document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=173156>Question 173156</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need to solve tis by graphing and have done this ten times and still can not find the points to graph.  <BR>\n" );
document.write( "3x-2y=6<BR>\n" );
document.write( "3x+2y=6<BR>\n" );
document.write( "Could somebody please help me figure out how to graph this?  Thank you in advance. Judy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #128010 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=128010\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Start with the given system of equations:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x-2y=6%2C3x%2B2y=6%29 ALIGN=MIDDLE  ALT=\"system%283x-2y=6%2C3x%2B2y=6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let's graph the first equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x-2y=6 ALIGN=MIDDLE  ALT=\"3x-2y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the first equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y=6-3x ALIGN=MIDDLE  ALT=\"-2y=6-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2y=-3x%2B6 ALIGN=MIDDLE  ALT=\"-2y=-3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3x%2B6%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"y=%28-3x%2B6%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-2 ALIGN=MIDDLE  ALT=\"-2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28-3%29%2F%28-2%29%29x%2B%286%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"y=%28%28-3%29%2F%28-2%29%29x%2B%286%29%2F%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%283%2F2%29x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "Now let's graph the equation:\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%283%2F2%29x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the equation is 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 the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=3%2F2 ALIGN=MIDDLE  ALT=\"m=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=-3 ALIGN=MIDDLE  ALT=\"b=-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-3\right)\">.Remember the y-intercept is the point where the graph intersects with the y-axis\r<BR>\n" );
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document.write( "So we have one point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-3\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C-3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now since the slope is comprised of the \"rise\" over the \"run\" this means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=rise%2Frun ALIGN=MIDDLE  ALT=\"slope=rise%2Frun\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, because the slope 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;\" >, this means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rise%2Frun=3%2F2 ALIGN=MIDDLE  ALT=\"rise%2Frun=3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which shows us that the rise is 3 and the run is 2. This means that to go from point to point, we can go up 3  and over 2\r<BR>\n" );
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document.write( "So starting at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-3\right)\">, go up 3 units \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C-3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%2C%0D%0A++blue%28arc%280%2C-3%2B%283%2F2%29%2C2%2C3%2C90%2C270%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "and to the right 2 units to get to the next point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C-3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C-3%2B%283%2F2%29%2C2%2C3%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%282%2F2%29%2C0%2C2%2C2%2C+180%2C360%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now draw a line through these points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%283%2F2%29x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%283%2F2%29x-3%29%2C%0D%0A++blue%28circle%280%2C-3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C-3%2C.15%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C-3%2B%283%2F2%29%2C2%2C3%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%282%2F2%29%2C0%2C2%2C2%2C+180%2C360%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=%283%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%283%2F2%29x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,-3\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">\r<BR>\n" );
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document.write( "Now let's graph the second equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B2y=6 ALIGN=MIDDLE  ALT=\"3x%2B2y=6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the second equation.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y=6-3x ALIGN=MIDDLE  ALT=\"2y=6-3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x ALIGN=MIDDLE  ALT=\"3x\"   DISABLED_style= \"max-width:90%; height:auto;\" > from both sides.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2y=-3x%2B6 ALIGN=MIDDLE  ALT=\"2y=-3x%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28-3x%2B6%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y=%28-3x%2B6%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate y.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%28%28-3%29%2F%282%29%29x%2B%286%29%2F%282%29 ALIGN=MIDDLE  ALT=\"y=%28%28-3%29%2F%282%29%29x%2B%286%29%2F%282%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Break up the fraction.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "Looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can see that the equation is 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 the slope is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=m=-3%2F2 ALIGN=MIDDLE  ALT=\"m=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=3 ALIGN=MIDDLE  ALT=\"b=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > this tells us that the y-intercept is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,3\right)\">.Remember the y-intercept is the point where the graph intersects with the y-axis\r<BR>\n" );
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document.write( "So we have one point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,3\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++blue%28circle%280%2C3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C3%2C.15%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now since the slope is comprised of the \"rise\" over the \"run\" this means\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=slope=rise%2Frun ALIGN=MIDDLE  ALT=\"slope=rise%2Frun\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Also, because the slope 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;\" >, this means:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=rise%2Frun=-3%2F2 ALIGN=MIDDLE  ALT=\"rise%2Frun=-3%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "which shows us that the rise is -3 and the run is 2. This means that to go from point to point, we can go down 3  and over 2\r<BR>\n" );
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document.write( "So starting at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,3\right)\">, go down 3 units \r<BR>\n" );
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document.write( "and to the right 2 units to get to the next point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">\r<BR>\n" );
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document.write( "Now draw a line through these points to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0A++grid%281%29%2C%0D%0A++graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-%283%2F2%29x%2B3%29%2C%0D%0A++blue%28circle%280%2C3%2C.1%29%29%2C%0D%0A++blue%28circle%280%2C3%2C.12%29%29%2C%0D%0A++blue%28circle%280%2C3%2C.15%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.15%2C1.5%29%29%2C%0D%0A++blue%28circle%282%2C0%2C.1%2C1.5%29%29%2C%0D%0A++blue%28arc%280%2C3%2B%28-3%2F2%29%2C2%2C-3%2C90%2C270%29%29%2C%0D%0A++blue%28arc%28%282%2F2%29%2C0%2C2%2C2%2C+0%2C180%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=-%283%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(0,3\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">\r<BR>\n" );
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document.write( "-------------------------------------------------------------------\r<BR>\n" );
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document.write( "Now let's graph the two equations together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28500%2C500%2C-10%2C10%2C-10%2C10%2C%0D%0Agrid%281%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%283%2F2%29x-3%2C-%283%2F2%29x%2B3%29%2C%0D%0Acircle%282%2C0%2C0.08%29%2C%0D%0Acircle%282%2C0%2C0.10%29%2C%0D%0Acircle%282%2C0%2C0.12%29%2C%0D%0Acircle%282%2C0%2C0.14%29%2C%0D%0Acircle%282%2C0%2C0.16%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%283%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%283%2F2%29x-3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red). Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%283%2F2%29x%2B3 ALIGN=MIDDLE  ALT=\"y=-%283%2F2%29x%2B3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (green)\r<BR>\n" );
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document.write( "From the graph, we can see that the two lines intersect at the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">. \r<BR>\n" );
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document.write( "So the solution to the system of equations is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(2,0\right)\">. \r<BR>\n" );
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document.write( "This tells us that the system of equations is consistent and independent.\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );