document.write( "Question 1148055: Find the maximum and minimum value of z = x+3y subject to the constraints
\n" ); document.write( "-x + 3y <= 6, x - 3y >= 6, and x+ y >= 6
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Algebra.Com's Answer #769433 by ikleyn(52788) $\"\"$ $\"About$

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document.write( "Let me, first, to rewrite your constraints in equivalent form\r\n" );
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document.write( "    y <=  2 +     (1)    (first  given constraint)\r\n" );
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document.write( "    y >= -2 +     (2)    (second given constraint) \r\n" );
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document.write( "    y >=  6 - x    (3)     (third  given constraint) \r\n" );
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document.write( "The plot is shown below.\r\n" );
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document.write( "    Plot  x-3y = -6  (red),  x-3y = 6  (green) and  x+y = 6 (blue).\r\n" );
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document.write( "So, your feasible area is the set of points between two parallel lines (red and green; constraint (1) and (2)), that are ABOVE blue line.\r\n" );
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document.write( "The given linear objective function  z = x + 3y,  OBVIOUSLY, HAS NO maximum\r\n" );
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document.write( "    (its value can be made arbitrary large).\r\n" );
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document.write( "It has a minimum value, instead.\r\n" );
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document.write( "This minimum value is its value in one of the two intersection (corner) points.\r\n" );
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document.write( "The intersection points are  (3,3)  and (6,0).\r\n" );
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document.write( "The values of the objective function  z = x + 3y  at these points are\r\n" );
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document.write( "    at (3,3)  z = 3 + 3*3 = 12\r\n" );
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document.write( "    at (0,6)  z = 6 + 3*0 = 6.\r\n" );
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document.write( "ANSWER.  The given objective function HAS NO maximum value.\r\n" );
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document.write( "         It has minimum value at the point (x,y) = (6,0).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved, answered, explained and completed.\r
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\n" ); document.write( "\n" ); document.write( "To see many other similar problems solved by the Linear Programming method and to learn how the method works, look into the lesson\r
\n" ); document.write( "\n" ); document.write( " - Solving minimax problems by the Linear Programming method \r
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\n" ); document.write( "\n" ); document.write( "Also, look into the solutions of other similar problems in the archive to this forum under the links\r
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\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Inequalities/Inequalities.faq.question.1142000.html \r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Sequences-and-series/Sequences-and-series.faq.question.1137172.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1136382.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Coordinate-system/Coordinate-system.faq.question.1134444.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Graphs/Graphs.faq.question.1131906.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1131043.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/word/finance/Money_Word_Problems.faq.question.1129285.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/playground/test.faq.question.1112482.html\r
\n" ); document.write( "\n" ); document.write( "https://www.algebra.com/algebra/homework/Finance/Finance.faq.question.1102103.html\r
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