document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=78908>Question 78908</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How Do I determine which two equations represent perpendicular lines.\r<BR>\n" );
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document.write( "(a) y = 2x – 9      (b)  y =  x + 9     (c) y =  - x + 9    (d) y =  x - 9\r<BR>\n" );
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document.write( "I'm lost on which one!<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 #56591 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=56591\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the four equations:<BR>\n" );
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document.write( "(a) y = 2x – 9      <BR>\n" );
document.write( "(b) y =  x + 9     <BR>\n" );
document.write( "(c) y =- x + 9    <BR>\n" );
document.write( "(d) y =  x - 9<BR>\n" );
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document.write( "Note that all four equations are in the slope-intercept form. And the slope-intercept<BR>\n" );
document.write( "form says that the slope of the graph equals the multiplier of the x in the equation.<BR>\n" );
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document.write( "Here's the important part.  For two lines to perpendicular to each other, they must have<BR>\n" );
document.write( "slopes that are the negative inverse of each other.  For example, if one has a slope of<BR>\n" );
document.write( "+5, the perpendicular must have a slope of -1/5.  If one has a slope of -9 the other must<BR>\n" );
document.write( "have a slope of +1/9.  If one has a slope of 5/6, the other must have a slope of -6/5. If<BR>\n" );
document.write( "one has a slope of -7/9, the other has a slope of +9/7.<BR>\n" );
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document.write( "That's the idea. Now let's look at the four equations and identify their slopes and the<BR>\n" );
document.write( "negative inverse of their slopes. (Note again that the slope equals the multiplier<BR>\n" );
document.write( "of the x in the equation.):<BR>\n" );
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document.write( "Equation (a) slope = +2 and negative inverse of slope = -1/2<BR>\n" );
document.write( "Equation (b) slope = +1 and negative inverse of slope = -1/1 = -1<BR>\n" );
document.write( "Equation (c) slope = -1 and negative inverse of slope = -1/(-1) = +1<BR>\n" );
document.write( "Equation (d) slope = +1 and the negative inverse of slope = -1/1 = -1<BR>\n" );
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document.write( "Note that the graph of (a) does not have a line that is perpendicular to it because<BR>\n" );
document.write( "the equation of any line perpendicular to (a) would need a slope of -1/2.<BR>\n" );
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document.write( "How about equation (b)? Any line perpendicular to it would need to have a slope of -1.<BR>\n" );
document.write( "Note that the graph of (c) has a slope of -1.  Therefore the graphs of equations (b) and (c)<BR>\n" );
document.write( "are perpendicular.<BR>\n" );
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document.write( "How about equation (c)? Its graph has a slope of -1, so any graph that is perpendicular<BR>\n" );
document.write( "to it will have a slope of +1.  We already know that (b) has a slope of +1 and is, therefore,<BR>\n" );
document.write( "perpendicular to (c).  But what about (d)? Its graph has a slope of +1 also, so its graph<BR>\n" );
document.write( "is also perpendicular to the graph of (c).<BR>\n" );
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document.write( "So you have two pairs whose graphs are perpendicular ... (b) and (c) plus (c) and (d).<BR>\n" );
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document.write( "Note also that this implies that (b) and (d) are parallel because they are both perpendicular<BR>\n" );
document.write( "to a common line. (They also both have the same slope which is another way of telling<BR>\n" );
document.write( "that they are parallel.)  <BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better.<BR>\n" );
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