document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=104002>Question 104002</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation, in standard form, with all integer coefficients, of the line perpendicular to x + 3y = 12 and passing through (9, -5).\r<BR>\n" );
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document.write( "I normally do not ask for help, but I am lost on this problem. If anyone can help, I would greatly appreciate it. <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 #75680 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=75680\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> \r<BR>\n" );
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document.write( "First convert the standard equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B3y=12 ALIGN=MIDDLE  ALT=\"x%2B3y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > into slope intercept form\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=converting_linear_equations.solver>Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Convert from standard form (Ax+By = C) to slope-intercept form (y = mx+b)<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2B3y=12 ALIGN=MIDDLE  ALT=\"1x%2B3y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2B3y-1x=12-1x ALIGN=MIDDLE  ALT=\"1x%2B3y-1x=12-1x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 1x from both sides<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3y=-1x%2B12 ALIGN=MIDDLE  ALT=\"3y=-1x%2B12\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y%29%2F%283%29=%28-1x%2B12%29%2F%283%29 ALIGN=MIDDLE  ALT=\"%283y%29%2F%283%29=%28-1x%2B12%29%2F%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by 3 to isolate y<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1x%29%2F%283%29%2B%2812%29%2F%283%29 ALIGN=MIDDLE  ALT=\"y+=+%28-1x%29%2F%283%29%2B%2812%29%2F%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction on the right hand side<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F3%29x%2B4 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F3%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce and simplify<br><br><br>The original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1x%2B3y=12 ALIGN=MIDDLE  ALT=\"1x%2B3y=12\"   DISABLED_style= \"max-width:90%; height:auto;\" > (standard form) is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F3%29x%2B4 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F3%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (slope-intercept form)<br><br><br>The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F3%29x%2B4 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F3%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the 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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-1%2F3 ALIGN=MIDDLE  ALT=\"m=-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the slope and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=4 ALIGN=MIDDLE  ALT=\"b=4\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the y intercept.<br><br><br></TD></TR>\n" );
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document.write( "Now let's find the equation of the line that is perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29x%2B4 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29x%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > which goes through (9,-5)\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=equation_parallel_or_perpendicular.solver>Finding the Equation of a Line Parallel or Perpendicular to a Given Line</A></TD></TR>\n" );
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document.write( "  Remember, any two perpendicular lines are negative reciprocals of each other. So if you're given the slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F3 ALIGN=MIDDLE  ALT=\"-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, you can find the perpendicular slope by this formula:<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5Bp%5D=-1%2Fm ALIGN=MIDDLE  ALT=\"m%5Bp%5D=-1%2Fm\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5Bp%5D ALIGN=MIDDLE  ALT=\"m%5Bp%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the perpendicular slope<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5Bp%5D=-1%2F%28-1%2F3%29 ALIGN=MIDDLE  ALT=\"m%5Bp%5D=-1%2F%28-1%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > So plug in the given slope to find the perpendicular slope<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5Bp%5D=%28-1%2F1%29%283%2F-1%29 ALIGN=MIDDLE  ALT=\"m%5Bp%5D=%28-1%2F1%29%283%2F-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > When you divide fractions, you multiply the first fraction (which is really <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F1 ALIGN=MIDDLE  ALT=\"1%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) by the reciprocal of the second<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m%5Bp%5D=3%2F1 ALIGN=MIDDLE  ALT=\"m%5Bp%5D=3%2F1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply the fractions.<BR>\n" );
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document.write( "  So the perpendicular slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So now we know the slope of the unknown line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (its the negative reciprocal of  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F3 ALIGN=MIDDLE  ALT=\"-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > from the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%2Ax%2B4 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%2Ax%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\n" );
