document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186326>Question 1186326</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the line parallel to 3x + 4y = 20 and at a distance of 5 units from this<BR>\n" );
document.write( " line. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #817231 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=817231\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "The solution from tutor @MathLover 1 is simply wrong.  Lines parallel to the given line with y-intercepts that are 5 more or less than the y-intercept of the given line are NOT 5 units from the given line.<br><BR>\n" );
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document.write( "The solution from tutor @Boreal is fine, except for a typo on one of the equation he shows as the answers.<br><BR>\n" );
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document.write( "Here is what I think is an easier way to get to the answers, using the basic idea of @Boreal's solution but in a different way.<br><BR>\n" );
document.write( "The y-intercept of the given line is (0,5); its slope is -3/4.<br><BR>\n" );
document.write( "To find the equations of the two lines parallel to the given line and a distance of 5 from the given line, do the following:<br><BR>\n" );
document.write( "(1) Pick a convenient point A on the given line; the obvious convenient point is the y-intercept of the given line, (0,5)<br><BR>\n" );
document.write( "(2) On the line perpendicular to the given line at (0,5), find points B and C that are a distance of 5 from (0,5).<br><BR>\n" );
document.write( "The slope of the given line is -3/4; the slope of the line perpendicular to the given line is 4/3.  Knowing that 3-4-5 is a Pythagorean Triple, to find a point 5 units from (0,5) on a line with slope 4/3, we can either (a) move 3 units right and 4 units up to reach B(3,9) or (b) move 3 units left and 4 units down to reach C(-3,1).<br><BR>\n" );
document.write( "(3) Find the equations of the two lines parallel to the given line and 5 units from the given line, knowing that any line parallel to the given line will have an equation of the form 3x+4y=C.  Plug in the values of the points B and C to find the equations we are looking for.<br><BR>\n" );
document.write( "through B(3,9): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=3%283%29%2B4%289%29=9%2B36=45 ALIGN=MIDDLE  ALT=\"3x%2B4y=3%283%29%2B4%289%29=9%2B36=45\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=45 ALIGN=MIDDLE  ALT=\"3x%2B4y=45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "through C(-3,1): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=3%28-3%29%2B4%281%29=-9%2B4=-5 ALIGN=MIDDLE  ALT=\"3x%2B4y=3%28-3%29%2B4%281%29=-9%2B4=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y=-5 ALIGN=MIDDLE  ALT=\"3x%2B4y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWERS:<BR>\n" );
document.write( "(1) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y+=+45 ALIGN=MIDDLE  ALT=\"3x%2B4y+=+45\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F4%29x%2B45%2F4 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F4%29x%2B45%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y-45+=+0 ALIGN=MIDDLE  ALT=\"3x%2B4y-45+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y+=+-5 ALIGN=MIDDLE  ALT=\"3x%2B4y+=+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-3%2F4%29x-5%2F4 ALIGN=MIDDLE  ALT=\"y+=+%28-3%2F4%29x-5%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%2B5+=+0 ALIGN=MIDDLE  ALT=\"3x%2B4y%2B5+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><hr><BR>\n" );
document.write( "And here is an easier way to find the answers....<br><BR>\n" );
document.write( "Given a point (m,n) and a line with equation Ax+By+C=0, the distance from the point to the line is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%28Am%2BBn%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29 ALIGN=MIDDLE  ALT=\"abs%28%28Am%2BBn%2BC%29%2Fsqrt%28A%5E2%2BB%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this problem, the equation of the given line -- in the form required for using the formula -- is 3x+4y-20=0; a convenient point on the given line is (0,5), and the desired distance is 5:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28%283%280%29%2B4%285%29%2BC%29%2Fsqrt%283%5E2%2B4%5E2%29%29=5 ALIGN=MIDDLE  ALT=\"abs%28%283%280%29%2B4%285%29%2BC%29%2Fsqrt%283%5E2%2B4%5E2%29%29=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%2820%2BC%29%2F5=5 ALIGN=MIDDLE  ALT=\"abs%2820%2BC%29%2F5=5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%2820%2BC%29=25 ALIGN=MIDDLE  ALT=\"abs%2820%2BC%29=25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2BC=25 ALIGN=MIDDLE  ALT=\"20%2BC=25\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20%2BC=-25 ALIGN=MIDDLE  ALT=\"20%2BC=-25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=5 ALIGN=MIDDLE  ALT=\"C=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C=-45 ALIGN=MIDDLE  ALT=\"C=-45\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "That gives us the same two equations we found earlier:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y%2B5=0 ALIGN=MIDDLE  ALT=\"3x%2B4y%2B5=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B4y-45=0 ALIGN=MIDDLE  ALT=\"3x%2B4y-45=0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );