document.write( "Question 1187741: Find the maximum value and the minimum value of the function and the values of x and y for which they occur. P = 17x - 4y​ + 61, subject to 7x + 9y​ ≤ 63, 0 ≤ y ≤ 4, and 0 ≤ x ≤ 7. \n" ); document.write( "
Algebra.Com's Answer #818778 by Edwin McCravy(20081)\"\" \"About 
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document.write( "Graph all the boundary lines' equations using = signs for ≤\r\n" );
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document.write( "7x + 9y = 63   <--that's a slanted line with intercepts (0,7) and (9,0)\r\n" );
document.write( "y = 0          <--that's the x-axis\r\n" );
document.write( "y = 4          <--that's a horizontal line thru 4 on the y-axis \r\n" );
document.write( "x = 0          <--that's the y-axis   \r\n" );
document.write( "x = 7          <--that's a vertical line through 7 on the x axis\r\n" );
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document.write( "Consider 0 ≤ y ≤ 4 as y ≥ 0 and y ≤ 4\r\n" );
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document.write( "Consider 0 ≤ x ≤ 7 as x ≥ 0 and x ≤ 7\r\n" );
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document.write( "We decide which sides of all these lines to shade:\r\n" );
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document.write( "7x + 9y ≤ 63   <--shade below the slanted line because (0,0) satisfies it\r\n" );
document.write( "y ≥ 0          <--shade above the x-axis (y greater than 0)\r\n" );
document.write( "y ≤ 4          <--shade below the horizontal line (less than 4) \r\n" );
document.write( "x ≥ 0          <--shade to the right of the y-axis (greater than 0)  \r\n" );
document.write( "x ≤ 7          <--shade left of the vertical line (less than 7)\r\n" );
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document.write( "We find all the corner points of the feasible region.\r\n" );
document.write( "We already have three of them, (0,0), (0,4), and (7,0).\r\n" );
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document.write( "We find the other two by \r\n" );
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document.write( "substituting y=4 into the slanted line's equation and solving for x, and\r\n" );
document.write( "substituting x=7 into the slanted line's equation and solving for y.\r\n" );
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document.write( "     7x + 9y​ = 63            7x + 9y = 63   \r\n" );
document.write( "   7x + 9(4) = 63          7(7) + 9y = 63\r\n" );
document.write( "     7x + 36 = 63            49 + 9y = 63 \r\n" );
document.write( "          7x = 27                 9y = 14\r\n" );
document.write( "           x = 27/7                y = 14/9\r\n" );
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document.write( "So the other two corner points are (27/7, 4) and (7, 14/9)\r\n" );
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document.write( "Now we find the maximum and minimum values by substituting each corner point\r\n" );
document.write( "into P = 17x - 4y​ + 61.\r\n" );
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document.write( "Corner point    P = 17x - 4y + 61 = value\r\n" );
document.write( "  (0,0)         P = 17(0)-4(0)+61 = 0-0+61 = 61\r\n" );
document.write( "  (7,0)         P = 17(7)-4(0)+61 = 119-0+61 = 180\r\n" );
document.write( "  (7,14/9)      P = 17(7)-4(14/9)+61 = 119-56/9+61 = 1564/9 = 173 7/9\r\n" );
document.write( "  (27/4,4)      P = 17(27/4)-4(4)+61 = 459/4-16+61 = 639/4 = 159 3/4\r\n" );
document.write( "  (0,4)         P = 17(0)-4(4)+61 = 0-16+61 = 45\r\n" );
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document.write( "So: \r\n" );
document.write( "The maximum value is P = 180 when x = 7 and y = 0.\r\n" );
document.write( "The minimum value is P = 45 when x = 0 and y = 4.\r\n" );
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document.write( "Edwin

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