document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81496>Question 81496</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve each of the following systems by substitution.\r<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
document.write( "y = 2x - 4 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58437 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=58437\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Solve the following system by substitution:<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
document.write( "y = 2x - 4<BR>\n" );
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document.write( "Since the bottom equation is already solved for y in terms of x, let's substitute the right<BR>\n" );
document.write( "side of the bottom equation for y in the top equation.  With that substitution the top equation<BR>\n" );
document.write( "becomes:<BR>\n" );
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document.write( "8x - 4(2x - 4) = 16<BR>\n" );
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document.write( "Do the distributed multiplication on the left side to get:<BR>\n" );
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document.write( "8x - 8x + 16 = 16<BR>\n" );
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document.write( "Notice that the two terms that contain x cancel each other.  That reduces the equation<BR>\n" );
document.write( "to just:<BR>\n" );
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document.write( "+16 = +16<BR>\n" );
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document.write( "What does this mean? It means that the two equations are always satisfied no matter what<BR>\n" );
document.write( "value of x you choose.  For that to be the case, both equations must be the same so that<BR>\n" );
document.write( "every solution of one of them is also a solution of the other.<BR>\n" );
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document.write( "Another way to look at this is to say to yourself, let me solve the top equation for y in<BR>\n" );
document.write( "terms of x. Start with the top equation:<BR>\n" );
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document.write( "8x - 4y = 16<BR>\n" );
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document.write( "Divide all the terms on both sides by the common factor of 4.  When you do that division,<BR>\n" );
document.write( "the top equation becomes:<BR>\n" );
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document.write( "2x - y = 4<BR>\n" );
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document.write( "Next subtract 2x from both sides and you get:<BR>\n" );
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document.write( "-y = -2x + 4<BR>\n" );
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document.write( "Finally, to solve for y, multiply all the terms on both sides by -1 and you get:<BR>\n" );
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document.write( "y = +2x - 4<BR>\n" );
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document.write( "Look at that! It turns out that the top equation is the same as the bottom equation.<BR>\n" );
document.write( "So every solution that works in the top equation must also work in the bottom equation.<BR>\n" );
document.write( "This also means that the graph of the top equation lies on top of the graph of the bottom<BR>\n" );
document.write( "equation.  Therefore, there are an infinite number of common solutions.<BR>\n" );
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document.write( "Hope this helps you to understand how the problem has misled you by asking you to solve<BR>\n" );
document.write( "using substitution, only for you to find out that all the variables in the equation after <BR>\n" );
document.write( "substitutions are made cancel out or disappear.<BR>\n" );
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