document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=245529>Question 245529</A>: <I><SPAN STYLE=\"word-break: break-all;\"> im doing linear equations and dont understand howto do it at all.<BR>\n" );
document.write( "2x+y=4<BR>\n" );
document.write( "x-y=2  ??<BR>\n" );
document.write( "it says use linear combinations to solve the systme of linear equations. use the graph to check your solution </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #179359 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=179359\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "The Linear Combinations Method is also known as the Addition Method which is also known as the Elimination Method.  I like to call it the Elimination Method because that name most aptly describes what you are trying to accomplish, namely the elimination of one of the variables so that you can easily solve for the other.\r<BR>\n" );
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document.write( "The first step of the method is to multiply one or the other (or sometimes both) of the equations by a number or numbers so that the coefficients on one of the variables becomes the additive inverse of the coefficient on that same variable in the other equation.\r<BR>\n" );
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document.write( "In your very simple example, you don't have to do the first step -- it is already done for you with respect to the variable <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">.  Note that the coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> in the first equation is 1 and the coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> in the second equation is -1.  1 and -1 are additive inverses because 1 + (-1) = 0.\r<BR>\n" );
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document.write( "The next step is to add the two equations together (that's where the Addition Method name comes from).  Just like any other simplification, you combine like terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ =\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ -\ y\ =\ 2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 2x\ +\ x\ =\ 3x\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\ +\ (-y)\ =\ 0y\">, and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 4\ +\ 2\ = 6\">, that is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3x\ +\ 0y\ =\ 6\">\r<BR>\n" );
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document.write( "Notice that we now have a zero coefficient on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">, so we can just eliminate (there's that name again) the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\"> variable and simply write:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 3x\ =\ 6\">\r<BR>\n" );
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document.write( "Which solves to:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ =\ 2\">\r<BR>\n" );
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document.write( "Now that you have a value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">, it is a simple matter to substitute that value back into one of the original equations and solve for the other variable.\r<BR>\n" );
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document.write( "However, I'm going to do this the hard way as a demonstration of the step 1 that you didn't have to do.  Let's eliminate the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> variable instead.\r<BR>\n" );
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document.write( "Begin with the given equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ =\ 4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \,x\ -\ y\ =\ 2\">\r<BR>\n" );
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document.write( "Multiply the second equation by -2:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ \ 2x\ +\ y\ \ \,=\ \ \,4\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ -2x\ +\ 2y\ =\ -4\">\r<BR>\n" );
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document.write( "Now the coefficients on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> are additive inverses, that is 2 + (-2) = 0.  Add the equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 0x\ +\ 3y\ =\ 0\">\r<BR>\n" );
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document.write( "and finally\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y\ =\ 0\">\r<BR>\n" );
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document.write( "That's all there is to it.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
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