document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=969176>Question 969176</A>: <I><SPAN STYLE=\"word-break: break-all;\"> what is  -x-10y=15<BR>\n" );
document.write( "         -x+4y=13  in elimination<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #592110 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=592110\">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=-1%2Ax-10%2Ay=15 ALIGN=MIDDLE  ALT=\"-1%2Ax-10%2Ay=15\"   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%2B4%2Ay=13 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B4%2Ay=13\"   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 -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 -1 and -1 is 1, we need to multiply both sides of the top equation by -1 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=-1%2A%28-1%2Ax-10%2Ay%29=%2815%29%2A-1 ALIGN=MIDDLE  ALT=\"-1%2A%28-1%2Ax-10%2Ay%29=%2815%29%2A-1\"   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=1%2A%28-1%2Ax%2B4%2Ay%29=%2813%29%2A1 ALIGN=MIDDLE  ALT=\"1%2A%28-1%2Ax%2B4%2Ay%29=%2813%29%2A1\"   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=1%2Ax%2B10%2Ay=-15 ALIGN=MIDDLE  ALT=\"1%2Ax%2B10%2Ay=-15\"   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%2B4%2Ay=13 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B4%2Ay=13\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Notice how 1 and -1 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2B-1=0 ALIGN=MIDDLE  ALT=\"1%2B-1=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=%281%2Ax-1%2Ax%29%2B%2810%2Ay%2B4%2Ay%29=-15%2B13 ALIGN=MIDDLE  ALT=\"%281%2Ax-1%2Ax%29%2B%2810%2Ay%2B4%2Ay%29=-15%2B13\"   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=%281-1%29%2Ax%2B%2810%2B4%29y=-15%2B13 ALIGN=MIDDLE  ALT=\"%281-1%29%2Ax%2B%2810%2B4%29y=-15%2B13\"   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%281%2B-1%29%2Ax%2B%2810%2B4%29%2Ay=-15%2B13 ALIGN=MIDDLE  ALT=\"cross%281%2B-1%29%2Ax%2B%2810%2B4%29%2Ay=-15%2B13\"   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=14%2Ay=-2 ALIGN=MIDDLE  ALT=\"14%2Ay=-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=y=-2%2F14 ALIGN=MIDDLE  ALT=\"y=-2%2F14\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14 ALIGN=MIDDLE  ALT=\"14\"   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=-1%2F7 ALIGN=MIDDLE  ALT=\"y=-1%2F7\"   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-10%2Ay=15 ALIGN=MIDDLE  ALT=\"-1%2Ax-10%2Ay=15\"   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-10%28-1%2F7%29=15 ALIGN=MIDDLE  ALT=\"-1%2Ax-10%28-1%2F7%29=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%2F7 ALIGN=MIDDLE  ALT=\"y=-1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B10%2F7=15 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B10%2F7=15\"   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%2B10%2F7=15 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B10%2F7=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax=15-10%2F7 ALIGN=MIDDLE  ALT=\"-1%2Ax=15-10%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F7 ALIGN=MIDDLE  ALT=\"10%2F7\"   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=105%2F7-10%2F7 ALIGN=MIDDLE  ALT=\"-1%2Ax=105%2F7-10%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make 15 into a fraction with a denominator of 7<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax=95%2F7 ALIGN=MIDDLE  ALT=\"-1%2Ax=95%2F7\"   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%2F-1%29%28-1%29%29%2Ax=%2895%2F7%29%281%2F-1%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F-1%29%28-1%29%29%2Ax=%2895%2F7%29%281%2F-1%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%2F-1 ALIGN=MIDDLE  ALT=\"1%2F-1\"   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( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-95%2F7 ALIGN=MIDDLE  ALT=\"x=-95%2F7\"   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=-95%2F7 ALIGN=MIDDLE  ALT=\"x=-95%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1%2F7 ALIGN=MIDDLE  ALT=\"y=-1%2F7\"   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=-95%2F7 ALIGN=MIDDLE  ALT=\"-95%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F7 ALIGN=MIDDLE  ALT=\"-1%2F7\"   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-10%2Ay=15 ALIGN=MIDDLE  ALT=\"-1%2Ax-10%2Ay=15\"   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%2B4%2Ay=13 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B4%2Ay=13\"   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+-23.5714285714286%2C+23.5714285714286%2C+-10.1428571428571%2C+10.1428571428571%2C%0D%0A++++graph%28+500%2C+600%2C+-23.5714285714286%2C+23.5714285714286%2C+-10.1428571428571%2C+10.1428571428571%2C+%2815--1%2Ax%29%2F-10%2C+%2813--1%2Ax%29%2F4+%29%2C%0D%0A++++blue%28++circle%28+-13.5714285714286%2C+-0.142857142857143%2C+0.314285714285714+%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-10%2Ay=15 ALIGN=MIDDLE  ALT=\"-1%2Ax-10%2Ay=15\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B4%2Ay=13 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B4%2Ay=13\"   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=-95%2F7 ALIGN=MIDDLE  ALT=\"-95%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F7 ALIGN=MIDDLE  ALT=\"-1%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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