document.write( "Question 1098453: Use the graphic method to solve this problem\r
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Algebra.Com's Answer #712818 by ikleyn(52786) $\"\"$ $\"About$

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document.write( "The feasible region is presented in the plot below.\r\n" );
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document.write( "It is the region in the first quadrant above the given straight lines - the constraints.\r\n" );
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document.write( "The lines  2x+y = 30 (red),  2x+5y = 50 (green)  and  x+y = 20 (blue).\r\n" );
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document.write( "This region has vertices P1, P2, P3 and P4\r\n" );
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document.write( "   P1 = (0,30),    which is y-intercept of the red line 2x+y = 30;\r\n" );
document.write( "   P2 = (10,10),   which is the intersection point of the red line 2x+y = 30 and blue line x+y = 20;\r\n" );
document.write( "   P3 = (, ), which is the intersection point of the blue line x+y = 20 and green line 2x+5y = 50;\r\n" );
document.write( "   P4 = (25,0),    which is x-intercept of the green line 2x+5y = 50.\r\n" );
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document.write( "        The graphic Linear Programming method includes finding the intersection points as the solutions of given constraint equations; these \r\n" );
document.write( "        procedures are considered as of atomic size components, and it is assumed that the student makes them automatically without special discussions.\r\n" );
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document.write( "Further, the Linear Programming method enact you to calculate the values of the objective function  F(x,y) = 3x + 2y at the vertices\r\n" );
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document.write( "    at P1:  F( 0,30) = 3*0 + 2*30 = 60;\r\n" );
document.write( "    at P2:  F(10,10) = 3*10 + 2*10 = 50;\r\n" );
document.write( "    at P3:  F(,) =  = ;\r\n" );
document.write( "    at P4:  F(25,0) = 3*25 + 2*0 = 75.\r\n" );
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document.write( "Then the Linear Programming method STATES that the objective function in the feasible region achieves its minimum \r\n" );
document.write( "at the vertex; namely at that vertex, where its value is minimal.\r\n" );
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document.write( "From the list above, the objective function has the minimum at P2 = (10,10), and this point is the minimum of the objective function \r\n" );
document.write( "over the entire feasible region.\r\n" );
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document.write( "Thus the solution of the problem is the point  P2 = (10,10).\r\n" );
document.write( "It means that the objective function under the given constraints achieves its minimum at the point  x = 10,  y = 10.\r\n" );
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document.write( "The value of this minimum is  = 50.\r\n" );
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document.write( "Answer.  The objective function under the given constraints achieves its minimum at the point  x = 10,  y = 10.\r\n" );
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document.write( "          The value of this minimum is   = 50.\r\n" );
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