Question 1187741
<pre>

Graph all the boundary lines' equations using = signs for ≤

7x + 9y = 63   <--that's a slanted line with intercepts (0,7) and (9,0)
y = 0          <--that's the x-axis
y = 4          <--that's a horizontal line thru 4 on the y-axis 
x = 0          <--that's the y-axis   
x = 7          <--that's a vertical line through 7 on the x axis

{{{drawing(400,8800/27,-.9,9.9,-.9,7.9, 
graph(400,8800/27,-.9,9.9,-.9,7.9), line(-9,14,18,-7),

line(-3,4,12,4),line(7,-12,7,12),line(0,-3,0,12),line(-3,0,12,0)

 )}}}

Consider 0 ≤ y ≤ 4 as y ≥ 0 and y ≤ 4

Consider 0 ≤ x ≤ 7 as x ≥ 0 and x ≤ 7

We decide which sides of all these lines to shade:

7x + 9y ≤ 63   <--shade below the slanted line because (0,0) satisfies it
y ≥ 0          <--shade above the x-axis (y greater than 0)
y ≤ 4          <--shade below the horizontal line (less than 4) 
x ≥ 0          <--shade to the right of the y-axis (greater than 0)  
x ≤ 7          <--shade left of the vertical line (less than 7)

{{{drawing(400,8800/27,-.9,9.9,-.9,7.9, 
graph(400,8800/27,-.9,9.9,-.9,7.9), line(-9,14,18,-7),
line(-3,4,12,4),line(7,-12,7,12),line(0,-3,0,12),line(-3,0,12,0),
green(
line(0,4,0,0),line(0.07,4,0.07,0),line(0.14,4,0.14,0),line(0.14,4,0.14,0),
line(0.21,4,0.21,0),line(0.28,4,0.28,0),line(0.35,4,0.35,0),line(0.35,4,0.35,0),
line(0.42,4,0.42,0),line(0.49,4,0.49,0),line(0.56,4,0.56,0),line(0.56,4,0.56,0),
line(0.63,4,0.63,0),line(0.7,4,0.7,0),line(0.77,4,0.77,0),line(0.77,4,0.77,0),
line(0.84,4,0.84,0),line(0.91,4,0.91,0),line(0.98,4,0.98,0),line(0.98,4,0.98,0),
line(1.05,4,1.05,0),line(1.12,4,1.12,0),line(1.19,4,1.19,0),line(1.19,4,1.19,0),
line(1.26,4,1.26,0),line(1.33,4,1.33,0),line(1.4,4,1.4,0),line(1.4,4,1.4,0),
line(1.47,4,1.47,0),line(1.54,4,1.54,0),line(1.61,4,1.61,0),line(1.61,4,1.61,0),
line(1.68,4,1.68,0),line(1.75,4,1.75,0),line(1.82,4,1.82,0),line(1.82,4,1.82,0),
line(1.89,4,1.89,0),line(1.96,4,1.96,0),line(2.03,4,2.03,0),line(2.03,4,2.03,0),
line(2.1,4,2.1,0),line(2.17,4,2.17,0),line(2.24,4,2.24,0),line(2.24,4,2.24,0),
line(2.31,4,2.31,0),line(2.38,4,2.38,0),line(2.45,4,2.45,0),line(2.45,4,2.45,0),
line(2.52,4,2.52,0),line(2.59,4,2.59,0),line(2.66,4,2.66,0),line(2.66,4,2.66,0),
line(2.73,4,2.73,0),line(2.8,4,2.8,0),line(2.87,4,2.87,0),line(2.87,4,2.87,0),
line(2.94,4,2.94,0),line(3.01,4,3.01,0),line(3.08,4,3.08,0),line(3.08,4,3.08,0),
line(3.15,4,3.15,0),line(3.22,4,3.22,0),line(3.29,4,3.29,0),line(3.29,4,3.29,0),
line(3.36,4,3.36,0),line(3.43,4,3.43,0),line(3.5,4,3.5,0),line(3.5,4,3.5,0),
line(3.57,4,3.57,0),line(3.64,4,3.64,0),line(3.71,4,3.71,0),line(3.71,4,3.71,0),
line(3.78,4,3.78,0),line(3.85,4,3.85,0),

