document.write( "Question 1207825: use graphical methods and the corner point theorem to maximize z=x+5y subject to 3x+2y<12
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Algebra.Com's Answer #845950 by Edwin McCravy(20060) $\"\"$ $\"About$

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document.write( "What she means is that you should have ≤, not <, and ≥, not >.\r\n" );
document.write( "Otherwise your inequalities will not include the boundary lines,\r\n" );
document.write( "and will have no maximum (or minimum) solutions.\r\n" );
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document.write( "Maximize z=x+5y \r\n" );
document.write( "subject to \r\n" );
document.write( "3x+2y≤12  \r\n" );
document.write( "2x+y≤7  \r\n" );
document.write( "x≥0, y≥0\r\n" );
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document.write( "Here is the first quadrant graphs of the lines whose equations\r\n" );
document.write( "are like the constraint inequalities with equal signs instead of\r\n" );
document.write( "the inequality symbols, 3x+2y≤12  and 2x+y≤7 . Use the origin \r\n" );
document.write( "(x,y) = (0,0) as a test point to find which side of each line the\r\n" );
document.write( "shading will appear on.\r\n" );
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document.write( "Testing 3x+2y≤12: 3(0)+2(0)≤12 is true so we shade the side of that\r\n" );
document.write( "line which the origin is on, which is underneath the line:\r\n" );
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document.write( "Testing 2x+y≤7: 2(0)+(0)≤7 is true so we shade the side of that\r\n" );
document.write( "line which the origin is on, which is also underneath the line.\r\n" );
document.write( "so we shade under BOTH lines.\r\n" );
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document.write( "The feasible region is the shaded area.\r\n" );
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document.write( "We find all the corner points. by finding the x and y intercepts of both\r\n" );
document.write( "lines and their point of intersection.  \r\n" );
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document.write( "The y-intercept of \r\n" );
document.write( "3x+2y=12 \r\n" );
document.write( "is found by substituting x=0 and solving for y,\r\n" );
document.write( "we get y-intercept for the red line is (0,6)\r\n" );
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document.write( "The x-intercept of \r\n" );
document.write( "2x+y=7 \r\n" );
document.write( "is found by substituting y=0 and solving for x,\r\n" );
document.write( "we get x-intercept for the blue line is (3.5,0)  \r\n" );
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document.write( "We get the corner point which is the intersection of\r\n" );
document.write( "the two lines:\r\n" );
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document.write( "Solve that by substitution or elimination and get (2,3)\r\n" );
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document.write( "Now we find the value of the objective function\r\n" );
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document.write( "z = x+5y at each of the 4 corner points:\r\n" );
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document.write( "At corner point (x,y) = (0,0),  z = x+5y = 0+5(0) = 0.\r\n" );
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document.write( "At corner point (x,y) = (0,6),  z = x+5y = 0+5(6) = 30.\r\n" );
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document.write( "At corner point (x,y) = (2,3),  z = x+5y = 2+5(3) = 17.\r\n" );
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document.write( "At corner point (x,y) = (3.5,0),  z = x+5y = 3.5+5(0) = 3.5.\r\n" );
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document.write( "So z has a maximum value of 30 when x=0 and y=6.\r\n" );
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document.write( "Edwin

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