document.write( "Also since the unknown line goes through (9,-5), we can find the equation by plugging in this info into the point-slope formula<BR>\n" );
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document.write( "  Point-Slope Formula:<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;\" > where m is the slope and (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D ALIGN=MIDDLE  ALT=\"y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is the given point<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B5=3%2A%28x-9%29 ALIGN=MIDDLE  ALT=\"y%2B5=3%2A%28x-9%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3 ALIGN=MIDDLE  ALT=\"m=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D=9 ALIGN=MIDDLE  ALT=\"x%5B1%5D=9\"   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;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%2B5=3%2Ax-%283%29%289%29 ALIGN=MIDDLE  ALT=\"y%2B5=3%2Ax-%283%29%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"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=y%2B5=3%2Ax-27 ALIGN=MIDDLE  ALT=\"y%2B5=3%2Ax-27\"   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=y=3%2Ax-27-5 ALIGN=MIDDLE  ALT=\"y=3%2Ax-27-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides to isolate y<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax-32 ALIGN=MIDDLE  ALT=\"y=3%2Ax-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Combine like terms<BR>\n" );
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document.write( "   So the equation of the line that is perpendicular to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%2Ax%2B4 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%2Ax%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and goes through (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax-32 ALIGN=MIDDLE  ALT=\"y=3%2Ax-32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  So here are the graphs of the equations <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%2Ax%2B4 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%2Ax%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax-32 ALIGN=MIDDLE  ALT=\"y=3%2Ax-32\"   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=drawing%28+500%2C+500%2C+-5%2C14%2C-10%2C9%2C+%0D%0A+++++++++++grid%281%29%2C%0D%0A+++++++++++graph%28+500%2C+500%2C+-5%2C14%2C-10%2C9%2C+-0.333333333333333%2Ax%2B4%2C+3%2Ax%2B-32%29%2C%0D%0A+++++++++++circle%289%2C-5%2C0.05%29%2C%0D%0A+++++++++++circle%289%2C-5%2C0.08%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > graph of the given equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%2Ax%2B4 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%2Ax%2B4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red) and graph of the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3%2Ax-32 ALIGN=MIDDLE  ALT=\"y=3%2Ax-32\"   DISABLED_style= \"max-width:90%; height:auto;\" >(green) that is perpendicular to the given graph and goes through (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=9 ALIGN=MIDDLE  ALT=\"9\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5 ALIGN=MIDDLE  ALT=\"-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "Now convert <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3x-32 ALIGN=MIDDLE  ALT=\"y=3x-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > into standard form\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=converting_linear_equations.solver>Converting Linear Equations in Standard form to Slope-Intercept Form (and vice versa)</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT>Convert from slope-intercept form (y = mx+b) to standard form (Ax+By = C)<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x-32 ALIGN=MIDDLE  ALT=\"y+=+3x-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1y-3x+=+3x-32-3x ALIGN=MIDDLE  ALT=\"1y-3x+=+3x-32-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract 3x from both sides<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x%2B1y+=+-32 ALIGN=MIDDLE  ALT=\"-3x%2B1y+=+-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Simplify<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2A%28-3x%2B1y%29+=+-1%2A%28-32%29 ALIGN=MIDDLE  ALT=\"-1%2A%28-3x%2B1y%29+=+-1%2A%28-32%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by -1 to make the A coefficient positive (note: this step may be optional; it will depend on your teacher and/or textbook)<br><br><br><IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1y+=+32 ALIGN=MIDDLE  ALT=\"3x-1y+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute and simplify<br><br><br>The original equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+3x-32 ALIGN=MIDDLE  ALT=\"y+=+3x-32\"   DISABLED_style= \"max-width:90%; height:auto;\" > (slope-intercept form) is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1y+=+32 ALIGN=MIDDLE  ALT=\"3x-1y+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > (standard form where A > 0)<br><br><br>The equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-1y+=+32 ALIGN=MIDDLE  ALT=\"3x-1y+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > is in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Ax%2BBy+=+C ALIGN=MIDDLE  ALT=\"Ax%2BBy+=+C\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+3 ALIGN=MIDDLE  ALT=\"A+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=B+=+-1 ALIGN=MIDDLE  ALT=\"B+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C+=+32 ALIGN=MIDDLE  ALT=\"C+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><br></TD></TR>\n" );
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document.write( "So the equation of the line that is perpendicular to x + 3y = 12 and passing through (9, -5) is\r<BR>\n" );
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