document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180313>Question 1180313</A>: <I><SPAN STYLE=\"word-break: break-all;\">              Solving for Two Variables using Elimination\r<BR>\n" );
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document.write( "        4x - 3y = -14      and     -x + 3y = -11 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #809940 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=809940\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <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=4%2Ax-3%2Ay=-14 ALIGN=MIDDLE  ALT=\"4%2Ax-3%2Ay=-14\"   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%2B3%2Ay=-11 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=-11\"   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 4 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 4 and -1 is -4, we need to multiply both sides of the top equation by -1 and multiply both sides of the bottom equation by -4 like this:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A%284%2Ax-3%2Ay%29=%28-14%29%2A-1 ALIGN=MIDDLE  ALT=\"-1%2A%284%2Ax-3%2Ay%29=%28-14%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=-4%2A%28-1%2Ax%2B3%2Ay%29=%28-11%29%2A-4 ALIGN=MIDDLE  ALT=\"-4%2A%28-1%2Ax%2B3%2Ay%29=%28-11%29%2A-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -4<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%2B3%2Ay=14 ALIGN=MIDDLE  ALT=\"-4%2Ax%2B3%2Ay=14\"   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-12%2Ay=44 ALIGN=MIDDLE  ALT=\"4%2Ax-12%2Ay=44\"   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%2B4=0 ALIGN=MIDDLE  ALT=\"-4%2B4=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=%28-4%2Ax%2B4%2Ax%29%2B%283%2Ay-12%2Ay%29=14%2B44 ALIGN=MIDDLE  ALT=\"%28-4%2Ax%2B4%2Ax%29%2B%283%2Ay-12%2Ay%29=14%2B44\"   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=%28-4%2B4%29%2Ax%2B%283-12%29y=14%2B44 ALIGN=MIDDLE  ALT=\"%28-4%2B4%29%2Ax%2B%283-12%29y=14%2B44\"   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%28-4%2B4%29%2Ax%2B%283-12%29%2Ay=14%2B44 ALIGN=MIDDLE  ALT=\"cross%28-4%2B4%29%2Ax%2B%283-12%29%2Ay=14%2B44\"   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=-9%2Ay=58 ALIGN=MIDDLE  ALT=\"-9%2Ay=58\"   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=58%2F-9 ALIGN=MIDDLE  ALT=\"y=58%2F-9\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9 ALIGN=MIDDLE  ALT=\"-9\"   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=-58%2F9 ALIGN=MIDDLE  ALT=\"y=-58%2F9\"   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=4%2Ax-3%2Ay=-14 ALIGN=MIDDLE  ALT=\"4%2Ax-3%2Ay=-14\"   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=4%2Ax-3%28-58%2F9%29=-14 ALIGN=MIDDLE  ALT=\"4%2Ax-3%28-58%2F9%29=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-58%2F9 ALIGN=MIDDLE  ALT=\"y=-58%2F9\"   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=4%2Ax%2B174%2F9=-14 ALIGN=MIDDLE  ALT=\"4%2Ax%2B174%2F9=-14\"   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=4%2Ax%2B58%2F3=-14 ALIGN=MIDDLE  ALT=\"4%2Ax%2B58%2F3=-14\"   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=4%2Ax=-14-58%2F3 ALIGN=MIDDLE  ALT=\"4%2Ax=-14-58%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=58%2F3 ALIGN=MIDDLE  ALT=\"58%2F3\"   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=4%2Ax=-42%2F3-58%2F3 ALIGN=MIDDLE  ALT=\"4%2Ax=-42%2F3-58%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Make -14 into a fraction with a denominator of 3<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2Ax=-100%2F3 ALIGN=MIDDLE  ALT=\"4%2Ax=-100%2F3\"   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%2F4%29%284%29%29%2Ax=%28-100%2F3%29%281%2F4%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F4%29%284%29%29%2Ax=%28-100%2F3%29%281%2F4%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%2F4 ALIGN=MIDDLE  ALT=\"1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   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=-25%2F3 ALIGN=MIDDLE  ALT=\"x=-25%2F3\"   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=-25%2F3 ALIGN=MIDDLE  ALT=\"x=-25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-58%2F9 ALIGN=MIDDLE  ALT=\"y=-58%2F9\"   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=-25%2F3 ALIGN=MIDDLE  ALT=\"-25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-58%2F9 ALIGN=MIDDLE  ALT=\"-58%2F9\"   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=4%2Ax-3%2Ay=-14 ALIGN=MIDDLE  ALT=\"4%2Ax-3%2Ay=-14\"   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%2B3%2Ay=-11 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=-11\"   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+-18.3333333333333%2C+18.3333333333333%2C+-16.4444444444444%2C+16.4444444444444%2C%0D%0A++++graph%28+500%2C+600%2C+-18.3333333333333%2C+18.3333333333333%2C+-16.4444444444444%2C+16.4444444444444%2C+%28-14-4%2Ax%29%2F-3%2C+%28-11--1%2Ax%29%2F3+%29%2C%0D%0A++++blue%28++circle%28+-8.33333333333333%2C+-6.44444444444444%2C+0.244444444444444+%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=4%2Ax-3%2Ay=-14 ALIGN=MIDDLE  ALT=\"4%2Ax-3%2Ay=-14\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2Ax%2B3%2Ay=-11 ALIGN=MIDDLE  ALT=\"-1%2Ax%2B3%2Ay=-11\"   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=-25%2F3 ALIGN=MIDDLE  ALT=\"-25%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-58%2F9 ALIGN=MIDDLE  ALT=\"-58%2F9\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\n" );
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