document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=901185>Question 901185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to solve the systems of equations by elimination and I do not understand how to solve by substitution. Can you help? <BR>\n" );
document.write( "y+5x=-3<BR>\n" );
document.write( "3y-2x=8<BR>\n" );
document.write( "Thank you so much! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #546522 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=546522\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B5x=-3 ALIGN=MIDDLE  ALT=\"y%2B5x=-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=5x%2By=-3 ALIGN=MIDDLE  ALT=\"5x%2By=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2x%2B3y=8 ALIGN=MIDDLE  ALT=\"-2x%2B3y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" >\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( "  <BR>\n" );
document.write( "  Lets start with the given system of linear equations<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B1%2Ay=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%2Ay=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax%2B3%2Ay=8 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B3%2Ay=8\"   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 5 and -2 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 5 and -2 is -10, we need to multiply both sides of the top equation by -2 and multiply both sides of the bottom equation by -5 like this:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%285%2Ax%2B1%2Ay%29=%28-3%29%2A-2 ALIGN=MIDDLE  ALT=\"-2%2A%285%2Ax%2B1%2Ay%29=%28-3%29%2A-2\"   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=-5%2A%28-2%2Ax%2B3%2Ay%29=%288%29%2A-5 ALIGN=MIDDLE  ALT=\"-5%2A%28-2%2Ax%2B3%2Ay%29=%288%29%2A-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the bottom equation (both sides) by -5<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
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-2%2Ay=6 ALIGN=MIDDLE  ALT=\"-10%2Ax-2%2Ay=6\"   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-15%2Ay=-40 ALIGN=MIDDLE  ALT=\"10%2Ax-15%2Ay=-40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Notice how -10 and 10 add to zero (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-10%2B10=0 ALIGN=MIDDLE  ALT=\"-10%2B10=0\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <BR>\n" );
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=%28-10%2Ax%2B10%2Ax%29-2%2Ay-15%2Ay%29=6-40 ALIGN=MIDDLE  ALT=\"%28-10%2Ax%2B10%2Ax%29-2%2Ay-15%2Ay%29=6-40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-10%2B10%29%2Ax-2-15%29y=6-40 ALIGN=MIDDLE  ALT=\"%28-10%2B10%29%2Ax-2-15%29y=6-40\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28-10%2B10%29%2Ax%2B%28-2-15%29%2Ay=6-40 ALIGN=MIDDLE  ALT=\"cross%28-10%2B10%29%2Ax%2B%28-2-15%29%2Ay=6-40\"   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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17%2Ay=-34 ALIGN=MIDDLE  ALT=\"-17%2Ay=-34\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-34%2F-17 ALIGN=MIDDLE  ALT=\"y=-34%2F-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-17 ALIGN=MIDDLE  ALT=\"-17\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for y<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  <BR>\n" );
document.write( "  Now plug this answer into the top equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B1%2Ay=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%2Ay=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > to solve for x<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B1%282%29=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%282%29=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B2=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B2=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax=-3-2 ALIGN=MIDDLE  ALT=\"5%2Ax=-3-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax=-5 ALIGN=MIDDLE  ALT=\"5%2Ax=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine the terms on the right side<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cross%28%281%2F5%29%285%29%29%2Ax=%28-5%29%281%2F5%29 ALIGN=MIDDLE  ALT=\"cross%28%281%2F5%29%285%29%29%2Ax=%28-5%29%281%2F5%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%2F5 ALIGN=MIDDLE  ALT=\"1%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This will cancel out <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left side.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2 ALIGN=MIDDLE  ALT=\"y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  which also looks like<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
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" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax%2B1%2Ay=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%2Ay=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax%2B3%2Ay=8 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B3%2Ay=8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  we get <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++drawing%28+500%2C+600%2C+-11%2C+11%2C+-12%2C+12%2C%0D%0A++++graph%28+500%2C+600%2C+-11%2C+11%2C+-12%2C+12%2C+%28-3-5%2Ax%29%2F1%2C+%288--2%2Ax%29%2F3+%29%2C%0D%0A++++blue%28++circle%28+-1%2C+2%2C+0.146666666666667+%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=5%2Ax%2B1%2Ay=-3 ALIGN=MIDDLE  ALT=\"5%2Ax%2B1%2Ay=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2Ax%2B3%2Ay=8 ALIGN=MIDDLE  ALT=\"-2%2Ax%2B3%2Ay=8\"   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( "  <BR>\n" );
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 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >). This verifies our answer.</TD></TR>\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=linear_substitution.solver>SOLVE linear system by SUBSTITUTION</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Solve:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+system%28+%0D%0A++++5%5Cx+%2B+1%5Cy+=+-3%2C%0D%0A++++-2%5Cx+%2B+3%5Cy+=+8+%29%0D%0A++ ALIGN=MIDDLE  ALT=\"+system%28+%0D%0A++++5%5Cx+%2B+1%5Cy+=+-3%2C%0D%0A++++-2%5Cx+%2B+3%5Cy+=+8+%29%0D%0A++\"   DISABLED_style= \"max-width:90%; height:auto;\" >We'll use substitution. After moving 1*y to the right, we get:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2Ax+=+-3+-+1%2Ay ALIGN=MIDDLE  ALT=\"5%2Ax+=+-3+-+1%2Ay\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F5+-+1%2Ay%2F5 ALIGN=MIDDLE  ALT=\"x+=+-3%2F5+-+1%2Ay%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substitute that<BR>\n" );
document.write( "  into another equation:<BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2%2A%28-3%2F5+-+1%2Ay%2F5%29+%2B+3%5Cy+=+8 ALIGN=MIDDLE  ALT=\"-2%2A%28-3%2F5+-+1%2Ay%2F5%29+%2B+3%5Cy+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and simplify: So, we know that y=2. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-3%2F5+-+1%2Ay%2F5 ALIGN=MIDDLE  ALT=\"x+=+-3%2F5+-+1%2Ay%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, x=-1.<BR>\n" );
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document.write( "  Answer: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x=-1%2C+y=2+%29 ALIGN=MIDDLE  ALT=\"system%28+x=-1%2C+y=2+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
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