document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186113>Question 1186113</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the equation of the line parallel to 3x + 4y= 20 and at a distance 5 from this line </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #816999 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=816999\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with 3x + 4y = 20<BR>\n" );
document.write( "subtract 3x from both sides of the equation to get:<BR>\n" );
document.write( "4y = -3x + 20<BR>\n" );
document.write( "divide both sides by 4 to get:<BR>\n" );
document.write( "y = -3/4 * x + 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "a line perpendicular to that line will have a slope that is is the negative reciprocal of the slope of that line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that slope will be 4/3.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if you want that perpendicular line to intersect the original line at (0,5), then the equation of that perpendicular line needs to be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 4/3 * x + 5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the y-intercept of both lines is at (0,5).<BR>\n" );
document.write( "this is where they will intersect.<BR>\n" );
document.write( "this is what it looks like.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/101403.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you now want to find another point on the perpendicular line that is 5 units from the point (0,5).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula for the length of a line is L = sqrt(x^2 + y^2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when y is equal to 4/3 * x, that formula becomes L = sqrt(x^2 + (4/3 * x)^2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when L = 5, that formula becomes 5 = sqrt(x^2 + (4/3 * x)^2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "square both sides of that equation to get 25 = x^2 + (4/3 * x)^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "simplify to get 25 = x^2 + 16/9 * x^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "multiply both sides of that equation by 9 to get 225 = 9x^2 + 16x^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "combine like terms to get 225 = 25x^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x^2 to get x^2 = 225/25 = 9\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solve for x to get x = 3\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "when x = 3, y = 4/3 * x becomes y = 4/3 * 3 = 4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the point on the perpendicular line that is 5 units from (0,5) will be at the point (0 + 3, 5 + 4) which is equal to (3,9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your two points on the perpendicular line that are of interest are (0,5) and (3,9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your line perpendicular to the perpendicular line will have a slope that is the negative reciprocal of the slope of the perpendicular line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the slope of the perpendicular line is 4/3, that slope will be the negative reciprocal of 4/3 = -3/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that is also the slope of the original line, because the perpendicular of the perpendicular is the original slope the line.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that parallel line will have a slope of -3/4 and will pass through the point (3,9).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to find the y-intercept of that line, replace x with 3 and y with 9 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -3/4 * x + b becomes:<BR>\n" );
document.write( "9 = -3/4 * 3 + b\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b is the y-intercept.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this comes from the general form of the slope intercept equation of a straight line which is y = mx + b.<BR>\n" );
document.write( "m is the slope<BR>\n" );
document.write( "b is the y-intercept.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "in the equation of 9 = -3/4 * 3 + b, solve for b to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "b = 9 + 3/4 * 3 = 9 + 9/4 = 36/4 + 9/4 = 45/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the equation of the line parallel to the original line and perpendicular to the perpendicular line and passing through the point (3,9) will be:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = -3/4 * x + 45/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that will look like this on the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2021/101404.jpg\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the line parallel to the original line and 5 units from the original line has the equation of y = -3/4 * x + 45/4.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "45/4 is the y-intercept equal to 11.25 as shown on the graph.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the original line is red.<BR>\n" );
document.write( "the line perpendicular to it is blue.<BR>\n" );
document.write( "the line parallel to it is green.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your solution is that the line parallel to the original line and 5 units from it is y = -3/4 * x + 45/4.<BR>\n" );
document.write( "this can also be shown as y = -3/4 * x + 11.25.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );