document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=81822>Question 81822</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system by addition or substitution.<BR>\n" );
document.write( "	–8x + 2y = 8<BR>\n" );
document.write( "	y = 4 + 4x <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #58636 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=58636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the equation set:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "	–8x + 2y = 8<BR>\n" );
document.write( "	y = 4 + 4x <BR>\n" );
document.write( ".<BR>\n" );
document.write( "Solve by addition or substitution.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "First by addition:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Rearrange the bottom equation so that it is in the same arrangement as the top equation.<BR>\n" );
document.write( "So get the bottom equation so the terms containing x and y are on the left side and the<BR>\n" );
document.write( "constants are on the right side. Do it using the following process:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "y = 4 + 4x<BR>\n" );
document.write( ".<BR>\n" );
document.write( "eliminate the 4x on the right side by subtracting 4x from both sides to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-4x + y = 4<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This is in the form of the original top equation. So now the equation set is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-8x + 2y = 8<BR>\n" );
document.write( "-4x +  y = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you now multiply both sides of the bottom equation by -2 the equation set becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-8x + 2y = 8<BR>\n" );
document.write( "+8x - 2y = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Now you can add the two equations. But notice that when you do that the result is that<BR>\n" );
document.write( "both the x and the y terms on the left side cancel each other and the result is:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "0 = 0<BR>\n" );
document.write( ".<BR>\n" );
document.write( "More significantly, look at what we did to the bottom equation. We rearranged it and then<BR>\n" );
document.write( "multiplied both sides by a common number. Suppose that instead of multiplying the rearranged<BR>\n" );
document.write( "bottom equation by -2 we had multiplied it by +2. The result would have been that it<BR>\n" );
document.write( "would become -8x + 2y = 8 and the equation set would then be:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-8x + 2y = 8 and<BR>\n" );
document.write( "-8x + 2y = 8<BR>\n" );
document.write( ".<BR>\n" );
document.write( "This shows that the bottom equation is actually the same as the top equation. And that<BR>\n" );
document.write( "means that any solution of the top equation is also a solution to the bottom equation.<BR>\n" );
document.write( "So for the given set of equations, there are an infinite number of common solutions, <BR>\n" );
document.write( "not just one. Any solution for one of the equations is common with the other equation<BR>\n" );
document.write( "because both equations are the same.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "If you try to solve this by substitution, you can notice that the original bottom <BR>\n" );
document.write( "equation is already solved for y. So you can substitute the right side of the bottom<BR>\n" );
document.write( "equation for y in the top equation.  When you do that substitution the top equation<BR>\n" );
document.write( "becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-8x + 2(4 + 4x) = 8<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Do the distributed multiplication on the left side by multiplying 2 times each of the<BR>\n" );
document.write( "terms in parentheses to get:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "-8x + 8 + 8x = 8<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Notice again how the two terms containing x cancel and the equation becomes:<BR>\n" );
document.write( ".<BR>\n" );
document.write( "8 = 8<BR>\n" );
document.write( ".<BR>\n" );
document.write( "And subtracting 8 from both sides again results in 0 = 0.  This again is a clue that something<BR>\n" );
document.write( "is amiss and that the two equations have to be identical for both sides to become <BR>\n" );
document.write( "0 = 0.  Again the conclusion is that if the equations are identical (or can be made so<BR>\n" );
document.write( "with some permissible algebraic manipulations) then every solution of one is a solution<BR>\n" );
document.write( "of the other ... and there are an infinite number of common solutions.<BR>\n" );
document.write( ".<BR>\n" );
document.write( "Hope this helps you to understand this \"tricky\" problem. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );