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<title>Solving Equations</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Solving Equations</h1>
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<h2>What is an Equation?</h2>
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<p>An equation says that two things are equal. It will have an equals sign "=" like this:</p>
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<table border="0" align="center">
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<tr valign="middle">
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<td align="center" width="50" class="huge"><b><i>x</i></b></td>
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<td align="center" width="50" class="huge"><b>−</b></td>
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<td align="center" width="50" class="huge"><b>2</b></td>
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<td align="center" width="50" class="huge"><b>=</b></td>
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<td align="center" width="50" class="huge"><b>4</b></td>
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</tr>
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</table>
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<p>That equations says: <b>what is on the left (x − 2) is equal to what is on the right (4)</b></p>
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<p>So an equation is like a <b>statement</b> "<i>this</i> equals <i>that</i>"</p>
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<h2>What is a Solution?</h2>
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<div class="def">
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<p>A Solution is a value we can put in place of a variable (such as <span class="times"><i>x</i></span>) that makes the equation <b>true</b>.</p>
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</div>
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<br />
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<div class="example">
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<h3>Example: <span class="large">x − 2 = 4</span></h3>
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<p>When we put <span class="large">6</span> in place of <span class="large">x</span> we get:</p>
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<p class="center"><span class="large">6 − 2 = 4</span></p>
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<p class="center">which is <b>true</b></p>
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<p>So <span class="large">x = 6</span> is a solution.</p>
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<p>How about other values for <span class="large">x</span> ?</p>
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<ul>
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<li>For x=5 we get "5−2=4" which is <b>not true</b>, so <b>x=5 is not a solution</b>.</li>
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<li>For x=9 we get "9−2=4" which is <b>not true</b>, so <b>x=9 is not a solution</b>.</li>
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<li>etc</li>
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</ul>
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<p>In this case <span class="large">x = 6</span> is the only solution. </p>
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</div>
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<p>You might like to <a href="add-subtract-balance.html">practice solving some animated equations</a>.</p>
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<h2>More Than One Solution</h2>
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<p>There can be <b>more than one</b> solution.</p>
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<div class="example">
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<h3>Example: (x−3)(x−2) = 0</h3>
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<p>When <span class="large">x</span> is <span class="large">3</span> we get:</p>
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<p align="center" class="larger"> (3−3)(3−2) = 0 × 1 = 0 </p>
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<p align="center">which is <b>true</b></p>
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<p>And when <span class="large">x</span> is <span class="large">2</span> we get:</p>
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<p align="center" class="larger"> (2−3)(2−2) = (−1) × 0 = 0 </p>
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<p align="center">which is also <b>true</b></p>
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<p>So the solutions are: </p>
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<p align="center" class="larger">x = <b>3</b>, or x = <b>2</b></p>
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</div>
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<p>When we gather all solutions together it is called a <b>Solution Set</b></p>
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<div class="example">
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<p>The above solution set is: <span class="large">{2, 3}</span></p>
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</div>
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<h2> Solutions Everywhere!</h2>
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<p>Some equations are true for all allowed values and are then called <b>Identities</b> </p>
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<div class="example">
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<h3>Example: <b>sin(−θ) = −sin(θ)</b> is one of the <a href="trigonometric-identities.html">Trigonometric Identities</a></h3>
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<p>Let's try θ = 30°:</p>
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<p class="so"><b>sin(−30°) = −0.5</b> and</p>
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<p class="so"> <b>−sin(30°) = −0.5</b></p>
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<p>So it is <b>true</b> for θ = 30°</p>
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<p>Let's try θ = 90°:</p>
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<p class="so"><b>sin(−90°) = −1</b> and</p>
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<p class="so"> <b>−sin(90°) = −1</b></p>
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<p>So it is also <b>true</b> for θ = 90°</p>
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<p>Is it true for <b>all values of θ</b>? Try some values for yourself!</p>
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</div><p> </p>
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<h2>How to Solve an Equation</h2>
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<p>There is no "one perfect way" to solve all equations. </p>
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<h3>A Useful Goal</h3>
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<p>But we often get success when <b>our goal</b> is to end up with:</p>
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<div class="center80">
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<p align="center" class="larger"><b>x</b> = <i>something</i></p>
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</div>
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<p>In other words, we want to move everything except "x" (or whatever name the variable has) over to the right hand side.</p>
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<div class="example">
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<h3>Example: Solve 3x−6 = 9</h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">3x−6 = 9</span></div>
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<div class="row"><span class="left">Add 6 to both sides:</span><span class="right">3x = 9+6</span></div>
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<div class="row"><span class="left">Divide by 3:</span><span class="right">x = (9+6)/3</span></div>
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</div>
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<p>Now we have <b>x = <i>something</i></b>, </p>
