document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=170422>Question 170422</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Plot the graph of the equations 4x - 6y = 2 and 2x - 3y = 1 and interpret the result. <BR>\n" );
document.write( "I am having such a difficult time solving this equation.  Can someone please help me understand how to do this problem, any help will be greatly appreciated. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #125817 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=125817\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <h3>Graphing the first equation</h3>\r<BR>\n" );
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document.write( "In order to graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-6y=2 ALIGN=MIDDLE  ALT=\"4x-6y=2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we need to solve for \"y\" first.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x-6y=2 ALIGN=MIDDLE  ALT=\"4x-6y=2\"   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=-6y=2-4x ALIGN=MIDDLE  ALT=\"-6y=2-4x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4x ALIGN=MIDDLE  ALT=\"4x\"   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=-6y=-4x%2B2 ALIGN=MIDDLE  ALT=\"-6y=-4x%2B2\"   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=%28-4x%2B2%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"y=%28-4x%2B2%29%2F%28-6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-6 ALIGN=MIDDLE  ALT=\"-6\"   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%28-4%29%2F%28-6%29%29x%2B%282%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"y=%28%28-4%29%2F%28-6%29%29x%2B%282%29%2F%28-6%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=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "In order to graph this equation, we only need two points to create a straight line\r<BR>\n" );
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document.write( "--------------------------------Let's find the first point--------------------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%285%29-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%285%29-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   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=y=10%2F3-1%2F3 ALIGN=MIDDLE  ALT=\"y=10%2F3-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%2F3 ALIGN=MIDDLE  ALT=\"10%2F3\"   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=y=9%2F3 ALIGN=MIDDLE  ALT=\"y=9%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract \r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=5 ALIGN=MIDDLE  ALT=\"x=5\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=3 ALIGN=MIDDLE  ALT=\"y=3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means we have the first point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(5,3\right)\">\r<BR>\n" );
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document.write( "--------------------------------Let's find the second point--------------------------------\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given equation\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29%28-1%29-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29%28-1%29-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   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=y=-2%2F3-1%2F3 ALIGN=MIDDLE  ALT=\"y=-2%2F3-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply <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;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1 ALIGN=MIDDLE  ALT=\"-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <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;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract\r<BR>\n" );
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document.write( "So when <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1 ALIGN=MIDDLE  ALT=\"x=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have the value <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-1 ALIGN=MIDDLE  ALT=\"y=-1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This means we have the second point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-1,-1\right)\">\r<BR>\n" );
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document.write( "So we have the two points: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(5,\frac{9}{3}\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-1,-\frac{3}{3}\right)\">\r<BR>\n" );
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document.write( "Now plot these two points on a coordinate system\r<BR>\n" );
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document.write( "Now draw a straight line through the two points. This line is the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   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-8%2C12%2C-8%2C12%2C%0D%0Agraph%28500%2C500%2C-8%2C12%2C-8%2C12%2C%282%2F3%29x-1%2F3%29%2C%0D%0Agrid%281%29%2C%0D%0Acircle%285%2C9%2F3%2C0.1%29%2C%0D%0Acircle%285%2C9%2F3%2C0.12%29%2C%0D%0Acircle%285%2C9%2F3%2C0.15%29%2C%0D%0Acircle%28-1%2C-3%2F3%2C0.1%29%2C%0D%0Acircle%28-1%2C-3%2F3%2C0.12%29%2C%0D%0Acircle%28-1%2C-3%2F3%2C0.15%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=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > through the two points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(5,3\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-1,-1\right)\">\r<BR>\n" );
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document.write( "<h3>Graphing the second equation</h3>\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=1 ALIGN=MIDDLE  ALT=\"2x-3y=1\"   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=-3y=1-2x ALIGN=MIDDLE  ALT=\"-3y=1-2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x ALIGN=MIDDLE  ALT=\"2x\"   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=-3y=-2x%2B1 ALIGN=MIDDLE  ALT=\"-3y=-2x%2B1\"   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=%28-2x%2B1%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"y=%28-2x%2B1%29%2F%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3 ALIGN=MIDDLE  ALT=\"-3\"   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%28-2%29%2F%28-3%29%29x%2B%281%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"y=%28%28-2%29%2F%28-3%29%29x%2B%281%29%2F%28-3%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=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Reduce.\r<BR>\n" );
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document.write( "Take note that this equation is EXACTLY identical to the equation we just plotted. So the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x-3y=1 ALIGN=MIDDLE  ALT=\"2x-3y=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is \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-8%2C12%2C-8%2C12%2C%0D%0Agraph%28500%2C500%2C-8%2C12%2C-8%2C12%2C%282%2F3%29x-1%2F3%29%2C%0D%0Agrid%281%29%0D%0A%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "Now graphing the two equations together gives you\r<BR>\n" );
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document.write( "Note: there are really two equations here. One is just right on top of the other (which hides it)\r<BR>\n" );
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document.write( "Since one equation is right on top of the other, this tells us that there are an infinite number of solutions (since there are an infinite number of intersections).\r<BR>\n" );
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document.write( "So the system is dependent.<BR>\n" );
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