document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=729547>Question 729547</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi there! I'm wondering what the best method would be to solve a system of three linear inequalities with only two variables each?\r<BR>\n" );
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document.write( "Example:\r<BR>\n" );
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document.write( "3x+5y ‹(or equal to) 180<BR>\n" );
document.write( "5x+2y ‹(or equal to) 90<BR>\n" );
document.write( "8x+3y ‹(or equal to) 240\r<BR>\n" );
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document.write( "My first instinct was to solve using the elimination method, but I don't know how to apply it to 3 equations.\r<BR>\n" );
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document.write( "I'm on my tablet, so I apologize that I can't use the less-than-or-equal-to symbols.\r<BR>\n" );
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document.write( "Any help is immensely appreciated! </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #446029 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><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/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=446029\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The way to show the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2B5y%3C=180%2C5x%2B2y%3C=90%2C8x%2B3y%3C=240%29 ALIGN=MIDDLE  ALT=\"system%283x%2B5y%3C=180%2C5x%2B2y%3C=90%2C8x%2B3y%3C=240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is graphing on a 2-dimensional graph.<BR>\n" );
document.write( "(It's lucky you have only 2 variables, because 3 dimensional graphs would be hard to do).<BR>\n" );
document.write( " <BR>\n" );
document.write( "The solution to each inequality would include the line represented by the equation with the equal sign, plus all the space to one side of that line.<BR>\n" );
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document.write( "The solution to the system is the space where the solutions to the 3 inequalities overlap (if they overlap).<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=blue%283x%2B5y=180%29 ALIGN=MIDDLE  ALT=\"blue%283x%2B5y=180%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > passes through the points<BR>\n" );
document.write( "(0,36), with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=0 ALIGN=MIDDLE  ALT=\"x=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2A0%2B5y=5y=180 ALIGN=MIDDLE  ALT=\"3%2A0%2B5y=5y=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=180%2F5 ALIGN=MIDDLE  ALT=\"y=180%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=36 ALIGN=MIDDLE  ALT=\"y=36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "(60,0) with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=0 ALIGN=MIDDLE  ALT=\"y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5%2A0=3x=180 ALIGN=MIDDLE  ALT=\"3x%2B5%2A0=3x=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=180%2F3 ALIGN=MIDDLE  ALT=\"x=180%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=60 ALIGN=MIDDLE  ALT=\"x=60\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The graph of the line is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-10%2C90%2C-10%2C90%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%280%2C36%2C2%29%29%2Cblue%28circle%2860%2C0%2C2%29%29%2C%0D%0Ablue%28line%28-10%2C42%2C90%2C-18%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5y%3C=180 ALIGN=MIDDLE  ALT=\"3x%2B5y%3C=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > graphs as (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28200%2C200%2C-10%2C90%2C-10%2C90%2C3x%2B5y%3C=180%29 ALIGN=MIDDLE  ALT=\"graph%28200%2C200%2C-10%2C90%2C-10%2C90%2C3x%2B5y%3C=180%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " An easy way to decide which side of the line is the solution to the inequality is to use a user-friendly \"test point\" such as (0,0) that is not on the line.<BR>\n" );
document.write( "Is (0,0), with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=y=0 ALIGN=MIDDLE  ALT=\"x=y=0\"   DISABLED_style= \"max-width:90%; height:auto;\" > part of the solution?<BR>\n" );
document.write( "It makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5y=3%2A0%2B5%2A0=0%3C180 ALIGN=MIDDLE  ALT=\"3x%2B5y=3%2A0%2B5%2A0=0%3C180\"   DISABLED_style= \"max-width:90%; height:auto;\" > so it is part of the solution, along with all the other points on the same side of the line.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Not all lines/inequalities are that easy to solve, but<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B2y%3C=90 ALIGN=MIDDLE  ALT=\"5x%2B2y%3C=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B3y%3C=240 ALIGN=MIDDLE  ALT=\"8x%2B3y%3C=240\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be solved in the same manner.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%285x%2B2y+=+90%29 ALIGN=MIDDLE  ALT=\"red%285x%2B2y+=+90%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > passes through (18,0) and (0,45);<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=green%288x%2B3y+=+240%29 ALIGN=MIDDLE  ALT=\"green%288x%2B3y+=+240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > passes through (30,0) and (0,80), and<BR>\n" );
