document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=83173>Question 83173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> #38\r<BR>\n" );
document.write( "\n" );
document.write( "Find an equation in slope-intercept form (where possible) for each line<BR>\n" );
document.write( "Use slopes to show that the square with vertices at (-2,5),(4,5),(4,-1),and <BR>\n" );
document.write( "(-2,-1)<BR>\n" );
document.write( "has diagonals that are perpendicular.\r<BR>\n" );
document.write( "\n" );
document.write( "#32<BR>\n" );
document.write( "Find an equation in slope-intercept form (where possible) for each line<BR>\n" );
document.write( "Through (-2,6),perpendicular to 2x -3y =5.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #59724 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=59724\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> #38\r<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+300%2C+300%2C+-7%2C+7%2C+-7%2C+7%2C%0D%0A++rectangle%28+-2%2C+-1%2C+4%2C+5+%29%2C%0D%0A++line%28-2%2C+-1%2C+4%2C+5+%29%2C%0D%0A++line%28-2%2C5%2C4%2C-1%29%2C%0D%0A++circle%28-2%2C-1%2C.1%29%2C%0D%0A++circle%28-2%2C-1%2C.15%29%2C%0D%0A++circle%28-2%2C5%2C.1%29%2C%0D%0A++circle%28-2%2C5%2C.15%29%2C%0D%0A++circle%284%2C5%2C.1%29%2C%0D%0A++circle%284%2C5%2C.15%29%2C%0D%0A++circle%284%2C-1%2C.1%29%2C%0D%0A++circle%284%2C-1%2C.15%29%2C%0D%0A++locate%28+-2.5%2C+-1.25%2C+%22%28-2%2C-1%29%22+%29%2C%0D%0A++locate%28+-2.5%2C+5.75%2C+%22%28-2%2C+5%29%22+%29%2C%0D%0A++locate%28+4%2C+-1%2C+%22%284%2C+-1%29%22+%29%2C%0D%0A++locate%28+4%2C+5%2C+%22%284%2C+5%29%22+%29%0D%0A%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If we construct the figure that the problem is describing, we get a square with the given vertices and 2 lines. We need to find the slopes of the 2 lines to see if they are perpendicular. So the first line goes through the points (-2,5) and (4,-1) (it is the line that is sloping downward). So lets find the slope of this line:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=slope.solver>Finding the slope</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Slope of the line through the points (-2, 5) and (4, -1)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+x%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+%28x%5B1%5D%29%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D+-+y%5B1%5D%29%2F%28x%5B2%5D+-+%28x%5B1%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-1+-+5%29%2F%284+-+%28-2%29%29 ALIGN=MIDDLE  ALT=\"m+=+%28-1+-+5%29%2F%284+-+%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-1+-+5%29%2F%284+%2B+2%29 ALIGN=MIDDLE  ALT=\"m+=+%28-1+-+5%29%2F%284+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28-6%29%2F%286%29 ALIGN=MIDDLE  ALT=\"m+=+%28-6%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-1 ALIGN=MIDDLE  ALT=\"m+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Answer: Slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+-1 ALIGN=MIDDLE  ALT=\"m+=+-1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now lets find the slope of the line going through (-2,-1) and (4,5)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=slope.solver>Finding the slope</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Slope of the line through the points (-2, -1) and (4, 5)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%28y%5B2%5D+-+%28y%5B1%5D%29%29%2F%28x%5B2%5D+-+%28x%5B1%5D%29%29 ALIGN=MIDDLE  ALT=\"m+=+%28y%5B2%5D+-+%28y%5B1%5D%29%29%2F%28x%5B2%5D+-+%28x%5B1%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%285+-+%28-1%29%29%2F%284+-+%28-2%29%29 ALIGN=MIDDLE  ALT=\"m+=+%285+-+%28-1%29%29%2F%284+-+%28-2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%285+%2B+1%29%2F%284+%2B+2%29 ALIGN=MIDDLE  ALT=\"m+=+%285+%2B+1%29%2F%284+%2B+2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+%286%29%2F%286%29 ALIGN=MIDDLE  ALT=\"m+=+%286%29%2F%286%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+1 ALIGN=MIDDLE  ALT=\"m+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Answer: Slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m+=+1 ALIGN=MIDDLE  ALT=\"m+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So we have one slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > and another of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1 ALIGN=MIDDLE  ALT=\"1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Since these slopes are negative reciprocals of each other (ie <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1=-%28-1%2F1%29 ALIGN=MIDDLE  ALT=\"1=-%28-1%2F1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1=-%281%2F1%29%29 ALIGN=MIDDLE  ALT=\"-1=-%281%2F1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, these two lines are perpendicular.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "#32\r<BR>\n" );
document.write( "\n" );
document.write( "Lets solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x+-3y+=5 ALIGN=MIDDLE  ALT=\"2x+-3y+=5\"   DISABLED_style= \"max-width:90%; height:auto;\" > first\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=graphing_linear_equations.solver>Graphing Linear Equations</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax-3%2Ay=5 ALIGN=MIDDLE  ALT=\"2%2Ax-3%2Ay=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given equation<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2Ay=5-2%2Ax ALIGN=MIDDLE  ALT=\"-3%2Ay=5-2%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax ALIGN=MIDDLE  ALT=\"2%2Ax\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%285-2%2Ax%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%285-2%2Ax%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by <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;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-1%2F3%29%285%29%2B%281%2F3%29%282%29x%29 