document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=694961>Question 694961</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve by he linear combination method \r<BR>\n" );
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document.write( "2x+3y=8.1\r<BR>\n" );
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document.write( "5x-2y=-1.6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #428273 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=428273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The Linear Combination Method or The Addition Method or the Elimination Method\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%2B3%2Ay=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=8.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax-2%2Ay=-1.6 ALIGN=MIDDLE  ALT=\"5%2Ax-2%2Ay=-1.6\"   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 5 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 5 is 10, we need to multiply both sides of the top equation by 5 and multiply both sides of the bottom equation by -2 like this:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2A%282%2Ax%2B3%2Ay%29=%288.1%29%2A5 ALIGN=MIDDLE  ALT=\"5%2A%282%2Ax%2B3%2Ay%29=%288.1%29%2A5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the top equation (both sides) by 5<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%285%2Ax-2%2Ay%29=%28-1.6%29%2A-2 ALIGN=MIDDLE  ALT=\"-2%2A%285%2Ax-2%2Ay%29=%28-1.6%29%2A-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -2<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=10%2Ax%2B15%2Ay=40.5 ALIGN=MIDDLE  ALT=\"10%2Ax%2B15%2Ay=40.5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2Ax%2B4%2Ay=3.2 ALIGN=MIDDLE  ALT=\"-10%2Ax%2B4%2Ay=3.2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 10 and -10 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2B-10=0 ALIGN=MIDDLE  ALT=\"10%2B-10=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=%2810%2Ax-10%2Ax%29%2B%2815%2Ay%2B4%2Ay%29=40.5%2B3.2 ALIGN=MIDDLE  ALT=\"%2810%2Ax-10%2Ax%29%2B%2815%2Ay%2B4%2Ay%29=40.5%2B3.2\"   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=%2810-10%29%2Ax%2B%2815%2B4%29y=40.5%2B3.2 ALIGN=MIDDLE  ALT=\"%2810-10%29%2Ax%2B%2815%2B4%29y=40.5%2B3.2\"   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%2810%2B-10%29%2Ax%2B%2815%2B4%29%2Ay=40.5%2B3.2 ALIGN=MIDDLE  ALT=\"cross%2810%2B-10%29%2Ax%2B%2815%2B4%29%2Ay=40.5%2B3.2\"   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=19%2Ay=43.7 ALIGN=MIDDLE  ALT=\"19%2Ay=43.7\"   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=43.7%2F19 ALIGN=MIDDLE  ALT=\"y=43.7%2F19\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19 ALIGN=MIDDLE  ALT=\"19\"   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=33.6153846153829%2F14.6153846153838 ALIGN=MIDDLE  ALT=\"y=33.6153846153829%2F14.6153846153838\"   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%2B3%2Ay=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=8.1\"   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%2B3%2833.6153846153829%2F14.6153846153838%29=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2833.6153846153829%2F14.6153846153838%29=8.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=33.6153846153829%2F14.6153846153838 ALIGN=MIDDLE  ALT=\"y=33.6153846153829%2F14.6153846153838\"   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%2B100.846153846149%2F14.6153846153838=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B100.846153846149%2F14.6153846153838=8.1\"   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%2B18.217777133666%2F2.64025755560377=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B18.217777133666%2F2.64025755560377=8.1\"   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=8.1-18.217777133666%2F2.64025755560377 ALIGN=MIDDLE  ALT=\"2%2Ax=8.1-18.217777133666%2F2.64025755560377\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=18.217777133666%2F2.64025755560377 ALIGN=MIDDLE  ALT=\"18.217777133666%2F2.64025755560377\"   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=21.3860862003906%2F2.64025755560377-18.217777133666%2F2.64025755560377 ALIGN=MIDDLE  ALT=\"2%2Ax=21.3860862003906%2F2.64025755560377-18.217777133666%2F2.64025755560377\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make 8.1 into a fraction with a denominator of 2.64025755560377<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax=3.16830906672453%2F2.64025755560377 ALIGN=MIDDLE  ALT=\"2%2Ax=3.16830906672453%2F2.64025755560377\"   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=%283.16830906672453%2F2.64025755560377%29%281%2F2%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F2%29%282%29%29%2Ax=%283.16830906672453%2F2.64025755560377%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=1.8310708654933%2F3.05178477582217 ALIGN=MIDDLE  ALT=\"x=1.8310708654933%2F3.05178477582217\"   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=1.8310708654933%2F3.05178477582217 ALIGN=MIDDLE  ALT=\"x=1.8310708654933%2F3.05178477582217\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=33.6153846153829%2F14.6153846153838 ALIGN=MIDDLE  ALT=\"y=33.6153846153829%2F14.6153846153838\"   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=1.8310708654933%2F3.05178477582217 ALIGN=MIDDLE  ALT=\"1.8310708654933%2F3.05178477582217\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33.6153846153829%2F14.6153846153838 ALIGN=MIDDLE  ALT=\"33.6153846153829%2F14.6153846153838\"   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%2B3%2Ay=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=8.1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax-2%2Ay=-1.6 ALIGN=MIDDLE  ALT=\"5%2Ax-2%2Ay=-1.6\"   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+-10.6%2C+10.6%2C+-12.3%2C+12.3%2C%0D%0A++++graph%28+500%2C+600%2C+-10.6%2C+10.6%2C+-12.3%2C+12.3%2C+%288.1-2%2Ax%29%2F3%2C+%28-1.6-5%2Ax%29%2F-2+%29%2C%0D%0A++++blue%28++circle%28+0.6%2C+2.3%2C+0.141333333333333+%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%2B3%2Ay=8.1 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=8.1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax-2%2Ay=-1.6 ALIGN=MIDDLE  ALT=\"5%2Ax-2%2Ay=-1.6\"   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=1.8310708654933%2F3.05178477582217 ALIGN=MIDDLE  ALT=\"1.8310708654933%2F3.05178477582217\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33.6153846153829%2F14.6153846153838 ALIGN=MIDDLE  ALT=\"33.6153846153829%2F14.6153846153838\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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