document.write( "point (0,0) is part of the solution to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B2y%3C=90 ALIGN=MIDDLE  ALT=\"5x%2B2y%3C=90\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B3y%3C=240 ALIGN=MIDDLE  ALT=\"8x%2B3y%3C=240\"   DISABLED_style= \"max-width:90%; height:auto;\" > too.<BR>\n" );
document.write( "Very conveniently, (0,0) is part of the solution to all 3 inequalities, so it is part of the solution to the system.<BR>\n" );
document.write( " <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28300%2C300%2C-15%2C135%2C-15%2C135%2C%0D%0Agrid%281%29%2C%0D%0Ablue%28circle%280%2C36%2C2%29%29%2Cblue%28circle%2860%2C0%2C2%29%29%2C%0D%0Ablue%28line%28-20%2C48%2C90%2C-18%29%29%2C%0D%0Ared%28circle%280%2C45%2C2%29%29%2Cred%28circle%2818%2C0%2C2%29%29%2C%0D%0Ared%28line%28-30%2C120%2C40%2C-55%29%29%2C%0D%0Agreen%28circle%280%2C80%2C2%29%29%2Cgreen%28circle%2830%2C0%2C2%29%29%2C%0D%0Agreen%28line%28-20%2C133.3%2C40%2C-26.7%29%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The 3 lines seemed so very easy to draw, and it was easy to see that the solution is the space to the left and below all 3 of the lines.<BR>\n" );
document.write( "Unfortunately, the drawing that was so easy does not properly show the solution.<BR>\n" );
document.write( "It does not show where the green and red lines intersect.<BR>\n" );
document.write( "The scale needs to be changed, and the points used to draw the line will not be very helpful with the new scale.<BR>\n" );
document.write( "I had to get a better way to calculate points on the lines:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3x%2B5y=180 ALIGN=MIDDLE  ALT=\"3x%2B5y=180\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-0.6x%2B36 ALIGN=MIDDLE  ALT=\"y=-0.6x%2B36\"   DISABLED_style= \"max-width:90%; height:auto;\" > ;<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x%2B2y+=+90 ALIGN=MIDDLE  ALT=\"5x%2B2y+=+90\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-2.5x%2B45 ALIGN=MIDDLE  ALT=\"y=-2.5x%2B45\"   DISABLED_style= \"max-width:90%; height:auto;\" > and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8x%2B3y+=+240 ALIGN=MIDDLE  ALT=\"8x%2B3y+=+240\"   DISABLED_style= \"max-width:90%; height:auto;\" > <--> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-%288%2F3%29x%2B80 ALIGN=MIDDLE  ALT=\"y=-%288%2F3%29x%2B80\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "I also had to solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%285x%2B2y+=+90%2C8x%2B3y+=+240%29 ALIGN=MIDDLE  ALT=\"system%285x%2B2y+=+90%2C8x%2B3y+=+240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to find that those two lines intersect at (210,-480), to figure out how far to extend the scale.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=drawing%28600%2C300%2C-50%2C450%2C-900%2C100%2C%0D%0Agrid%280%29%2C%0D%0Aline%28-50%2C66%2C4.73%2C33.2%29%2Cblue%28line%284.73%2C33.2%2C450%2C-234%29%29%2C%0D%0Ared%28line%28-30%2C120%2C4.73%2C33.2%29%29%2Cline%284.73%2C33.2%2C210%2C-480%29%2Cred%28line%28210%2C-480%2C390%2C-930%29%29%2C%0D%0Agreen%28line%28-15%2C120%2C210%2C-480%29%29%2Cline%28210%2C-480%2C390%2C-960%29%2C%0D%0Aline%28-50%2C50%2C-23.33%2C50%29%2Cline%28-50%2C20%2C26.7%2C20%29%2Cline%28-50%2C-20%2C26%2C-20%29%2C%0D%0Aline%28-50%2C-60%2C42%2C-60%29%2Cline%28-50%2C-100%2C58%2C-100%29%2Cline%28-50%2C-140%2C74%2C-140%29%2C%0D%0Aline%28-50%2C-180%2C90%2C-180%29%2Cline%28-50%2C-220%2C106%2C-220%29%2Cline%28-50%2C-260%2C122%2C-260%29%2C%0D%0Aline%28-50%2C-300%2C138%2C-300%29%2Cline%28-50%2C-340%2C154%2C-340%29%2Cline%28-50%2C-380%2C170%2C-380%29%2C%0D%0Aline%28-50%2C-420%2C186%2C-420%29%2Cline%28-50%2C-460%2C202%2C-460%29%2Cline%28-50%2C-500%2C217.5%2C-500%29%2C%0D%0Aline%28-50%2C-540%2C232.5%2C-540%29%2Cline%28-50%2C-580%2C247.5%2C-580%29%2Cline%28-50%2C-620%2C262.5%2C-620%29%2C%0D%0Aline%28-50%2C-660%2C277.5%2C-660%29%2Cline%28-50%2C-700%2C292.5%2C-700%29%2Cline%28-50%2C-740%2C307.5%2C-740%29%2C%0D%0Aline%28-50%2C-780%2C322.5%2C-780%29%2Cline%28-50%2C-820%2C337.5%2C-820%29%2Cline%28-50%2C-860%2C352.5%2C-860%29%0D%0A%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > The solution is the shaded area, including the blackened parts of the colored lines.<BR>\n" );
document.write( "On those blackened parts of the colored lines, either<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2B5y=180%2C5x%2B2y%3C=90%2C8x%2B3y%3C=240%29 ALIGN=MIDDLE  ALT=\"system%283x%2B5y=180%2C5x%2B2y%3C=90%2C8x%2B3y%3C=240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2B5y%3C=180%2C5x%2B2y=90%2C8x%2B3y%3C=240%29 ALIGN=MIDDLE  ALT=\"system%283x%2B5y%3C=180%2C5x%2B2y=90%2C8x%2B3y%3C=240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%283x%2B5y%3C=180%2C5x%2B2y%3C=90%2C8x%2B3y=240%29 ALIGN=MIDDLE  ALT=\"system%283x%2B5y%3C=180%2C5x%2B2y%3C=90%2C8x%2B3y=240%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
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