document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111294>Question 111294</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Are the following lines parallel, perpendicular, or neither?<BR>\n" );
document.write( "   L1 with equation x – 6y = 12<BR>\n" );
document.write( "   L2 with equation 6x + y = 6<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 #81167 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=81167\">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=solve_by_graphing.solver>Solve the System of Equations by Graphing</A></TD></TR>\n" );
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document.write( "  Start with the given system of equations:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x-6y=12 ALIGN=MIDDLE  ALT=\"1x-6y=12\"   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=6x%2By=6 ALIGN=MIDDLE  ALT=\"6x%2By=6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  In order to graph these equations, we need to solve for y for each equation.<BR>\n" );
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document.write( "  So let's solve for y on the first equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x-6y=12 ALIGN=MIDDLE  ALT=\"1x-6y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6y=12-x ALIGN=MIDDLE  ALT=\"-6y=12-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x ALIGN=MIDDLE  ALT=\"+x\"   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=-6y=-x%2B12 ALIGN=MIDDLE  ALT=\"-6y=-x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-x%2B12%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"y=%28-x%2B12%29%2F%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-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=y=%28-1%2F-6%29x%2B%2812%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F-6%29x%2B%2812%29%2F%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F6%29x-2 ALIGN=MIDDLE  ALT=\"y=%281%2F6%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now lets graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F6%29x-2 ALIGN=MIDDLE  ALT=\"y=%281%2F6%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (note: 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=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F6%29x-2%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F6%29x-2%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F6%29x-2 ALIGN=MIDDLE  ALT=\"y=%281%2F6%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So let's solve for y on the second equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x%2By=6 ALIGN=MIDDLE  ALT=\"6x%2By=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y=6-6x ALIGN=MIDDLE  ALT=\"1y=6-6x\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6+x ALIGN=MIDDLE  ALT=\"6+x\"   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=1y=-6x%2B6 ALIGN=MIDDLE  ALT=\"1y=-6x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-6x%2B6%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28-6x%2B6%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   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=%28-6%2F1%29x%2B%286%29%2F%281%29 ALIGN=MIDDLE  ALT=\"y=%28-6%2F1%29x%2B%286%29%2F%281%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6x%2B6 ALIGN=MIDDLE  ALT=\"y=-6x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  Now lets add the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6x%2B6 ALIGN=MIDDLE  ALT=\"y=-6x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > to our first plot to get:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F6%29x-2%2C-6x%2B6%29+ ALIGN=MIDDLE  ALT=\"+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C+%281%2F6%29x-2%2C-6x%2B6%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F6%29x-2 ALIGN=MIDDLE  ALT=\"y=%281%2F6%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >(red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-6x%2B6 ALIGN=MIDDLE  ALT=\"y=-6x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" >(green)<BR>\n" );
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document.write( "  From the graph, we can see that the two lines intersect at the point (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=48%2F37 ALIGN=MIDDLE  ALT=\"48%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-66%2F37 ALIGN=MIDDLE  ALT=\"-66%2F37\"   DISABLED_style= \"max-width:90%; height:auto;\" >) (note: you might have to adjust the window to see the intersection)</TD></TR>\n" );
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document.write( "Since you have  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m1+=+1%2F6 ALIGN=MIDDLE  ALT=\"m1+=+1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >  and    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m2+=+-6 ALIGN=MIDDLE  ALT=\"m2+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and perpendicular lines have slope that are negative reciprocals  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m1+=+-1%2Fm2 ALIGN=MIDDLE  ALT=\"m1+=+-1%2Fm2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6+=+-1%2F-6 ALIGN=MIDDLE  ALT=\"1%2F6+=+-1%2F-6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F6+=+1%2F6 ALIGN=MIDDLE  ALT=\"1%2F6+=+1%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" >…so, your lines are perpendicular<BR>\n" );
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