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Delete this to disable. //listMenu.animations[listMenu.animations.length] = animFade; //listMenu.animations[listMenu.animations.length] = animClipDown; //listMenu.animSpeed = 20; var arrow = null; if (document.createElement && document.documentElement) { arrow = document.createElement('span'); arrow.appendChild(document.createTextNode('-->')); arrow.className = 'subind'; } addEvent(window, 'load', new Function('listMenu.activateMenu("listMenuRoot", arrow)')); addEvent(window, 'load', new Function('listMenu2.activateMenu("listMenuRoot2", arrow)')); addEvent(window, 'load', new Function('listMenu3.activateMenu("listMenuRoot3", arrow)')); <p> </p> </td><td width="10"></td> <td valign="top"> <h2 class="linkedH1"><a target="_blank" href="http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq.htm">Simultaneous Linear Equations</a></h2> <h2>The Elimination Method</h2> <p> This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable. Once this has been done, the solution is the same as that for when one line was vertical or parallel. This method is known as the <em><strong>Gaussian elimination method</strong></em>.</p> <p> </p> <h3>Example 2. </h3> <p>Solve the following pair of simultaneous linear equations:</p> Equation 1: <font color="#FF0000">2</font><em>x</em> + 3<em>y</em> = 8<br> Equation 2: <font color="#00FF00">3</font><em>x</em> + 2<em>y</em> = 7 <p><strong>Step 1:</strong> Multiply each equation by a suitable number so that the two equations have the same leading coefficient. An easy choice is to multiply Equation 1 by <font color="#00FF00">3</font><em></em>, the coefficient of<em> x</em> in Equation 2, and multiply Equation 2 by <font color="#FF0000">2</font>, the <em>x</em> coefficient in Equation 1:</p> <p> </p><table width="100%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="11%" height="20" nowrap=""><font color="#00FF00">3</font> * (Eqn 1) ---> </td> <td width="10%"><table width="100%" border="0" cellpadding="0" cellspacing="0" bgcolor="#CCCCCC"> <tr> <td nowrap="" bgcolor="#FFFFFF"><font color="#00FF00">3</font>* (<font color="#FF0000">2</font><em>x</em> + 3<em>y</em> = 8) </td> </tr> </table></td> <td width="16%" nowrap=""> ---> <font color="#0000FF">6</font><em>x</em> + 9<em>y</em> = 24</td> <td width="63%"> </td> </tr> </table> <table width="100%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="12%" height="20" nowrap=""><font color="#FF0000">2</font> * (Eqn 2) ---> </td> <td width="11%"><table width="12%" border="0" cellpadding="0" cellspacing="0" bgcolor="#CCCCCC"> <tr> <td nowrap="" bgcolor="#FFFFFF"><font color="#FF0000">2</font> * (<font color="#00FF00">3</font><em>x</em> + 2<em>y</em> = 7) </td> </tr> </table></td> <td width="15%" nowrap=""> ---> <font color="#0000FF">6</font><em>x</em> + 4<em>y</em> = 14</td> <td width="62%" nowrap=""> Both equations now have the same leading coefficient = <font color="#0000FF">6</font></td> </tr> </table> <p> </p> <p><strong>Step 2:</strong> Subtract the second equation from the first.</p> <table width="14%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="23%" height="19" nowrap=""> </td> <td width="77%" nowrap=""><font color="#FFFFFF">-(</font>6<em>x</em> + 9<em>y</em> = 24</td> </tr> <tr> <td nowrap=""> </td> <td nowrap="">-(6<em>x</em> + 4<em>y</em> = 14)</td> </tr> <tr> <td nowrap=""> </td> <td nowrap=""><hr></td> </tr> <tr> <td nowrap=""> </td> <td nowrap=""> 5<em>y</em> = 10</td> </tr> </table> <p><strong>Step 3:</strong> Solve this new equation for <em>y</em>.</p> <table width="21%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="23%" nowrap=""> </td> <td width="77%" nowrap=""> <em>y</em> = 10/5 = 2</td> </tr> </table> <p><strong>Step 4:</strong> Substitute <em>y</em> = 2 into either Equation 1 or Equation 2 above and solve for <em>x</em>. We'll use Equation 1.</p> <table width="57%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="23%" height="19" nowrap=""> </td> <td width="38%" nowrap="">2<em>x</em> + 3(2) = 8</td> <td width="39%" nowrap=""> </td> </tr> <tr> <td nowrap=""> </td> <td nowrap="">2<em>x</em> + 6 = 8</td> <td nowrap="">Subtract 6 from both sides</td> </tr> <tr> <td nowrap=""> </td> <td nowrap="">2<em>x</em> = 2</td> <td nowrap="">Divide both sides by 2</td> </tr> <tr> <td nowrap=""> </td> <td nowrap=""><em>x</em> = 1</td> <td nowrap=""> </td> </tr> </table> <p><strong>Solution:</strong> <em>x</em> = 1, <em>y</em> = 2 or (1,2).