document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=260757>Question 260757</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Perpendicular Lines: Write the equation of the line through (4,5) and perpendicular to the line with equation 3x + 2y = -1.\r<BR>\n" );
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document.write( "Please explain how the equation came out to y = 2/3 x + 7 as shown in my example answer key.\r<BR>\n" );
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document.write( "Thank you,<BR>\n" );
document.write( "Melissa </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #192099 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=192099\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First, we must solve for y to get the equation in slope intercept form.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x+%2B+2y+=+-1 ALIGN=MIDDLE  ALT=\"3x+%2B+2y+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y+=+-1-3x ALIGN=MIDDLE  ALT=\"2y+=+-1-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3x from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y+=+-3x-1 ALIGN=MIDDLE  ALT=\"2y+=+-3x-1\"   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-1%29%2F2 ALIGN=MIDDLE  ALT=\"y+=+%28-3x-1%29%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 2 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+=+%28-3%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction and simplify.\r<BR>\n" );
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document.write( "So the equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > is now in slope intercept form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=mx%2Bb ALIGN=MIDDLE  ALT=\"y=mx%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > where the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-3%2F2 ALIGN=MIDDLE  ALT=\"m=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the y intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-1%2F2 ALIGN=MIDDLE  ALT=\"b=-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Recall that perpendicular lines have slopes that are negative reciprocals of one another. So because the slope of the given line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, the perpendicular slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just flip the fraction and change the sign).\r<BR>\n" );
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document.write( "So we now know that the slope of the perpendicular line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2%2F3 ALIGN=MIDDLE  ALT=\"m=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and it goes through the point (4,5).\r<BR>\n" );
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document.write( "Now let's use the point-slope formula to find the perpendicular equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=m%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=m%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the point-slope formula\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-y%5B1%5D=%282%2F3%29%28x-x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y-y%5B1%5D=%282%2F3%29%28x-x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2%2F3 ALIGN=MIDDLE  ALT=\"m=2%2F3\"   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=y-5=%282%2F3%29%28x-4%29 ALIGN=MIDDLE  ALT=\"y-5=%282%2F3%29%28x-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=4 ALIGN=MIDDLE  ALT=\"x%5B1%5D=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D=5 ALIGN=MIDDLE  ALT=\"y%5B1%5D=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This is given from (4,5). Our goal now is to solve for y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=%282%2F3%29x-%282%2F3%294 ALIGN=MIDDLE  ALT=\"y-5=%282%2F3%29x-%282%2F3%294\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y-5=%282%2F3%29x-8%2F3 ALIGN=MIDDLE  ALT=\"y-5=%282%2F3%29x-8%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-8%2F3%2B5 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-8%2F3%2B5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add 5 to both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%2B7%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%2B7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms.\r<BR>\n" );
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document.write( "So the equation of the perpendicular line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%2B7%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%2B7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (make sure that there isn't a typo). Notice that if we graph the two lines, we get\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%28-1-3x%29%2F2%2C%282%2F3%29x%2B7%2F3%29%2C%0D%0Acircle%284%2C5%2C0.05%29%2Ccircle%284%2C5%2C0.08%29%2Ccircle%284%2C5%2C0.10%29%2Ccircle%284%2C5%2C0.12%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3%2F2%29x-1%2F2 ALIGN=MIDDLE  ALT=\"y=%28-3%2F2%29x-1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x%2B7%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x%2B7%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green) going through the point (4,5). Notice how the red line is perpendicular to the green line. </SPAN>\n" );
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