document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=112087>Question 112087</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use elimination to solve:    x/2+2y=9<BR>\n" );
document.write( "                             2x-y=18 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #81809 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=81809\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=solving_linear_system_by_elimination.solver>Solving a System of Linear Equations by Elimination/Addition</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>TEST<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29%2Ax%2B%281%29%2Ay=9 ALIGN=MIDDLE  ALT=\"%281%2F2%29%2Ax%2B%281%29%2Ay=9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28%281%2F2%29%2Ax%2B%281%29%2Ay%29=%282%29%2A%289%29 ALIGN=MIDDLE  ALT=\"2%28%281%2F2%29%2Ax%2B%281%29%2Ay%29=%282%29%2A%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by the LCD 2<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" >Distribute and simplify<BR>\n" );
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document.write( "  Lets start with the given system of linear equations<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-1%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax-1%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for y, we would have to eliminate x (or vice versa).<BR>\n" );
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document.write( "  So lets eliminate x. In order to do that, we need to have both x coefficients that are equal but have opposite signs (for instance 2 and -2 are equal but have opposite signs). This way they will add to zero.<BR>\n" );
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document.write( "  So to make the x coefficients equal but opposite, we need to multiply both x coefficients by some number to get them to an equal number. So if we wanted to get 1 and 2 to some equal number, we could try to get them to the LCM.<BR>\n" );
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document.write( "  Since the LCM of 1 and 2 is 2, we need to multiply both sides of the top equation by 2 and multiply both sides of the bottom equation by -1 like this:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%281%2Ax%2B2%2Ay%29=%2818%29%2A2 ALIGN=MIDDLE  ALT=\"2%2A%281%2Ax%2B2%2Ay%29=%2818%29%2A2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 2<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A%282%2Ax-1%2Ay%29=%2818%29%2A-1 ALIGN=MIDDLE  ALT=\"-1%2A%282%2Ax-1%2Ay%29=%2818%29%2A-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -1<BR>\n" );
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document.write( "  So after multiplying we get this:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B4%2Ay=36 ALIGN=MIDDLE  ALT=\"2%2Ax%2B4%2Ay=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax%2B1%2Ay=-18 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B1%2Ay=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 2 and -2 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2B-2=0 ALIGN=MIDDLE  ALT=\"2%2B-2=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  Now add the equations together. In order to add 2 equations, group like terms and combine them<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Ax-2%2Ax%29%2B%284%2Ay%2B1%2Ay%29=36-18 ALIGN=MIDDLE  ALT=\"%282%2Ax-2%2Ax%29%2B%284%2Ay%2B1%2Ay%29=36-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282-2%29%2Ax%2B%284%2B1%29y=36-18 ALIGN=MIDDLE  ALT=\"%282-2%29%2Ax%2B%284%2B1%29y=36-18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%282%2B-2%29%2Ax%2B%284%2B1%29%2Ay=36-18 ALIGN=MIDDLE  ALT=\"cross%282%2B-2%29%2Ax%2B%284%2B1%29%2Ay=36-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Notice the x coefficients add to zero and cancel out. This means we've eliminated x altogether.<BR>\n" );
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document.write( "  So after adding and canceling out the x terms we're left with:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ay=18 ALIGN=MIDDLE  ALT=\"5%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F5 ALIGN=MIDDLE  ALT=\"y=18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F5 ALIGN=MIDDLE  ALT=\"y=18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2818%2F5%29=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2818%2F5%29=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F5 ALIGN=MIDDLE  ALT=\"y=18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B36%2F5=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B36%2F5=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B36%2F5=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B36%2F5=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax=18-36%2F5 ALIGN=MIDDLE  ALT=\"1%2Ax=18-36%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=36%2F5 ALIGN=MIDDLE  ALT=\"36%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax=90%2F5-36%2F5 ALIGN=MIDDLE  ALT=\"1%2Ax=90%2F5-36%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make 18 into a fraction with a denominator of 5<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax=54%2F5 ALIGN=MIDDLE  ALT=\"1%2Ax=54%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F1%29%281%29%29%2Ax=%2854%2F5%29%281%2F1%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F1%29%281%29%29%2Ax=%2854%2F5%29%281%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1 ALIGN=MIDDLE  ALT=\"1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=54%2F5 ALIGN=MIDDLE  ALT=\"x=54%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the terms on the right side<BR>\n" );
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document.write( "  So our answer is <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=54%2F5 ALIGN=MIDDLE  ALT=\"x=54%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F5 ALIGN=MIDDLE  ALT=\"y=18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  which also looks like<BR>\n" );
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document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=54%2F5 ALIGN=MIDDLE  ALT=\"54%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F5 ALIGN=MIDDLE  ALT=\"18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  Notice if we graph the equations (if you need help with graphing, check out this <a  rel=nofollow HREF=http://www.algebra.com/algebra/homework/Linear-equations/graphing-linear-equations.solver>solver</a>)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-1%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax-1%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  we get <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-20.8%2C+20.8%2C+-13.6%2C+13.6%2C%0D%0A++++graph%28+500%2C+600%2C+-20.8%2C+20.8%2C+-13.6%2C+13.6%2C+%2818-1%2Ax%29%2F2%2C+%2818-2%2Ax%29%2F-1+%29%2C%0D%0A++++blue%28++circle%28+10.8%2C+3.6%2C+0.277333333333333+%29+%29+%0D%0A++%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B2%2Ay=18 ALIGN=MIDDLE  ALT=\"1%2Ax%2B2%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-1%2Ay=18 ALIGN=MIDDLE  ALT=\"2%2Ax-1%2Ay=18\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) (hint: you may have to solve for y to graph these) and the intersection of the lines (blue circle).<BR>\n" );
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document.write( "  and we can see that the two equations intersect at (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=54%2F5 ALIGN=MIDDLE  ALT=\"54%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18%2F5 ALIGN=MIDDLE  ALT=\"18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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