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linear programming applications using graphical method to maximize?

p=5x1+9x2
2x1+x2 less than or equal to 4
x1+2x2 less than or equal to 6
x1 greater than or = to 0 x2 greater than or =to 0

I can't really show you a graphic depiction, but hoping I can walk you through it nonetheless.

Starting with the last constraint (x1, x2 >= 0), we are in the first quadrant with the x- and y-axis as boundaries. Then with (x1 + 2x2 <= 6) we draw a line from (0,3) to (6,0) which satisfies the inequality such that when x1 = 0, x2 is <= 3, and when x2 = 0, x1 <= 6. We would shade underneath it to satisfy the less than constraint. Next we have (2x1 + x2 <= 4) and draw a line from (0,4) to (2,0) to satisfy that inequality for when x1 = 0, x2 <=4 and when x2 = 0, x1 <= 2. Again, the area underneath since it is a less than constraint. Before we can solve, we have to figure out the coordinates for the point where the lines cross. Since the visual representation does not easily lend itself to seeing where they cross, we most solve for when the inequalities are equal. To do this, I solved both for x1 giving me (x1 <= 2 - .5x2) and (x1 <= 6 - 2x2). Set them equal to each other, (6 - 2x2 = 2 - .5x2) and solve for x2. We then have x2 = 8/3, which we can plug into either equation to find that x1 = 2/3. We now have our 3 points to test to see which one maximizes profit.
P = 5(2) + 9(0) = 10
P = 5(0) + 9(3) = 15
P = 5(2/3) + 9(8/3) = 27 1/3

Profit is maximized when x1 = 2/3 and x2 = 8/3.

Hope that helped!

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