document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=170807>Question 170807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2.Plot the graph of the equations 4x-6y=2 and 2x-3y=1 and interpret the results. <BR>\n" );
document.write( "  4x-6y=2<BR>\n" );
document.write( "-4x+6y=-2<BR>\n" );
document.write( "0=0<BR>\n" );
document.write( "These lines are the same and intersect at every point. <BR>\n" );
document.write( "Y intercept:<BR>\n" );
document.write( "2x-3y=1<BR>\n" );
document.write( "2(0)-3y=1<BR>\n" );
document.write( "0-3y/3=1/3    y intercept = (0,1/3)<BR>\n" );
document.write( "Y=0: 2x-3(0)=1<BR>\n" );
document.write( "2x/2-0=1/2    x intercept = (1/2,0)<BR>\n" );
document.write( "4x-6y=2<BR>\n" );
document.write( "X=0: 4(0)-6y=2<BR>\n" );
document.write( "            0-6y/6=2/6 = 1/3   y intercept = (0,1/3)<BR>\n" );
document.write( "Y=0: 4x-6(0)=2<BR>\n" );
document.write( "4x/4-0=2/4=1/2            x intercept = (1/2,0)<BR>\n" );
document.write( "Could you please tell me if I did the above problem correctly and could you show me how to graph it. \r<BR>\n" );
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document.write( "3.Plot the graph of the equations 10x-4y=3 and 5x-2y=6<BR>\n" );
document.write( "Can you please show me how to do number 3. Thanks Nichole <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 #126104 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=126104\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 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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document.write( "# 3\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%2810x-4y=3%2C5x-2y=6%29 ALIGN=MIDDLE  ALT=\"system%2810x-4y=3%2C5x-2y=6%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=10x-4y=3 ALIGN=MIDDLE  ALT=\"10x-4y=3\"   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=-4y=3-10x ALIGN=MIDDLE  ALT=\"-4y=3-10x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10x ALIGN=MIDDLE  ALT=\"10x\"   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=%283-10x%29%2F%28-4%29 ALIGN=MIDDLE  ALT=\"y=%283-10x%29%2F%28-4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   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=%285%2F2%29x-3%2F4 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3%2F4\"   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%285%2F2%29x-3%2F4%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=%285%2F2%29x-3%2F4 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3%2F4\"   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=5x-2y=6 ALIGN=MIDDLE  ALT=\"5x-2y=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=6-5x ALIGN=MIDDLE  ALT=\"-2y=6-5x\"   DISABLED_style= \"max-width:90%; height:auto;\" > Subtract <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x ALIGN=MIDDLE  ALT=\"5x\"   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=%286-5x%29%2F%28-2%29 ALIGN=MIDDLE  ALT=\"y=%286-5x%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=%285%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3\"   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%285%2F2%29x-3%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=%285%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3\"   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%285%2F2%29x-3%2F4%2C%285%2F2%29x-3%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=%285%2F2%29x-3%2F4 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" > (red). Graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=%285%2F2%29x-3 ALIGN=MIDDLE  ALT=\"y=%285%2F2%29x-3\"   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 are parallel, which means that they will <font size=\"4\"><b>NEVER</b></font> intersect. So there are no solutions. This means that the system of equations is inconsistent.\r<BR>\n" );
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