document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=468975>Question 468975</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How do I minimize: z=13x+9y<BR>\n" );
document.write( "subject to: 3x+4y>=21; 8x+4y>=32; x>=0; y>=0 ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #321766 by </B><A HREF=/tutors/aboutme.mpl?userid=ccs2011><B>ccs2011(207)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ccs2011><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=ccs2011><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=321766\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> First graph your bounds which are defined as:<BR>\n" );
document.write( "3x+4y>=21; 8x+4y>=32; x>=0; y>=0<BR>\n" );
document.write( "Each of these represents an area above a line.<BR>\n" );
document.write( "Determine slope and y_intercept of each line by rewriting in slope-intercept form.<BR>\n" );
document.write( "y = mx + b<BR>\n" );
document.write( "x = 0 and y =0 are just the axis of the graph<BR>\n" );
document.write( "Rewrite 3x+4y>=21 by solving for y:<BR>\n" );
document.write( "Subtract 3x on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y+%3E=+-3x+%2B+21 ALIGN=MIDDLE  ALT=\"4y+%3E=+-3x+%2B+21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 4 on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+%28-3%2F4%29x+%2B+%2821%2F4%29 ALIGN=MIDDLE  ALT=\"y+%3E=+%28-3%2F4%29x+%2B+%2821%2F4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Rewrite 8x+4y>=32 by solving for y:<BR>\n" );
document.write( "Subtract 8x on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4y+%3E=+-8x+%2B+32 ALIGN=MIDDLE  ALT=\"4y+%3E=+-8x+%2B+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 4 on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+%3E=+-2x+%2B+8 ALIGN=MIDDLE  ALT=\"y+%3E=+-2x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now graph these lines<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28300%2C300%2C0%2C9%2C0%2C9%2C-.75x+%2B+5.25%2C+-2x+%2B+8%29 ALIGN=MIDDLE  ALT=\"graph%28300%2C300%2C0%2C9%2C0%2C9%2C-.75x+%2B+5.25%2C+-2x+%2B+8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "All the possible (x,y) points are in the region that is on or above both lines<BR>\n" );
document.write( "Lets look at z=13x+9y<BR>\n" );
document.write( "The goal is to minimize z, there is a direct variation with z to x and y so to decrease z you must decrease x and/or y.<BR>\n" );
document.write( "If you pick any point in the possible region and move it straight down until you hit one of the 2 lines, then you kept the same x but decreased y thus decreasing z.<BR>\n" );
document.write( "From that logic the smallest z values will be on the line.<BR>\n" );
document.write( "Typically the smallest z values will be on the endpoints.<BR>\n" );
document.write( "There are 3 end-points, x-intercept (7,0) y-intercept (0,8) and point where lines intersect.<BR>\n" );
document.write( "To find where they intersect, set 2 equations equal to each other.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-3%2F4%29x+%2B+%2821%2F4%29+=+-2x+%2B+8 ALIGN=MIDDLE  ALT=\"%28-3%2F4%29x+%2B+%2821%2F4%29+=+-2x+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Eliminate fractions by multiplying everything by 4<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-3x+%2B+21+=+-8x+%2B+32 ALIGN=MIDDLE  ALT=\"-3x+%2B+21+=+-8x+%2B+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Add 8x on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+%2B+21+=+32 ALIGN=MIDDLE  ALT=\"5x+%2B+21+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Subtract 21 on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5x+=+11 ALIGN=MIDDLE  ALT=\"5x+=+11\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 5 on both sides<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+11%2F5 ALIGN=MIDDLE  ALT=\"x+=+11%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substitute back into equation to find y<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+-2%2811%2F5%29+%2B+8 ALIGN=MIDDLE  ALT=\"y+=+-2%2811%2F5%29+%2B+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+%28-22%2F5%29+%2B+%2840%2F5%29+=+18%2F5 ALIGN=MIDDLE  ALT=\"y+=+%28-22%2F5%29+%2B+%2840%2F5%29+=+18%2F5\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Finally evaluate z at each end-point, smallest value will be minimum<BR>\n" );
document.write( "For (7,0): z = 13*7 = 91<BR>\n" );
document.write( "For (0,8): z = 9*8 = 72<BR>\n" );
document.write( "For (11/5,18/5): z = 13*(11/5) + 9*(18/5) = 305/5 = 61<BR>\n" );
document.write( "Therefore z has minimum of 61 at point(11/5,18/5)<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );