document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=153292>Question 153292</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Plot the graph of the equations and interpret the result.\r<BR>\n" );
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document.write( "1. 3x - 8y = 5<BR>\n" );
document.write( "   4x - 2y = 11\r<BR>\n" );
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document.write( "2. 4x - 6y = 2 <BR>\n" );
document.write( "   2x - 3y = 1\r<BR>\n" );
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document.write( "3. 10x - 4y = 3<BR>\n" );
document.write( "    5x - 2y = 2<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 #112833 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=112833\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'll do the first two to get you started\r<BR>\n" );
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document.write( "# 1\r<BR>\n" );
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document.write( "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-8y=5%2C4x-2y=11%29 ALIGN=MIDDLE  ALT=\"system%283x-8y=5%2C4x-2y=11%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to graph these equations, we <font size=\"4\"><b>must</b></font> solve for y first.\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-8y=5 ALIGN=MIDDLE  ALT=\"3x-8y=5\"   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=-8y=5-3x ALIGN=MIDDLE  ALT=\"-8y=5-3x\"   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=y=%285-3x%29%2F%28-8%29 ALIGN=MIDDLE  ALT=\"y=%285-3x%29%2F%28-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-8 ALIGN=MIDDLE  ALT=\"-8\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=%283%2F8%29x-5%2F8 ALIGN=MIDDLE  ALT=\"y=%283%2F8%29x-5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms and simplify.\r<BR>\n" );
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document.write( "Now let's graph the equation:\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%280%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%283%2F8%29x-5%2F8%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%2F8%29x-5%2F8 ALIGN=MIDDLE  ALT=\"y=%283%2F8%29x-5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\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=4x-2y=11 ALIGN=MIDDLE  ALT=\"4x-2y=11\"   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=11-4x ALIGN=MIDDLE  ALT=\"-2y=11-4x\"   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=y=%2811-4x%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"y=%2811-4x%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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=2x-11%2F2 ALIGN=MIDDLE  ALT=\"y=2x-11%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms and simplify.\r<BR>\n" );
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document.write( "Now let's graph the equation:\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%280%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x-11%2F2%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=2x-11%2F2 ALIGN=MIDDLE  ALT=\"y=2x-11%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\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%2F8%29x-5%2F8%2C2x-11%2F2%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%2F8%29x-5%2F8 ALIGN=MIDDLE  ALT=\"y=%283%2F8%29x-5%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red). Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=2x-11%2F2 ALIGN=MIDDLE  ALT=\"y=2x-11%2F2\"   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(3,\frac{1}{2}\right)\">. So the solution to the system of equations is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(3,\frac{1}{2}\right)\">. This tells us that the system of equations is consistent and independent.\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "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%284x-6y=2%2C2x-3y=1%29 ALIGN=MIDDLE  ALT=\"system%284x-6y=2%2C2x-3y=1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In order to graph these equations, we <font size=\"4\"><b>must</b></font> solve for y first.\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=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=y=%282-4x%29%2F%28-6%29 ALIGN=MIDDLE  ALT=\"y=%282-4x%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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms and simplify.\r<BR>\n" );
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document.write( "Now let's graph the equation:\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%280%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%282%2F3%29x-1%2F3%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;\" >.\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=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=y=%281-2x%29%2F%28-3%29 ALIGN=MIDDLE  ALT=\"y=%281-2x%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 <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y ALIGN=MIDDLE  ALT=\"y\"   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=%282%2F3%29x-1%2F3 ALIGN=MIDDLE  ALT=\"y=%282%2F3%29x-1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rearrange the terms and simplify.\r<BR>\n" );
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document.write( "Now let's graph the equation:\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%280%29%2C%0D%0Agraph%28500%2C500%2C-10%2C10%2C-10%2C10%2C%282%2F3%29x-1%2F3%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;\" >.\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%282%2F3%29x-1%2F3%2C%282%2F3%29x-1%2F3%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;\" > (red). 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;\" > (green)\r<BR>\n" );
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document.write( "From the graph, we can see that one line is right on top of the other one, which means that they intersect an infinite number of times. So there are an infinite number of solutions. This means that the system of equations is consistent and dependent.\r<BR>\n" );
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