line(3.86,0,3.86,3.99777778),line(3.93,0,3.93,3.94333333),line(4.0,0,4.0,3.88888889),line(4.0,0,4.0,3.88888889),
line(4.07,0,4.07,3.83444444),line(4.14,0,4.14,3.78),line(4.21,0,4.21,3.72555556),line(4.21,0,4.21,3.72555556),
line(4.28,0,4.28,3.67111111),line(4.35,0,4.35,3.61666667),line(4.42,0,4.42,3.56222222),line(4.42,0,4.42,3.56222222),
line(4.49,0,4.49,3.50777778),line(4.56,0,4.56,3.45333333),line(4.63,0,4.63,3.39888889),line(4.63,0,4.63,3.39888889),
line(4.7,0,4.7,3.34444444),line(4.77,0,4.77,3.29),line(4.84,0,4.84,3.23555556),line(4.84,0,4.84,3.23555556),
line(4.91,0,4.91,3.18111111),line(4.98,0,4.98,3.12666667),line(5.05,0,5.05,3.07222222),line(5.05,0,5.05,3.07222222),
line(5.12,0,5.12,3.01777778),line(5.19,0,5.19,2.96333333),line(5.26,0,5.26,2.90888889),line(5.26,0,5.26,2.90888889),
line(5.33,0,5.33,2.85444444),line(5.4,0,5.4,2.8),line(5.47,0,5.47,2.74555556),line(5.47,0,5.47,2.74555556),
line(5.54,0,5.54,2.69111111),line(5.61,0,5.61,2.63666667),line(5.68,0,5.68,2.58222222),line(5.68,0,5.68,2.58222222),
line(5.75,0,5.75,2.52777778),line(5.82,0,5.82,2.47333333),line(5.89,0,5.89,2.41888889),line(5.89,0,5.89,2.41888889),
line(5.96,0,5.96,2.36444444),line(6.03,0,6.03,2.31),line(6.1,0,6.1,2.25555556),line(6.1,0,6.1,2.25555556),
line(6.17,0,6.17,2.20111111),line(6.24,0,6.24,2.14666667),line(6.31,0,6.31,2.09222222),line(6.31,0,6.31,2.09222222),
line(6.38,0,6.38,2.03777778),line(6.45,0,6.45,1.98333333),line(6.52,0,6.52,1.92888889),line(6.52,0,6.52,1.92888889),
line(6.59,0,6.59,1.87444444),line(6.66,0,6.66,1.82),line(6.73,0,6.73,1.76555556),line(6.73,0,6.73,1.76555556),
line(6.8,0,6.8,1.71111111),line(6.87,0,6.87,1.65666667),line(6.94,0,6.94,1.60222222),line(6.94,0,6.94,1.60222222)), line(0,-3,0,12),line(-3,0,12,0)

 )}}}

We find all the corner points of the feasible region.
We already have three of them, (0,0), (0,4), and (7,0).

We find the other two by 

substituting y=4 into the slanted line's equation and solving for x, and
substituting x=7 into the slanted line's equation and solving for y.

     7x + 9y​ = 63            7x + 9y = 63   
   7x + 9(4) = 63          7(7) + 9y = 63
     7x + 36 = 63            49 + 9y = 63 
          7x = 27                 9y = 14
           x = 27/7                y = 14/9

So the other two corner points are (27/7, 4) and (7, 14/9)

{{{drawing(400,8800/27,-.9,9.9,-.9,7.9, 
graph(400,8800/27,-.9,9.9,-.9,7.9), line(-9,14,18,-7),
line(-3,4,12,4),line(7,-12,7,12),line(0,-3,0,12),line(-3,0,12,0),
green(
line(0,4,0,0),line(0.07,4,0.07,0),line(0.14,4,0.14,0),line(0.14,4,0.14,0),
line(0.21,4,0.21,0),line(0.28,4,0.28,0),line(0.35,4,0.35,0),line(0.35,4,0.35,0),
line(0.42,4,0.42,0),line(0.49,4,0.49,0),line(0.56,4,0.56,0),line(0.56,4,0.56,0),
line(0.63,4,0.63,0),line(0.7,4,0.7,0),line(0.77,4,0.77,0),line(0.77,4,0.77,0),
line(0.84,4,0.84,0),line(0.91,4,0.91,0),line(0.98,4,0.98,0),line(0.98,4,0.98,0),
line(1.05,4,1.05,0),line(1.12,4,1.12,0),line(1.19,4,1.19,0),line(1.19,4,1.19,0),
line(1.26,4,1.26,0),line(1.33,4,1.33,0),line(1.4,4,1.4,0),line(1.4,4,1.4,0),
line(1.47,4,1.47,0),line(1.54,4,1.54,0),line(1.61,4,1.61,0),line(1.61,4,1.61,0),
line(1.68,4,1.68,0),line(1.75,4,1.75,0),line(1.82,4,1.82,0),line(1.82,4,1.82,0),
line(1.89,4,1.89,0),line(1.96,4,1.96,0),line(2.03,4,2.03,0),line(2.03,4,2.03,0),
line(2.1,4,2.1,0),line(2.17,4,2.17,0),line(2.24,4,2.24,0),line(2.24,4,2.24,0),
line(2.31,4,2.31,0),line(2.38,4,2.38,0),line(2.45,4,2.45,0),line(2.45,4,2.45,0),
line(2.52,4,2.52,0),line(2.59,4,2.59,0),line(2.66,4,2.66,0),line(2.66,4,2.66,0),
line(2.73,4,2.73,0),line(2.8,4,2.8,0),line(2.87,4,2.87,0),line(2.87,4,2.87,0),
line(2.94,4,2.94,0),line(3.01,4,3.01,0),line(3.08,4,3.08,0),line(3.08,4,3.08,0),
line(3.15,4,3.15,0),line(3.22,4,3.22,0),line(3.29,4,3.29,0),line(3.29,4,3.29,0),
line(3.36,4,3.36,0),line(3.43,4,3.43,0),line(3.5,4,3.5,0),line(3.5,4,3.5,0),
line(3.57,4,3.57,0),line(3.64,4,3.64,0),line(3.71,4,3.71,0),line(3.71,4,3.71,0),
line(3.78,4,3.78,0),line(3.85,4,3.85,0),

