document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198952>Question 1198952</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Maximize P=10x+40y<BR>\n" );
document.write( "2x+y≤100<BR>\n" );
document.write( "80x+10y≤500<BR>\n" );
document.write( "30x+10y≤2000<BR>\n" );
document.write( "X≥0<BR>\n" );
document.write( "y≥0 \r<BR>\n" );
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document.write( "Use the simplex method to solve the problem. Use s, t, and u as your slack variables for the first, second, and third inequalities respectively. Use the final simplex tableau to identify the values below.<BR>\n" );
document.write( "X= </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832655 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=832655\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "I recommend checking out this page to help refresh your memory how the Simplex Method works.<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://people.richland.edu/james/ictcm/2006/simplex.html\">https://people.richland.edu/james/ictcm/2006/simplex.html</a>\r<BR>\n" );
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document.write( "We wish to maximize P = 10x+40y\r<BR>\n" );
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document.write( "The constraints are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2By+%3C=+100%2C80x%2B10y%3C=500%2C30x%2B10y%3C=2000%2Cx%3E=0%2Cy%3E=0%29 ALIGN=MIDDLE  ALT=\"system%282x%2By+%3C=+100%2C80x%2B10y%3C=500%2C30x%2B10y%3C=2000%2Cx%3E=0%2Cy%3E=0%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Focus on this subset of constraints<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2By+%3C=+100%2C80x%2B10y%3C=500%2C30x%2B10y%3C=2000%29 ALIGN=MIDDLE  ALT=\"system%282x%2By+%3C=+100%2C80x%2B10y%3C=500%2C30x%2B10y%3C=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Convert to their corresponding equation format<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2By%2Bs+=+100%2C80x%2B10y%2Bt=500%2C30x%2B10y%2Bu=2000%29 ALIGN=MIDDLE  ALT=\"system%282x%2By%2Bs+=+100%2C80x%2B10y%2Bt=500%2C30x%2B10y%2Bu=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where s,t,u are the slack variables mentioned. They are nonnegative real numbers.\r<BR>\n" );
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document.write( "Let's have the slack variables present in each equation. <BR>\n" );
document.write( "A coefficient of '1' means the slack variable is actually present in that particular equation.<BR>\n" );
document.write( "A coefficient of '0' means it is not present.\r<BR>\n" );
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document.write( "This system<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2By%2Bs+=+100%2C80x%2B10y%2Bt=500%2C30x%2B10y%2Bu=2000%29 ALIGN=MIDDLE  ALT=\"system%282x%2By%2Bs+=+100%2C80x%2B10y%2Bt=500%2C30x%2B10y%2Bu=2000%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "is equivalent to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=system%282x%2B1y%2B1s%2B0t%2B0u+=+100%2C80x%2B10y%2B0s%2B1t%2B0u=500%2C30x%2B10y%2B0s%2B0t%2B1u=2000%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Then an equation like<BR>\n" );
document.write( "2x+1y+1s+0t+0u = 100<BR>\n" );
document.write( "will have the coefficients: 2,1,1,0,0<BR>\n" );
document.write( "which will make up part of row1 in the simplex tableau.\r<BR>\n" );
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document.write( "This is our starting simplex tableau<BR>\n" );
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document.write( "where C represents the right-hand-side constants of each equation mentioned above.\r<BR>\n" );
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document.write( "The bottom row represents the equation -10x-40y+P = 0<BR>\n" );
document.write( "This is a rephrasing of P = 10x+40y where everything is to one side. \r<BR>\n" );
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document.write( "Write 0's in the slack entries of the bottom row<BR>\n" );
document.write( "This is because <BR>\n" );
document.write( "-10x-40y+P = 0<BR>\n" );
document.write( "is the same as<BR>\n" );
document.write( "-10x-40y+0s+0t+0u+1P = 0\r<BR>\n" );
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document.write( "Currently this tableau is not optimized because we have negative items in the bottom row.<BR>\n" );
document.write( "Select the negative item furthest from zero, which is -40. Highlight or circle this entire column. This is the current pivot column.\r<BR>\n" );
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document.write( "To find the pivot row, we'll need to divide each C value over each item in that pivot column.<BR>\n" );
document.write( "row1: 100/1 = 100<BR>\n" );
document.write( "row2: 500/10 = 50<BR>\n" );
document.write( "row3: 2000/10 = 200\r<BR>\n" );
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document.write( "The smallest ratio will point us to the pivot row. In this case, that would be row 2.\r<BR>\n" );
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document.write( "The item \"10\" in this pivot column and pivot row must be made into a \"1\".<BR>\n" );
document.write( "We do this by multiplying everything in this row by 1/10, i.e. divide everything in this row by 10.\r<BR>\n" );
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document.write( "I'll write the notation (1/10)*R2 --> R2 to indicate \"multiply everything in row 2 (aka R2) by 1/10, then store the result in R2\".\r<BR>\n" );
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document.write( "This is what we get after that row operation<BR>\n" );
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document.write( "Now we must zero out everything but this '1' in that column. <BR>\n" );
document.write( "This pivot item is highlighted in yellow.<BR>\n" );
document.write( "We are clearing out this column so to speak.<BR>\n" );
document.write( "Use row operations to get this done.\r<BR>\n" );
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document.write( "Something like R1-R2 --> R1 means we subtract the corresponding entries of row1 and row2 in that order, then store the result in row1.<BR>\n" );
document.write( "For more information, check out the concept of \"row reduction\".\r<BR>\n" );
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document.write( "This is the result of clearing out the pivot column mentioned<BR>\n" );
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document.write( "The row operations performed are shown on the right.\r<BR>\n" );
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document.write( "At this point, there aren't any negative items in the bottom row. Therefore, this tableau is optimized.\r<BR>\n" );
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document.write( "The variables y, s, u, and P are considered basic variables. <BR>\n" );
document.write( "This is because they contain zeros in the non-pivot entries for the given column. These columns have been \"cleared out\" so to speak.<BR>\n" );
document.write( "Refer to the link I posted above.\r<BR>\n" );
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document.write( "The nonbasic variables are x and t, as they don't have such zeros.<BR>\n" );
document.write( "Nonbasic variables will take on the value of zero.<BR>\n" );
document.write( "Therefore: x = 0 and t = 0\r<BR>\n" );
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document.write( "Going back to the basic variables, we have <BR>\n" );
document.write( "y = 50<BR>\n" );
document.write( "s = 50<BR>\n" );
document.write( "u = 1500<BR>\n" );
document.write( "P = 2000<BR>\n" );
document.write( "Each value mentioned is in the column C at the far right side. <BR>\n" );
document.write( "We have u = 1500 for instance because of the '1' in that corresponding row.\r<BR>\n" );
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document.write( "To put everything together:<BR>\n" );
document.write( "x = 0, y = 50<BR>\n" );
document.write( "s = 50, t = 0, u = 1500<BR>\n" );
document.write( "P = 2000\r<BR>\n" );
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document.write( "The nonzero slack values for variable s and variable u indicate that the point (x,y) = (0,50) is not on the boundaries of 2x+y≤100 and 30x+10y≤2000 respectively.\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "The max is P = 2000<BR>\n" );
document.write( "It occurs when x = 0 and y = 50\r<BR>\n" );
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document.write( "You can use graphing to confirm the answer.<BR>\n" );
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