this case an x) just means "we don't know this yet", and is often called the <b>unknown</b> or the <b>variable</b>.</p>
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<p>And when we solve it we write:</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>6</b></td>
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</tr>
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</tbody></table>
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<h2>Why Use a Letter?</h2>
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<table align="center" width="90%" cellpadding="5" border="0">
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<tbody>
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<tr>
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<td> </td>
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<td><i>Because:</i></td>
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</tr>
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<tr>
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<td><img src="../images/style/right-arrow.gif" alt="arrow" height="46" width="46"></td>
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<td>it is easier to write "x" than drawing empty boxes (and easier to say "x" than "the empty box").</td>
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</tr>
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<tr>
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<td><img src="../images/style/right-arrow.gif" alt="arrow" height="46" width="46"></td>
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<td>if there are several empty
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boxes (several "unknowns") we can use a different letter for each one.</td>
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</tr>
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</tbody></table>
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<div class="words">
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<p>So <b><span class="times">x</span></b> is simply better than having an empty box. We aren't trying to make words with it!</p>
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<p>And it doesn't have to be <span class="times"><b>x</b></span>, it could be <span class="times">y</span> or <span class="times">w</span> ... or any letter or symbol we like.</p>
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</div>
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<h2>How to Solve</h2>
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<p>Algebra is just like a puzzle where we start with something like "x − 2 = 4" and we want to end
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up with something like "x = 6".</p>
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<p>But instead of saying "<i>obviously</i> x=6", 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> (adding is the opposite of subtracting)</li>
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<li>Do that to<b> both sides</b></li>
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</ul>
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<p>Here is an 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 "−2"</div>
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<img src="images/x24a.gif" alt="x - 2 = 4" height="42" width="114"></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 add 2</p>
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<img src="images/x24b.gif" alt="x - 2 = 4 add 2 to left" height="100" width="114"></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/x24c.gif" alt="x - 2 = 4 add 2 to left and right" height="97" width="122"></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/x24d.gif" alt="x + 0 = 6" height="29" width="113"></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/x24e.gif" alt="x = 6" height="29" width="77"></div>
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</div>
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<h2>Why did we add 2 to both sides?</h2>
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<h3>To "keep the balance"...</h3>
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<div class="center simple">
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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-1.gif" alt="balance x - 2 vs 4" height="109" width="200"></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;">Add 2 to Left Side</td>
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</tr>
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<tr>
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<td><img src="images/balance-2.gif" alt="unbalanced x - 2 + 2 vs 4" height="107" width="200"></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> <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;">Add 2 to Right Side Also</td>
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</tr>
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<tr>
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<td><img src="images/balance-3.gif" alt="balanced x - 2 + 2 vs 4 + 2" height="109" width="200"></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>
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<div style="clear:both"></div>
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<p>Just remember this:</p>
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<div class="beach">
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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 class="larger" align="center">
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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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<p><i>See this in action at the <a href="add-subtract-balance.html">Algebra Balance Animation</a>.</i></p>
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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" width="50"> <b>+</b></td>
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<td class="huge" align="center" width="50"> <b>5</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>12</b></td>
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</tr>
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</tbody></table><br>
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<p>What we want is an answer like "x = ...",<br>
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but the <b>+5</b> is in the way of that!<br>
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We can cancel out the <b>+5</b> with a <b>−5</b> (because 5−5=0)</p>
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<div class="tbl">
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<div class="row"><span class="left">So, let us have a go at subtracting 5 from <b>both sides</b>:</span><span class="center"><span class="right ">x + 5 <b>−5</b> = 12 <b>−5</b></span></span></div>
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<div class="row"><span class="left">A little arithmetic (5−5 = 0 and 12−5 = 7) becomes:</span><span class="center"><span class="right ">x + 0 = 7</span></span></div>
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<div class="row"><span class="left">Which is just:</span><span style="text-align: center;"><span class="right"><b>x = 7</b></span></span></div>
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<div class="row"><span class="left"> </span><span class="center"><span class="right"><b><i>Solved!</i></b></span></span></div>
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<div class="row"><span class="left">(Quick Check: 7+5=12)</span><span class="right center"> </span></div>
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</div>
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<h2>Have a Try Yourself</h2>
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<p>Now practice on this <a href="../worksheets/ws-algebra5319.html?steps=1&n=10&amin=2&amax=9&ops=a&seed=5394">Simple Algebra Worksheet</a> and then check your answers. Try to use the steps we have shown you here, rather than just guessing!</p>
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<p>Try the questions below, then read <a href="introduction-multiply.html">Introduction to Algebra - Multiplication</a></p>
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<p> </p>
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<div class="questions">1725,1726,1727,1728,3135,3136,3137,3138,3850,3851</div>
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<div class="related">
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<a href="introduction-multiply.html">Introduction to Algebra - Multiplication</a>
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<a href="index.html">Algebra Menu</a>
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</div>
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<div id="copyrt">Copyright © 2021 MathsIsFun.com</div>
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