ALIGN=MIDDLE  ALT=\"y=%28-1%2F3%29%285%29%2B%281%2F3%29%282%29x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Distribute <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;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5%2F3%2B%282%2F3%29x ALIGN=MIDDLE  ALT=\"y=-5%2F3%2B%282%2F3%29x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2Ax-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2Ax-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2Ax-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2Ax-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So the equation is now 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;\" >) where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=2%2F3 ALIGN=MIDDLE  ALT=\"m=2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the slope) and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-5%2F3 ALIGN=MIDDLE  ALT=\"b=-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the y-intercept)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So to graph this equation lets plug in some points<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Plug in x=-8<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2A%28-8%29-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2A%28-8%29-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-16%2F3-5%2F3 ALIGN=MIDDLE  ALT=\"y=-16%2F3-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-21%2F3 ALIGN=MIDDLE  ALT=\"y=-21%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-7 ALIGN=MIDDLE  ALT=\"y=-7\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So here's one point (-8,-7)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C5%2C+-17%2C+5%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now lets find another point <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Plug in x=-5<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2A%28-5%29-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2A%28-5%29-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-10%2F3-5%2F3 ALIGN=MIDDLE  ALT=\"y=-10%2F3-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-15%2F3 ALIGN=MIDDLE  ALT=\"y=-15%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-5 ALIGN=MIDDLE  ALT=\"y=-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So here's another point (-5,-5). Add this to our graph<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C5%2C+-17%2C+5%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-5%2C-5%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-5%2C-5%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now draw a line through these points<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-18%2C5%2C+-17%2C+5%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-18-10%2C0.666666666666667%2A%28-18-10%29%2B-1.66666666666667%2C5%2B10%2C0.666666666666667%2A%285%2B10%29%2B-1.66666666666667%29%2C%0D%0A++++blue%28+circle%28+-5%2C-5%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-5%2C-5%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-8%2C-7%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2Ax-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2Ax-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-8,-7) and (-5,-5)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So from the graph we can see that the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (which tells us that in order to go from point to point we have to start at one point and go  up 2 units and to the right 3 units to get to the next point) the y-intercept is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1.66666666666667 ALIGN=MIDDLE  ALT=\"-1.66666666666667\"   DISABLED_style= \"max-width:90%; height:auto;\" >) ,or (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F3 ALIGN=MIDDLE  ALT=\"-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >), and the x-intercept is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.5 ALIGN=MIDDLE  ALT=\"2.5\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0)  ,or (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%2F2 ALIGN=MIDDLE  ALT=\"5%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0) . So all of this information verifies our graph.<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  We could graph this equation another way. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-5%2F3 ALIGN=MIDDLE  ALT=\"b=-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept (the point where the graph intersects with the y-axis) is (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F3 ALIGN=MIDDLE  ALT=\"-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  So we have one point (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F3 ALIGN=MIDDLE  ALT=\"-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C13%2C+-11.6666666666667%2C+10.3333333333333%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now since the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, this means that in order to go from point to point we can use the slope to do so. So starting at (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-5%2F3 ALIGN=MIDDLE  ALT=\"-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we can go up 2 units <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C13%2C+-11.6666666666667%2C+10.3333333333333%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-1.66666666666667%2B%282%2F2%29%2C+2%2C+2%2C+90%2C+270+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  and to the right 3 units to get to our next point<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C13%2C+-11.6666666666667%2C+10.3333333333333%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+3%2C0.333333333333333%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+3%2C0.333333333333333%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-1.66666666666667%2B%282%2F2%29%2C+2%2C+2%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%283%2F2%29%2C+0.333333333333333%2C+3%2C+2%2C+180%2C+360+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  Now draw a line through those points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2Ax-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2Ax-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C13%2C+-11.6666666666667%2C+10.3333333333333%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-10-10%2C0.666666666666667%2A%28-10-10%29%2B-1.66666666666667%2C13%2B10%2C0.666666666666667%2A%2813%2B10%29%2B-1.66666666666667%29%2C%0D%0A++++blue%28+circle%28+3%2C0.333333333333333%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+3%2C0.333333333333333%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C-1.66666666666667%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+-1.66666666666667%2B%282%2F2%29%2C+2%2C+2%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%283%2F2%29%2C+0.333333333333333%2C+3%2C+2%2C+180%2C+360+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > So this is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%2Ax-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%2Ax-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (0,-1.66666666666667) and (3,0.333333333333333)<BR>\n" );
document.write( "  </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So now we have the equation\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-5%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-5%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So this means the perpendicular slope is the negative reciprocal of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So the negative reciprocal of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F3 ALIGN=MIDDLE  ALT=\"2%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we have a line with a slope of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3%2F2 ALIGN=MIDDLE  ALT=\"-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and goes through (-2,6)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  BORDER=1>\n" );
document.write( "\n" );
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=find_line_by_slope_and_point.solver>FIND a line by slope and one point</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><BR>\n" );
document.write( "  What we know about the line whose equation we are trying to find out:<BR>\n" );
document.write( "    <UL><BR>\n" );
document.write( "      <LI>it goes through point (-2, 6)</LI><BR>\n" );
document.write( "      <LI>it has a slope of -1.5</LI><BR>\n" );
document.write( "    </UL><BR>\n" );
document.write( "  <BR>\n" );
document.write( "  First, let's draw a diagram of the coordinate system with point (-2, 6) plotted with a little blue dot:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++blue%28+circle%28+-2%2C+6%2C+0.2+%29+%29+%29%0D%0A++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <B>Write this down</B>: the formula for the equation, given point <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B1%5D%2C+y%5B1%5D ALIGN=MIDDLE  ALT=\"x%5B1%5D%2C+y%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and intercept a, is<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > (see a paragraph below explaining why this formula is correct)<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Given that a=-1.5, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%28+x%5B1%5D+=+-2%2C+y%5B1%5D+=+6+%29+ ALIGN=MIDDLE  ALT=\"system%28+x%5B1%5D+=+-2%2C+y%5B1%5D+=+6+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have the equation of the line:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "    <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1.5%2Ax+%2B+3 ALIGN=MIDDLE  ALT=\"y=-1.5%2Ax+%2B+3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Explanation: <I>Why did we use formula <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ALIGN=MIDDLE  ALT=\"y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (<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;\" >) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (<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;\" >): <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5B1%5D+=+a%2Ax%5B1%5D%2Bb ALIGN=MIDDLE  ALT=\"y%5B1%5D+=+a%2Ax%5B1%5D%2Bb\"   DISABLED_style= \"max-width:90%; height:auto;\" > Here, we know a, <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;\" >, and <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;\" >, and do not know b. It is easy to find out: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=y%5B1%5D-a%2Ax%5B1%5D ALIGN=MIDDLE  ALT=\"b=y%5B1%5D-a%2Ax%5B1%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So, then, the equation of the line is: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+ ALIGN=MIDDLE  ALT=\"+y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </I><BR>\n" );
document.write( "  <BR>\n" );
document.write( "    Here's the graph:<BR>\n" );
document.write( "  <BR>\n" );
document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0A++++drawing%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C%0D%0A+++++++++++++grid%281%29%2C%0D%0A+++++++++++++graph%28+400%2C+400%2C+-10%2C+10%2C+-10%2C+10%2C+-1.5%2Ax%2B3+%29%2C%0D%0A+++++++++++++blue%28+circle%28+-2%2C+6%2C+0.2+%29+%29%2C%0D%0A+++++++++++++locate%28+-2%2C+6%2C+%22%28-2%2C+6%29%22+%29+%29%0D%0A+++++++++++++%29%0D%0A++++++++++ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "   </TD></TR>\n" );
document.write( "</TABLE><BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "So the equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=%28-3%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );