document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=911966>Question 911966</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine the solution sets of the following systems of linear equation by elimination.\r<BR>\n" );
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document.write( "3x-8y=-18<BR>\n" );
document.write( "x+y=1 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #553468 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=553468\">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" );
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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=3%2Ax-8%2Ay=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%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=1%2Ax%2B1%2Ay=1 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=1\"   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 3 and 1 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 3 and 1 is 3, we need to multiply both sides of the top equation by 1 and multiply both sides of the bottom equation by -3 like this:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2A%283%2Ax-8%2Ay%29=%28-18%29%2A1 ALIGN=MIDDLE  ALT=\"1%2A%283%2Ax-8%2Ay%29=%28-18%29%2A1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 1<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2A%281%2Ax%2B1%2Ay%29=%281%29%2A-3 ALIGN=MIDDLE  ALT=\"-3%2A%281%2Ax%2B1%2Ay%29=%281%29%2A-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -3<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=3%2Ax-8%2Ay=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%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=-3%2Ax-3%2Ay=-3 ALIGN=MIDDLE  ALT=\"-3%2Ax-3%2Ay=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 3 and -3 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2B-3=0 ALIGN=MIDDLE  ALT=\"3%2B-3=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=%283%2Ax-3%2Ax%29-8%2Ay-3%2Ay%29=-18-3 ALIGN=MIDDLE  ALT=\"%283%2Ax-3%2Ax%29-8%2Ay-3%2Ay%29=-18-3\"   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=%283-3%29%2Ax-8-3%29y=-18-3 ALIGN=MIDDLE  ALT=\"%283-3%29%2Ax-8-3%29y=-18-3\"   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%283%2B-3%29%2Ax%2B%28-8-3%29%2Ay=-18-3 ALIGN=MIDDLE  ALT=\"cross%283%2B-3%29%2Ax%2B%28-8-3%29%2Ay=-18-3\"   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=-11%2Ay=-21 ALIGN=MIDDLE  ALT=\"-11%2Ay=-21\"   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=-21%2F-11 ALIGN=MIDDLE  ALT=\"y=-21%2F-11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-11 ALIGN=MIDDLE  ALT=\"-11\"   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=21%2F11 ALIGN=MIDDLE  ALT=\"y=21%2F11\"   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=3%2Ax-8%2Ay=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%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=3%2Ax-8%2821%2F11%29=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%2821%2F11%29=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=21%2F11 ALIGN=MIDDLE  ALT=\"y=21%2F11\"   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=3%2Ax-168%2F11=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-168%2F11=-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=3%2Ax-168%2F11=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-168%2F11=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax=-18%2B168%2F11 ALIGN=MIDDLE  ALT=\"3%2Ax=-18%2B168%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-168%2F11 ALIGN=MIDDLE  ALT=\"-168%2F11\"   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=3%2Ax=-198%2F11%2B168%2F11 ALIGN=MIDDLE  ALT=\"3%2Ax=-198%2F11%2B168%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make -18 into a fraction with a denominator of 11<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ax=-30%2F11 ALIGN=MIDDLE  ALT=\"3%2Ax=-30%2F11\"   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%2F3%29%283%29%29%2Ax=%28-30%2F11%29%281%2F3%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F3%29%283%29%29%2Ax=%28-30%2F11%29%281%2F3%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%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   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=-10%2F11 ALIGN=MIDDLE  ALT=\"x=-10%2F11\"   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=-10%2F11 ALIGN=MIDDLE  ALT=\"x=-10%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=21%2F11 ALIGN=MIDDLE  ALT=\"y=21%2F11\"   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=-10%2F11 ALIGN=MIDDLE  ALT=\"-10%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2F11 ALIGN=MIDDLE  ALT=\"21%2F11\"   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=3%2Ax-8%2Ay=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%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=1%2Ax%2B1%2Ay=1 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=1\"   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+-10.9090909090909%2C+10.9090909090909%2C+-11.9090909090909%2C+11.9090909090909%2C%0D%0A++++graph%28+500%2C+600%2C+-10.9090909090909%2C+10.9090909090909%2C+-11.9090909090909%2C+11.9090909090909%2C+%28-18-3%2Ax%29%2F-8%2C+%281-1%2Ax%29%2F1+%29%2C%0D%0A++++blue%28++circle%28+-0.909090909090909%2C+1.90909090909091%2C+0.145454545454545+%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=3%2Ax-8%2Ay=-18 ALIGN=MIDDLE  ALT=\"3%2Ax-8%2Ay=-18\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Ax%2B1%2Ay=1 ALIGN=MIDDLE  ALT=\"1%2Ax%2B1%2Ay=1\"   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=-10%2F11 ALIGN=MIDDLE  ALT=\"-10%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=21%2F11 ALIGN=MIDDLE  ALT=\"21%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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