Solve the following LP problem graphically

MAX 2X + 7Y
Subject to:
5X + 9Y >= 90
9X + 8Y <= 144
Y <= 8
X, Y >= 0

well do it.

graph the two lines in the first quadrant.

y = -(5/9)x + 10 must be above and right
points (0,10) and (18,0)

y = -(9/8)x + 18 must be below and left
points (0,18) and (16,0)

intersection at
(72*8/41 , -64*5/41 + 10)
which is about
(14,2)

at 0,18 --> 126
at (14,2) -> 42

so solution is at (0,18)
and is 2(0) + 7(18) = 126

To solve the given linear programming problem (LP) graphically, we need to plot the constraints and determine the feasible region. Then, we can find the corner points of the feasible region and evaluate the objective function at each corner point to determine the maximum value.

Let's start by plotting the constraints:

Constraint 1: 5X + 9Y >= 90
Rearrange the inequality: 9Y >= -5X + 90
Divide both sides by 9: Y >= (-5/9)X + 10

Constraint 2: 9X + 8Y <= 144
Rearrange the inequality: 8Y <= -9X + 144
Divide both sides by 8: Y <= (-9/8)X + 18

Constraint 3: Y <= 8 (no rearrangement necessary)

Now, let's plot these three constraints on a graph with X on the horizontal axis and Y on the vertical axis. Remember that X, Y >= 0 in this problem, so we only need to plot the positive region.

1. Plot the inequality Y >= (-5/9)X + 10 as a dashed line with a positive slope (start from the y-intercept of 10 and move downward as the slope indicates).

2. Plot the inequality Y <= (-9/8)X + 18 as a solid line with a negative slope (start from the y-intercept of 18 and move downward as the slope indicates).

3. Plot the line Y = 8 as a horizontal line at y = 8.

The feasible region is the area that satisfies all three constraints. It is the region below the dashed line, above the solid line, and below the line Y = 8.

Now, we need to find the corner points of the feasible region. These are the points where the lines intersect. In this case, there are three corner points:

1. The intersection of the dashed line and the line Y = 8.
2. The intersection of the dashed line and the solid line.
3. The intersection of the solid line and the line Y = 8.

To find the coordinates of each corner point, solve the simultaneous equations for each pair of intersecting lines.

After finding the corner points, substitute the coordinates into the objective function (2X + 7Y) to evaluate the objective function value at each corner point.

Finally, identify and select the corner point that gives the maximum objective function value. That point represents the optimal solution to the LP problem.

Please note that I have explained the graphical method to solve the LP problem, but you can also use mathematical methods like the Simplex algorithm to solve LP problems more efficiently.