line(3.86,0,3.86,3.99777778),line(3.93,0,3.93,3.94333333),line(4.0,0,4.0,3.88888889),line(4.0,0,4.0,3.88888889),
line(4.07,0,4.07,3.83444444),line(4.14,0,4.14,3.78),line(4.21,0,4.21,3.72555556),line(4.21,0,4.21,3.72555556),
line(4.28,0,4.28,3.67111111),line(4.35,0,4.35,3.61666667),line(4.42,0,4.42,3.56222222),line(4.42,0,4.42,3.56222222),
line(4.49,0,4.49,3.50777778),line(4.56,0,4.56,3.45333333),line(4.63,0,4.63,3.39888889),line(4.63,0,4.63,3.39888889),
line(4.7,0,4.7,3.34444444),line(4.77,0,4.77,3.29),line(4.84,0,4.84,3.23555556),line(4.84,0,4.84,3.23555556),
line(4.91,0,4.91,3.18111111),line(4.98,0,4.98,3.12666667),line(5.05,0,5.05,3.07222222),line(5.05,0,5.05,3.07222222),
line(5.12,0,5.12,3.01777778),line(5.19,0,5.19,2.96333333),line(5.26,0,5.26,2.90888889),line(5.26,0,5.26,2.90888889),
line(5.33,0,5.33,2.85444444),line(5.4,0,5.4,2.8),line(5.47,0,5.47,2.74555556),line(5.47,0,5.47,2.74555556),
line(5.54,0,5.54,2.69111111),line(5.61,0,5.61,2.63666667),line(5.68,0,5.68,2.58222222),line(5.68,0,5.68,2.58222222),
line(5.75,0,5.75,2.52777778),line(5.82,0,5.82,2.47333333),line(5.89,0,5.89,2.41888889),line(5.89,0,5.89,2.41888889),
line(5.96,0,5.96,2.36444444),line(6.03,0,6.03,2.31),line(6.1,0,6.1,2.25555556),line(6.1,0,6.1,2.25555556),
line(6.17,0,6.17,2.20111111),line(6.24,0,6.24,2.14666667),line(6.31,0,6.31,2.09222222),line(6.31,0,6.31,2.09222222),
line(6.38,0,6.38,2.03777778),line(6.45,0,6.45,1.98333333),line(6.52,0,6.52,1.92888889),line(6.52,0,6.52,1.92888889),
line(6.59,0,6.59,1.87444444),line(6.66,0,6.66,1.82),line(6.73,0,6.73,1.76555556),line(6.73,0,6.73,1.76555556),
line(6.8,0,6.8,1.71111111),line(6.87,0,6.87,1.65666667),line(6.94,0,6.94,1.60222222),line(6.94,0,6.94,1.60222222)), line(0,-3,0,12),line(-3,0,12,0),
locate(0,.5,"(0,0)"), locate(7,.5,"(7,0)"), locate(7,14/9+.5,"(7,14/9)"),
locate(0,4.5,"(0,4)"),locate(27/7,4.5,"(27/4,4)")



 )}}}

Now we find the maximum and minimum values by substituting each corner point
into P = 17x - 4y​ + 61.

Corner point    P = 17x - 4y + 61 = value
  (0,0)         P = 17(0)-4(0)+61 = 0-0+61 = 61
  (7,0)         P = 17(7)-4(0)+61 = 119-0+61 = 180
  (7,14/9)      P = 17(7)-4(14/9)+61 = 119-56/9+61 = 1564/9 = 173 7/9
  (27/4,4)      P = 17(27/4)-4(4)+61 = 459/4-16+61 = 639/4 = 159 3/4
  (0,4)         P = 17(0)-4(4)+61 = 0-16+61 = 45

So: 
The maximum value is P = 180 when x = 7 and y = 0.
The minimum value is P = 45 when x = 0 and y = 4.

Edwin</pre>