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<p>and a short calculation reveals that <b>x = 5</b></p>
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</div>
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<h2>Like a Puzzle</h2>
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<p>In fact, solving an equation is just like solving a puzzle. And like puzzles, there are things we can (and cannot) do. </p>
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<p>Here are some things we can do:</p>
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<ul>
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<li><a href="introduction.html">Add or Subtract</a> the same value from both sides</li>
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<li>Clear out any fractions by <a href="introduction-multiply.html">Multiplying</a> every term by the bottom parts</li>
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<li>Divide every term by the same nonzero value</li>
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<li><a href="like-terms.html">Combine Like Terms</a></li>
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<li><a href="factoring.html">Factoring</a></li>
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<li><a href="expanding.html">Expanding</a> (the opposite of factoring) may also help</li>
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<li>Recognizing a pattern, such as the <a href="special-binomial-products.html">difference of squares</a></li>
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<li>Sometimes we can apply a function to both sides (e.g. square both sides) </li>
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</ul>
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<div class="example">
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<h3>Example: Solve √(x/2) = 3</h3>
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<div class="tbl">
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<div class="row"><span class="left">Start with:</span><span class="right">√(x/2) = 3</span></div>
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<div class="row"><span class="left">Square both sides:</span><span class="right">x/2<sup> </sup>= 3<sup>2</sup></span></div>
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<div class="row"><span class="left">Calculate 3<sup>2</sup> = 9:</span><span class="right">x/2 = 9</span></div>
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<div class="row"><span class="left">Multiply both sides by 2:</span><span class="right">x = 18</span></div>
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</div>
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</div>
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<p>And the more "tricks" and techniques you learn the better you will get.</p>
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<h2>Special Equations</h2>
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<p>There are special ways of solving some types of equations. Learn how to ...</p>
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<ul>
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<li>solve <a href="quadratic-equation.html">Quadratic Equations</a></li>
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<li>solve <a href="radical-equations-solving.html">Radical Equations</a></li>
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<li>solve <a href="../sine-cosine-tangent.html">Equations with Sine, Cosine and Tangent </a></li>
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</ul>
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<h2>Check Your Solutions</h2>
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<p>You should always check that your "solution" really <b>is</b> a solution.</p>
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<h3>How To Check</h3>
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<p>Take the solution(s) and put them in the <b>original equation</b> to see if they really work.</p>
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<div class="example">
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<h3>Example: solve for x:</h3>
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<p class="center large"><span class="intbl"><em>2x</em><strong>x − 3</strong></span> + 3 = <span class="intbl"><em>6</em><strong>x − 3</strong></span> (x≠3)</p>
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<p>We have said <span class="large">x≠3</span> to avoid a division by zero.</p>
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<p>Let's multiply through by <span class="larger">(x − 3)</span>:</p>
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<p align="center" class="larger">2x + 3(x−3) = 6</p>
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<p>Bring the 6 to the left:</p>
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<p align="center" class="larger">2x + 3(x−3) − 6 = 0</p>
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<p>Expand and solve:</p>
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<p align="center" class="larger">2x + 3x − 9 − 6 = 0</p>
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<p align="center" class="larger">5x − 15 = 0</p>
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<p align="center" class="larger">5(x − 3) = 0</p>
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<p align="center" class="larger">x − 3 = 0 </p>
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<p>That can be solved by having <span class="large">x=3</span></p>
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<p>Let us check:</p>
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<p class="center large"><span class="intbl">
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<em>2 × 3</em>
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<strong>3 − 3</strong>
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</span> + 3 = <span class="intbl">
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<em>6</em>
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<strong>3 − 3</strong>
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</span></p>
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<p><b>Hang On! <br>
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That means Dividing by Zero! </b></p>
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<p>And anyway, we said at the top that <span class="large">x≠3</span>, so ...</p>
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<p align="center">x = 3 does not actually work, and so:</p>
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<p align="center" class="larger">There is <b>No</b> Solution!</p>
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</div>
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<p>That was interesting ... we <b>thought</b> we had found a solution, but when we looked back at the question we found it wasn't allowed!</p>
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<p>This gives us a moral lesson:</p>
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<div class="def">
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<p>"Solving" only gives us possible solutions, they need to be checked!</p>
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</div>
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<h2>Tips</h2>
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<ul>
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<li>Note down where an expression is <b>not defined</b> (due to a division by zero, the square root of a negative number, or some other reason)</li>
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<li>Show <b>all the steps</b>, so it can be checked later (by you or someone else)</li>
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</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(493, 2324, 494, 2325, 495, 1137, 2326, 1138, 1139, 2327);</script>
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</div>
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<div class="related"><a href="definitions.html">Algebra - Basic Definitions</a> <a href="index.html">Algebra Index</a></div>
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Copyright © 2018 MathsIsFun.com
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