</p> <p> </p> <p>Now study some more worked examples:</p> <form> <table align="CENTER" cellpadding="5" cellspacing="0" bgcolor="#003366"> <tr> <td align="CENTER"><div align="left"> <p align="center"><font face="Arial" color="#FFFFFF"><b> <input name="button2322" type="button" onclick="setup3(this.form);" value="Get New Example"> </b></font></p> <table align="center" border="0" cellspacing="0" cellpadding="0"> <tr> <td><div align="right"><font face="Arial" color="#FFFFFF"><b> Equation 1: </b></font></div></td> <td><font face="Arial" color="#FFFFFF"><b> </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="ONE"> <em>x</em> + <input type="text" size="3" name="TWO"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="THREE"> </b></font><font face="Arial" color="#FFFFFF"><b> </b></font></td> </tr> <tr> <td><div align="right"><font face="Arial" color="#FFFFFF"><b>Equation 2: </b></font></div></td> <td><font face="Arial" color="#FFFFFF"><b> </b></font><font face="Arial" color="#FFFFFF"><b> </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="FOUR"> <em>x</em> + <input type="text" size="3" name="FIVE"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="SIX"> </b></font><font face="Arial" color="#FFFFFF"><b> </b></font><font face="Arial" color="#FFFFFF"><b> </b></font></td> </tr> </table> <p align="center"> </p> <hr width="400"> <p><font face="Arial" color="#FFFFFF"><b>Click on the buttons below to see how to solve these equations.</b></font></p> <p align="left"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <font color="#FFFFFF" face="Arial"><b> <input name="button3222" type="button" onclick="operation(this.form);" value="1. Combine Equations 1 and 2 to find a simpler equation"> </b></font> </b></font></b></font> <font face="Arial" color="#FFFFFF"><b> <input type="text" size="100" name="SEVEN"> </b></font> <font face="Arial" color="#FFFFFF"><b> <input type="text" size="100" name="SEVEN2"> <input type="text" size="100" name="SEVEN3"> </b></font></p> <p align="left"> </p> <table width="100%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="27%" nowrap=""><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="coef2"> ( <input type="text" size="3" name="eq1a"> <em>x</em> + <input type="text" size="3" name="eq1b"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="eq1c"> )</b></font></td> <td width="36%" nowrap=""><font face="Arial" color="#FFFFFF"><b> --------> <input type="text" size="3" name="EQ1a2"> <em>x</em> + <input type="text" size="3" name="EQ1b2"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ1c2"> --------> </b></font></td> <td width="5%" rowspan="2" nowrap=""><font face="Arial" color="#FFFFFF"> </font></td> <td width="9%" nowrap=""><font face="Arial" color="#FFFFFF"><b>Subtract:</b></font></td> <td width="23%" nowrap=""><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ1a"> <em>x</em> + <input type="text" size="3" name="EQ1b"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ1c"> </b></font></td> </tr> <tr> <td nowrap=""> <div align="left"><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="coef1"> ( <input type="text" size="3" name="eq2a"> <em>x</em> + <input type="text" size="3" name="eq2b"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="eq2c"> ) </b></font></div></td> <td><font face="Arial" color="#FFFFFF"><b> --------><font size="+1"> </font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ2a2"> <em>x</em> + <input type="text" size="3" name="EQ2b2"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ2c2"> --------><font size="+1"> </font><font face="Arial" color="#FFFFFF"><b> </b></font> </b></font><font size="+1"> </font></b></font></td> <td width="9%" nowrap=""><div align="right"><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="m2"> [ </b></font></div></td> <td nowrap=""><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ2a"> <em>x</em> + <input type="text" size="3" name="EQ2b"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="EQ2c"> ] </b></font></td> </tr> <tr> <td> </td> <td> </td> <td colspan="2" rowspan="2" align="right" valign="bottom" nowrap=""><font face="Arial" color="#FFFFFF"><b> </b></font></td> <td><hr></td> </tr> <tr> <td> </td> <td> </td> <td nowrap=""> <div align="left"><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="FIVE2"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="SIX2"> </b></font></div></td> </tr> </table> <p align="left"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <input name="button32222" type="button" onclick="solvey(this.form);" value="2. Solve for y"> </b></font></b></font></p> <p align="center"><font face="Arial" color="#FFFFFF"><b><font face="Arial" color="#FFFFFF"><b><em> y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input name="EIGHT" type="text" id="EIGHT" size="10"> (2 d.p.) </b></font></b></font></p> <p align="left"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <font color="#FFFFFF" face="Arial"><b> <input name="button322222" type="button" onclick="solvex(this.form);" value="3. Solve for x"> </b></font> </b></font></b></font></p> <p align="center"><font face="Arial" color="#FFFFFF"><b> Substitute <em>y</em> into Equation 1 and solve for x: <input type="text" size="3" name="ONE2"> <em>x</em> + <input type="text" size="3" name="TWO2"> <font face="Arial" color="#FFFFFF"><b> ( <input name="EIGHT2" type="text" id="EIGHT2" size="4"> ) </b></font></b></font><font face="Arial" color="#FFFFFF"><b>= </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="THREE2"> </b></font></p> <p align="center"> </p> <p align="center"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><em>x </em>=</font></b></font> <font face="Arial" color="#FFFFFF"><b> <input name="NINE" type="text" id="NINE" size="10"> <font face="Arial" color="#FFFFFF"><b>(2 d.p.)</b></font> </b></font></p> </div> <p><font color="#FFFFFF" size="2" face="Arial"><b><br> </b></font></p></td> </tr> </table> </form> <h3> </h3> <h3>Exercise 2.</h3> <form> <table align="CENTER" cellpadding="5" cellspacing="0" bgcolor="#003366"> <tr> <td align="CENTER"><div align="left"> <p align="center"><font face="Arial" color="#FFFFFF"><b> <input name="button23222" type="button" onclick="setup4(this.form);" value="Get New Exercise"> </b></font></p> <p align="center"><font face="Arial" color="#FFFFFF"><b> Equation 1: <input type="text" size="3" name="b1"> <em>x</em> + <input type="text" size="3" name="b2"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="b3"> </b></font></p> <p align="center"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b>Equation 2</b></font>: <input type="text" size="3" name="b4"> <em>x</em> + <input type="text" size="3" name="b5"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="b6"> </b></font></p> </div></td> </tr> <tr> <td align="CENTER" bgcolor="#669900"><p><font face="Arial" color="#FFFFFF"><b>First determine what you will multiply each of the above equations by to get the same leading coefficients:</b></font> <font face="Arial" color="#FFFFFF"><b> </b></font></p> <div align="center"> <table width="61%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="46%" nowrap=""><font face="Arial" color="#FFFFFF"><b>Multiply Equation 1 by <input type="text" size="2" name="b7"> </b></font></td> <td width="54%"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> </b></font></b></font></td> </tr> <tr> <td><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <font color="#FFFFFF" face="Arial"></font> Multiply Equation 2 by <input type="text" size="2" name="b8"> </b></font></b></font></td> <td><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <input name="button32223" type="button" onclick="check_op(this.form);" value="Check"> </b></font></b></font></td> </tr> </table> <p align="left"><font face="Arial" color="#FFFFFF"><b>Subtract:</b></font> <font face="Arial" color="#FFFFFF"><b> </b></font></p> <table width="61%" border="0" cellspacing="0" cellpadding="0"> <tr> <td width="46%" nowrap=""><font face="Arial" color="#FFFFFF"><b> </b></font><font face="Arial" color="#FFFFFF"><b> </b></font></td> <td width="54%" nowrap=""><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <input type="text" size="4" name="B1"> <em>x</em> + <input type="text" size="4" name="B2"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="4" name="B3"> </b></font><font color="#FFFFFF" face="Arial"><b> </b></font></b></font></td> </tr> <tr> <td><div