document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=179447>Question 179447</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system of equations using the addition (elimination) method.<BR>\n" );
document.write( "If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is \"no solution\" or \"infinitely many solutions\" and state how you arrived at that conclusion.<BR>\n" );
document.write( "      3x - 11y = 9<BR>\n" );
document.write( "     -9x + 33y = -27<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #134387 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=134387\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+1\">\r<BR>\n" );
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document.write( "See the solution to problem number 179448.  The process is the same.\r<BR>\n" );
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document.write( "http://www.algebra.com/algebra/homework/Linear-equations/Linear-equations.faq.question.179448.html\r<BR>\n" );
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document.write( "However, in this case, when you apply the elimination process, the result of adding the two equations will be the trivial identity <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0 = 0\">\r<BR>\n" );
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document.write( "What this indicates is that you have a pair of equations that represent the exact same set of ordered pairs, in other words, any ordered pair that satisfies one of the equations will satisfy the other.  You have an infinite number of solutions.\r<BR>\n" );
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document.write( "Graphically, each of these equations represents a straight line in the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large R^2\"> plane.  This is a situation where the two equations represent the same line.  You can verify this by solving each one of the equations for <i>y</i>, meaning arranging them so that <i>y</i> is by itself on the left and everything else is on the right of the equals sign.  You will see that the coefficient on the <i>x</i> terms are identical and the constant terms are identical as well meaning that the lines have the same slope and intersect the <i>y</i>-axis at the same point -- hence they are the same line.\r<BR>\n" );
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document.write( "Now consider the situation where you have the same two equations except that you change the constant term in one of them, like this:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {        } \math 3x - 11y = 10\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \text {        } \math -9x + 33y = -27\">\r<BR>\n" );
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document.write( "Now when you perform the elimination process, you arrive at the absurdity <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large 0 = 3\">.\r<BR>\n" );
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document.write( "This situation represents a situation where you have two equations that have no ordered pair that satisfies them both simultaneously, hence the solution set for the system of equations is the empty set.\r<BR>\n" );
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document.write( "The graphic analogue is that these two equations represent two sets of ordered pairs describing two distinct parallel lines.  The results of solving the equations for <i>y</i> in this case would be identical coefficients on <i>x</i> but different constant terms.  The slopes are the same indicating parallelism but the <i>y</i>-intercepts are different indicating different lines.\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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