document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196448>Question 1196448</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I've been working on this for hours, but keep taking the wrong path. I need to solve the 4x4 equations:<BR>\n" );
document.write( "2w + x - y + 2z = 10<BR>\n" );
document.write( "-w - 3x + y - 5z = -11<BR>\n" );
document.write( "3w + 5x - 3y + z = 35<BR>\n" );
document.write( "-4w - 2x - 3y - z = 2 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #829433 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=829433\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The other tutors have great solutions, so I don't have much to add. Though I would like to mention relevant software tools.\r<BR>\n" );
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document.write( "For problems like this, the Linear Algebra Toolkit is an invaluable resource<BR>\n" );
document.write( "The link is here<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"http://www.math.odu.edu/~bogacki/lat/\">http://www.math.odu.edu/~bogacki/lat/</a><BR>\n" );
document.write( "Click on \"Enter\". Then click on \"Solving a system of linear equations\"\r<BR>\n" );
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document.write( "We have m = 4 equations and n = 4 unknowns\r<BR>\n" );
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document.write( "The variables x1,x2,x3,x4 represent w,x,y,z in that order.<BR>\n" );
document.write( "The boxes to the left of each represent the coefficients.<BR>\n" );
document.write( "Type those coefficients in, as well as the right hand sides.\r<BR>\n" );
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document.write( "Once everything is typed in, double-check that the numbers have been entered correctly. Then hit the \"submit\" button.\r<BR>\n" );
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document.write( "This will provide the final answer. The step by step process is also provided. Click on \"Show Details\" to see the steps.\r<BR>\n" );
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document.write( "You should end up with<BR>\n" );
document.write( "x1 = 2<BR>\n" );
document.write( "x2 = 3<BR>\n" );
document.write( "x3 = -5<BR>\n" );
document.write( "x4 = -1\r<BR>\n" );
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document.write( "which means<BR>\n" );
document.write( "w = 2<BR>\n" );
document.write( "x = 3<BR>\n" );
document.write( "y = -5<BR>\n" );
document.write( "z = -1\r<BR>\n" );
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document.write( "An alternative calculator is WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.wolframalpha.com/\">https://www.wolframalpha.com/</a><BR>\n" );
document.write( "This is even easier to use since all you need to do is type the following shown in red<BR>\n" );
document.write( "<font color=red>2w + x - y + 2z = 10,-w - 3x + y - 5z = -11,3w + 5x - 3y + z = 35,-4w - 2x - 3y - z = 2</font><BR>\n" );
document.write( "Basically list each equation separated by commas.\r<BR>\n" );
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document.write( "WolframAlpha is great, but it doesn't show steps freely. You would need to become a paid subscriber to their service if you wanted to see the steps. \r<BR>\n" );
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document.write( "The first link (the linear algebra toolkit) is completely free for both steps and final answer. \r<BR>\n" );
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document.write( "Another useful tool, that I personally use all the time, is GeoGebra. <BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.geogebra.org/\">https://www.geogebra.org/</a><BR>\n" );
document.write( "It is completely free software. The drawback is that it doesn't show the step-by-step process.\r<BR>\n" );
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document.write( "To solve this system using GeoGebra, you would need to open the CAS (computer algebra system) panel.\r<BR>\n" );
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document.write( "Then in a CAS input box, you'll type the following shown in red<BR>\n" );
document.write( "<font color=red>Solve[{2w + x - y + 2z = 10,-w - 3x + y - 5z = -11,3w + 5x - 3y + z = 35,-4w - 2x - 3y - z = 2}]</font>\r<BR>\n" );
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document.write( "The input structure is very similar to WolframAlpha's input. Though we have \"Solve\" up front, and a pair of square brackets and curly braces surrounding the equations.\r<BR>\n" );
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document.write( "The template is Solve[{SYSTEM}] where \"SYSTEM\" is replaced with the system of equations in question.\r<BR>\n" );
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document.write( "There are numerous other calculators and tools dealing with this topic. Feel free to explore to find your favorites.\r<BR>\n" );
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document.write( "Keep in mind that these tools are meant to help you, and not do your homework for you. You should still do the problems yourself to get practice.<BR>\n" );
document.write( "When it comes to exams, it's likely your teacher will not allow such calculators to be used. <BR>\n" );
document.write( "S/he would probably only allow things like a TI83 but may not even allow that. \r<BR>\n" );
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document.write( "In short, it's best to get practice doing the problem yourself and then using the calculators mentioned as a way to check your answer. <BR>\n" );
document.write( "They are also helpful tools to see how to do something if you get stuck.<BR>\n" );
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