document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=111919>Question 111919</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780073016023/algebrahomeworkh>Beginning Algebra</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> Graph the equation.\r<BR>\n" );
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document.write( "2x + 3y = 6\r<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 #81661 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=81661\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <TABLE  BORDER=1>\n" );
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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" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2Ax%2B3%2Ay=6 ALIGN=MIDDLE  ALT=\"2%2Ax%2B3%2Ay=6\"   DISABLED_style= \"max-width:90%; height:auto;\" >Start with the given equation<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2Ay=6-2%2Ax ALIGN=MIDDLE  ALT=\"3%2Ay=6-2%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" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%281%2F3%29%286-2%2Ax%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%286-2%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=%281%2F3%29%286%29-%281%2F3%29%282%29x%29 ALIGN=MIDDLE  ALT=\"y=%281%2F3%29%286%29-%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" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6%2F3-%282%2F3%29x ALIGN=MIDDLE  ALT=\"y=6%2F3-%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=%28-2%2F3%29%2Ax%2B6%2F3 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2Ax%2B6%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2F3%29%2Ax%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2Ax%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce any fractions<BR>\n" );
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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=2 ALIGN=MIDDLE  ALT=\"b=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (the y-intercept)<BR>\n" );
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document.write( "  So to graph this equation lets plug in some points<BR>\n" );
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document.write( "  Plug in x=-9<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2F3%29%2A%28-9%29%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2A%28-9%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F3%2B2 ALIGN=MIDDLE  ALT=\"y=18%2F3%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=24%2F3 ALIGN=MIDDLE  ALT=\"y=24%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=8 ALIGN=MIDDLE  ALT=\"y=8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  So here's one point (-9,8)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-19%2C4%2C+-4%2C+18%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now lets find another point <BR>\n" );
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document.write( "  Plug in x=-6<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2F3%29%2A%28-6%29%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2A%28-6%29%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=12%2F3%2B2 ALIGN=MIDDLE  ALT=\"y=12%2F3%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=18%2F3 ALIGN=MIDDLE  ALT=\"y=18%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Add<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=6 ALIGN=MIDDLE  ALT=\"y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce<BR>\n" );
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document.write( "  So here's another point (-6,6). Add this to our graph<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-19%2C4%2C+-4%2C+18%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now draw a line through these points<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-19%2C4%2C+-4%2C+18%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-19-10%2C-0.666666666666667%2A%28-19-10%29%2B2%2C4%2B10%2C-0.666666666666667%2A%284%2B10%29%2B2%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-6%2C6%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+-9%2C8%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=%28-2%2F3%29%2Ax%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2Ax%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (-9,8) and (-6,6)<BR>\n" );
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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  down -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=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)and the x-intercept is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" >,0) . So all of this information verifies our graph.<BR>\n" );
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document.write( "  We could graph this equation another way. Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2 ALIGN=MIDDLE  ALT=\"b=2\"   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=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >). <BR>\n" );
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document.write( "  So we have one point (0,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C13%2C+-10%2C+12%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++%0D%0A++++blue%28+circle%28+0%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.1%2C+1.5+%29+%29%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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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=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" >), we can go down 2 units <BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C13%2C+-10%2C+12%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+2%2B%28-2%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" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28++600%2C+600%2C+-10%2C13%2C+-10%2C+12%2C%0D%0A++++grid%281%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+3%2C0%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+3%2C0%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+2%2B%28-2%2F2%29%2C+2%2C+-2%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%283%2F2%29%2C+0%2C+3%2C+2%2C+0%2C+180+%29+%29%0D%0A++%0D%0A++%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  Now draw a line through those points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-2%2F3%29%2Ax%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2Ax%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28+600%2C+600%2C+-10%2C13%2C+-10%2C+12%2C%0D%0A++++grid%28+1+%29%2C%0D%0A++++line%28-10-10%2C-0.666666666666667%2A%28-10-10%29%2B2%2C13%2B10%2C-0.666666666666667%2A%2813%2B10%29%2B2%29%2C%0D%0A++++blue%28+circle%28+3%2C0%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+3%2C0%2C+.1%2C+1.5+%29+%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.15%2C+1.5+%29%29%2C%0D%0A++++blue%28+circle%28+0%2C2%2C+.1%2C+1.5+%29+%29%2C%0D%0A++blue%28+arc%28+0%2C+2%2B%28-2%2F2%29%2C+2%2C+-2%2C+90%2C+270+%29+%29%2C%0D%0A++blue%28+arc%28+%283%2F2%29%2C+0%2C+3%2C+2%2C+0%2C+180+%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=%28-2%2F3%29%2Ax%2B2 ALIGN=MIDDLE  ALT=\"y=%28-2%2F3%29%2Ax%2B2\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points (0,2) and (3,0)<BR>\n" );
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