document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=970535>Question 970535</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the system { 2x-3y=5<BR>\n" );
document.write( "                   4x+5y=6\r<BR>\n" );
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document.write( "using the elimination method ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #593165 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20850)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=593165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\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=2%2Ax-3%2Ay=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%2Ay=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax%2B5%2Ay=6 ALIGN=MIDDLE  ALT=\"4%2Ax%2B5%2Ay=6\"   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 2 and 4 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 2 and 4 is 4, 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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2A%282%2Ax-3%2Ay%29=%285%29%2A2 ALIGN=MIDDLE  ALT=\"2%2A%282%2Ax-3%2Ay%29=%285%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%284%2Ax%2B5%2Ay%29=%286%29%2A-1 ALIGN=MIDDLE  ALT=\"-1%2A%284%2Ax%2B5%2Ay%29=%286%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=4%2Ax-6%2Ay=10 ALIGN=MIDDLE  ALT=\"4%2Ax-6%2Ay=10\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2Ax-5%2Ay=-6 ALIGN=MIDDLE  ALT=\"-4%2Ax-5%2Ay=-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 4 and -4 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2B-4=0 ALIGN=MIDDLE  ALT=\"4%2B-4=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=%284%2Ax-4%2Ax%29-6%2Ay-5%2Ay%29=10-6 ALIGN=MIDDLE  ALT=\"%284%2Ax-4%2Ax%29-6%2Ay-5%2Ay%29=10-6\"   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=%284-4%29%2Ax-6-5%29y=10-6 ALIGN=MIDDLE  ALT=\"%284-4%29%2Ax-6-5%29y=10-6\"   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%284%2B-4%29%2Ax%2B%28-6-5%29%2Ay=10-6 ALIGN=MIDDLE  ALT=\"cross%284%2B-4%29%2Ax%2B%28-6-5%29%2Ay=10-6\"   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=4 ALIGN=MIDDLE  ALT=\"-11%2Ay=4\"   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=4%2F-11 ALIGN=MIDDLE  ALT=\"y=4%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=-4%2F11 ALIGN=MIDDLE  ALT=\"y=-4%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=2%2Ax-3%2Ay=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%2Ay=5\"   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=2%2Ax-3%28-4%2F11%29=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%28-4%2F11%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4%2F11 ALIGN=MIDDLE  ALT=\"y=-4%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=2%2Ax%2B12%2F11=5 ALIGN=MIDDLE  ALT=\"2%2Ax%2B12%2F11=5\"   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=2%2Ax%2B12%2F11=5 ALIGN=MIDDLE  ALT=\"2%2Ax%2B12%2F11=5\"   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=2%2Ax=5-12%2F11 ALIGN=MIDDLE  ALT=\"2%2Ax=5-12%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=12%2F11 ALIGN=MIDDLE  ALT=\"12%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=2%2Ax=55%2F11-12%2F11 ALIGN=MIDDLE  ALT=\"2%2Ax=55%2F11-12%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make 5 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=2%2Ax=43%2F11 ALIGN=MIDDLE  ALT=\"2%2Ax=43%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%2F2%29%282%29%29%2Ax=%2843%2F11%29%281%2F2%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F2%29%282%29%29%2Ax=%2843%2F11%29%281%2F2%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%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=43%2F22 ALIGN=MIDDLE  ALT=\"x=43%2F22\"   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=43%2F22 ALIGN=MIDDLE  ALT=\"x=43%2F22\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4%2F11 ALIGN=MIDDLE  ALT=\"y=-4%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=43%2F22 ALIGN=MIDDLE  ALT=\"43%2F22\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F11 ALIGN=MIDDLE  ALT=\"-4%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=2%2Ax-3%2Ay=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%2Ay=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax%2B5%2Ay=6 ALIGN=MIDDLE  ALT=\"4%2Ax%2B5%2Ay=6\"   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+-11.9545454545455%2C+11.9545454545455%2C+-10.3636363636364%2C+10.3636363636364%2C%0D%0A++++graph%28+500%2C+600%2C+-11.9545454545455%2C+11.9545454545455%2C+-10.3636363636364%2C+10.3636363636364%2C+%285-2%2Ax%29%2F-3%2C+%286-4%2Ax%29%2F5+%29%2C%0D%0A++++blue%28++circle%28+1.95454545454545%2C+-0.363636363636364%2C+0.159393939393939+%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=2%2Ax-3%2Ay=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%2Ay=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax%2B5%2Ay=6 ALIGN=MIDDLE  ALT=\"4%2Ax%2B5%2Ay=6\"   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=43%2F22 ALIGN=MIDDLE  ALT=\"43%2F22\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4%2F11 ALIGN=MIDDLE  ALT=\"-4%2F11\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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