align="right"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="3" name="b92"> <font face="Arial" color="#FFFFFF"><b>( </b></font> </b></font></b></font></div></td> <td nowrap=""><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b><font color="#FFFFFF" face="Arial"><b><font color="#FFFFFF" face="Arial"></font> <input type="text" size="4" name="B12"> <em>x</em> + <input type="text" size="4" name="B22"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="4" name="B32"> <font face="Arial" color="#FFFFFF"></font>) </b></font> </b></font></b></font></td> </tr> <tr> <td> </td> <td><hr></td> </tr> <tr> <td><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <input name="button322223" type="button" onclick="check_work(this.form);" value="Check Working"> </b></font></b></font></td> <td><font face="Arial" color="#FFFFFF"><b> <input type="text" size="4" name="b9"> </b></font><font face="Arial" color="#FFFFFF"><b><em>y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input type="text" size="4" name="b10"> </b></font></td> </tr> </table> <p> </p> </div> <div align="left"><font face="Arial" color="#FFFFFF"><b>Solve for <em>y.</em> Then substitute <em>y</em> in either Equation 1 or Equation 2 and solve for <em>x</em>. Give your solutions for both <em>x</em> and <em>y</em> to two decimal places:</b></font> <font face="Arial" color="#FFFFFF"><b> </b></font></div> <p align="center"><font face="Arial" color="#FFFFFF"><b><font face="Arial" color="#FFFFFF"><b><em> y</em> = </b></font><font face="Arial" color="#FFFFFF"><b> <input name="b11" type="text" size="10"> <b>to </b>2 d.p.<font color="#FFFFFF" face="Arial"><b> <input name="button3222222" type="button" onclick="check_y(this.form);" value="Check y value"> </b></font> </b></font><font color="#FFFFFF" face="Arial"><b> </b></font></b></font></p> <p align="left"><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> </b></font></b></font></p> <p align="center"><font face="Arial" color="#FFFFFF"><b><font face="Arial" color="#FFFFFF"><b></b></font></b></font><font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><em>x </em>=</font></b></font> <font face="Arial" color="#FFFFFF"><b> <input name="b12" type="text" size="10"> <font face="Arial" color="#FFFFFF"><b><b>to </b>2 d.p.</b></font> <font face="Arial" color="#FFFFFF"><b><font color="#FFFFFF" face="Arial"><b> <input name="button32222222" type="button" onclick="check_x(this.form);" value="Check x value"> </b></font></b></font></b></font></p></td> </tr> </table> </form> <p align="center"> << <a target="_blank" href="http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq/Systems.htm">Simultaneous Linear Equations (definition)</a> | <a target="_blank" href="http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq.htm">Simultaneous Linear Equations Index </a> | <a target="_blank" href="http://mathsfirst.massey.ac.nz/Algebra/SystemsofLinEq/SMeth.htm">The Substitution Method</a> >></p> <p align="center"> </p> <p align="center"> </p> </td> </tr> </table> <table width="100%" border="0" cellpadding="0" cellspacing="2"> <tr bgcolor="#003366"> <td height="30"> <span class="nav"><a target="_blank" href="http://aboutmassey.massey.ac.nz/massey/about-massey/contacts.cfm">Contact Us</a></span> <span class="nav"> | <a target="_blank" href="http://aboutmassey.massey.ac.nz/">About Massey University</a> | <a target="_blank" href="http://sitemap.massey.ac.nz/">Sitemap</a> | <a target="_blank" href="http://www.massey.ac.nz/disclaim.htm">Disclaimer</a> | Last updated: November 21, 2012 © Massey University 2003</span> </td> </tr> </table> </body> </html> How do you do the elimination method?The Elimination Method. Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. ... . Step 2: Subtract the second equation from the first.. Step 3: Solve this new equation for y.. Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x.. What are the four steps to solving systems of equations by elimination?Step 1: Put the equations in Standard Form. Step 2: Determine which variable to eliminate. Step 3: Multiply the equations and solve. Step 4: Plug back in to find the other variable.
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How to do elimination method for systems of equations
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