document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=107423>Question 107423</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I solve by elimination: X-4Y=-23<BR>\n" );
document.write( "                               2X+3Y=9 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #78188 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=78188\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the two linear equations:<BR>\n" );
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document.write( "X-4Y=-23<BR>\n" );
document.write( "2X+3Y=9<BR>\n" );
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document.write( "To solve these two equations by elimination we must get the size of one term in the top<BR>\n" );
document.write( "equation to be equal to the size of the corresponding term in the bottom equation. <BR>\n" );
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document.write( "Let's suppose we decide to eliminate the Y terms in the two given equations. We can do that<BR>\n" );
document.write( "by multiplying the entire top equation (all terms on both sides) by 3 to convert it to:<BR>\n" );
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document.write( "3X-12Y=-69<BR>\n" );
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document.write( "Next let's multiply the entire bottom equation (all terms on both sides) by 4 to convert<BR>\n" );
document.write( "it to:<BR>\n" );
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document.write( "8X+12Y=36<BR>\n" );
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document.write( "So now our two equations are:<BR>\n" );
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document.write( "3X-12Y=-69<BR>\n" );
document.write( "8X+12Y=36<BR>\n" );
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document.write( "Suppose now that we add the two equations vertically in columns. Notice that the 3X and the<BR>\n" );
document.write( "8X add to 11X. But more important, notice that in the \"Y\" column the -12Y and the +12Y<BR>\n" );
document.write( "sum to zero because of the difference in their signs. So they disappear. On the right side<BR>\n" );
document.write( "the -69 and the +36 sum to -33. So what we are left with if we add the two equations<BR>\n" );
document.write( "vertically is:<BR>\n" );
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document.write( "11X = -33<BR>\n" );
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document.write( "Solve for X by dividing both sides of this equation by 11 (the multiplier of X) to get:<BR>\n" );
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document.write( "X = -33/11 = -3<BR>\n" );
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document.write( "So we have part of the answer ... X = -3<BR>\n" );
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document.write( "Now we can return to either one of the original equations, substitute -3 for X, and solve for<BR>\n" );
document.write( "Y. Let's return to X - 4Y = -23. Substitute -3 for X and this equation becomes:<BR>\n" );
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document.write( "-3 - 4Y = -23<BR>\n" );
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document.write( "Get rid of the -3 on the left side by adding +3 to both sides. When we do the equation becomes:<BR>\n" );
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document.write( "-4Y = -20<BR>\n" );
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document.write( "Solve for Y by dividing both sides of this by -4 (the multiplier of Y) to get:<BR>\n" );
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document.write( "Y = -20/-4 = +5<BR>\n" );
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document.write( "So the answer to this problem is X = -3 and Y = +5. This means that the point (-3, 5) is on<BR>\n" );
document.write( "the graphs of both given equations.<BR>\n" );
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document.write( "Hope this helps you to get the \"hang\" of finding the common solution for two linear equations<BR>\n" );
document.write( "that have a common solution.<BR>\n" );
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document.write( " </SPAN>\n" );
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