document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1159868>Question 1159868</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Line A is perpendicular to Line B. Determine<BR>\n" );
document.write( "the equation of Line A if it passes through Line B<BR>\n" );
document.write( "at (5, -1), where Line B is represented by the<BR>\n" );
document.write( "equation x + 2y = 3. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #782971 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=782971\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "First, a solution using elementary methods; you should understand it and be able to solve similar problems by this method.<br><BR>\n" );
document.write( "(1) Put the equation of B 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;\" > to find the slope.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y+=+3 ALIGN=MIDDLE  ALT=\"x%2B2y+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y+=+-x%2B3 ALIGN=MIDDLE  ALT=\"2y+=+-x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-1%2F2%29x%2B3%2F2 ALIGN=MIDDLE  ALT=\"y+=+%28-1%2F2%29x%2B3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The slope is (-1/2).<br><BR>\n" );
document.write( "(2) Find the slope of a line perpendicular to B -- it is the negative reciprocal of the slope of B: +(2/1) = 2<br><BR>\n" );
document.write( "(3) Find the equation of line A knowing it has a slope of 2 and passes through (5,-1).<BR>\n" );
document.write( "<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;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+2%285%29%2Bb ALIGN=MIDDLE  ALT=\"-1+=+2%285%29%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1+=+10%2Bb ALIGN=MIDDLE  ALT=\"-1+=+10%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b+=+-1-10+=+-11 ALIGN=MIDDLE  ALT=\"b+=+-1-10+=+-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2x-11 ALIGN=MIDDLE  ALT=\"y+=+2x-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "A graph of the two lines -- B red, A green....<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28400%2C400%2C-12%2C12%2C-12%2C12%2C%28-1%2F2%29x%2B3%2F2%2C2x-11%29 ALIGN=MIDDLE  ALT=\"graph%28400%2C400%2C-12%2C12%2C-12%2C12%2C%28-1%2F2%29x%2B3%2F2%2C2x-11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Alternatively, a shortcut using ideas about vectors.  This is a quick path to the answer you can use if a formal algebraic solution is not required.<br><BR>\n" );
document.write( "Given line B with equation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=3 ALIGN=MIDDLE  ALT=\"x%2B2y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, every line parallel to B will have an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2B2y=c ALIGN=MIDDLE  ALT=\"x%2B2y=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some constant c; and every line perpendicular to B will have an equation of the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-y=c ALIGN=MIDDLE  ALT=\"2x-y=c\"   DISABLED_style= \"max-width:90%; height:auto;\" > for some constant c.<br><BR>\n" );
document.write( "Without understanding vectors, what you need to see is that the coefficients of x and y have switched places, and one of them has changed signs.  In this example, the coefficients 1 and 2 become 2 and -1.<br><BR>\n" );
document.write( "So the equation of line A is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-y=c ALIGN=MIDDLE  ALT=\"2x-y=c\"   DISABLED_style= \"max-width:90%; height:auto;\" >; and we can determine the constant c knowing that (x,y)=(5,-1) satisfies the equation.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-y=c ALIGN=MIDDLE  ALT=\"2x-y=c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%285%29-%28-1%29+=+c ALIGN=MIDDLE  ALT=\"2%285%29-%28-1%29+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11+=+c ALIGN=MIDDLE  ALT=\"11+=+c\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "The equation of A is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-y+=+11 ALIGN=MIDDLE  ALT=\"2x-y+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That is a different form of the same equation found earlier, which was <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2x-11 ALIGN=MIDDLE  ALT=\"y+=+2x-11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );