document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=237558>Question 237558</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine whether the graphs of the equations are parallel lines, perpendicular lines, or neither.\r<BR>\n" );
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document.write( "3x - 4y = 8<BR>\n" );
document.write( "8x + 6y = 8  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #174689 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=174689\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-4y=8 ALIGN=MIDDLE  ALT=\"3x-4y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4y=8-3x ALIGN=MIDDLE  ALT=\"-4y=8-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4y=-3x%2B8 ALIGN=MIDDLE  ALT=\"-4y=-3x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3x%2B8%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y=%28-3x%2B8%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28-3%29%2F%28-4%29%29x%2B%288%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y=%28%28-3%29%2F%28-4%29%29x%2B%288%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "So we can see that the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2F4 ALIGN=MIDDLE  ALT=\"m=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a y-intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-2 ALIGN=MIDDLE  ALT=\"b=-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B6y=8 ALIGN=MIDDLE  ALT=\"8x%2B6y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Now move onto the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6y=8-8x ALIGN=MIDDLE  ALT=\"6y=8-8x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x ALIGN=MIDDLE  ALT=\"8x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6y=-8x%2B8 ALIGN=MIDDLE  ALT=\"6y=-8x%2B8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-8x%2B8%29%2F%286%29 ALIGN=MIDDLE  ALT=\"y=%28-8x%2B8%29%2F%286%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;\" > to isolate y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28-8%29%2F%286%29%29x%2B%288%29%2F%286%29 ALIGN=MIDDLE  ALT=\"y=%28%28-8%29%2F%286%29%29x%2B%288%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x%2B4%2F3 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x%2B4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "So we can see that the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x%2B4%2F3 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x%2B4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > has a slope <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-4%2F3 ALIGN=MIDDLE  ALT=\"m=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and a y-intercept <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4%2F3 ALIGN=MIDDLE  ALT=\"b=4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the slope of the first line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2F4 ALIGN=MIDDLE  ALT=\"m=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the slope of the second line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-4%2F3 ALIGN=MIDDLE  ALT=\"m=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Notice how the slope of the second line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-4%2F3 ALIGN=MIDDLE  ALT=\"m=-4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is simply the negative reciprocal of the slope of the first line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2F4 ALIGN=MIDDLE  ALT=\"m=3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "In other words, if you flip the fraction of the second slope and change its sign, you'll get the first slope. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F4%29x-2 ALIGN=MIDDLE  ALT=\"y=%283%2F4%29x-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%284%2F3%29x%2B4%2F3 ALIGN=MIDDLE  ALT=\"y=-%284%2F3%29x%2B4%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > are perpendicular lines. Consequently, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-4y=8 ALIGN=MIDDLE  ALT=\"3x-4y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B6y=8 ALIGN=MIDDLE  ALT=\"8x%2B6y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > are also perpendicular lines. </SPAN>\n" );
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