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Problem 10

# Use the method of this section to solve each linear programming problem. $$\begin{array}{ll} \text { Maximize } & P=5 x+y \\ \text { subject to } & 2 x+y \leq 8 \\ & -x+y \geq 2 \\ & x \geq 0, y \geq 0 \end{array}$$

Expert verified
To maximize the given linear programming problem, first graph the constraints and identify the feasible region vertices: (0, 0), (0, 2), (3, 2), and (4, 0). Compute P = 5x + y for each vertex: P(0, 0) = 0, P(0, 2) = 2, P(3, 2) = 17, and P(4, 0) = 20. The maximum value of P is 20, achieved at the point (4, 0).
See the step by step solution

## Step 1: Graph the constraints

First, we need to graph the constraint inequalities. It is helpful to rewrite them as equalities: 1. $$2x + y = 8$$ 2. $$-x + y = 2$$ Now, we'll graph these lines along with the constraints of x and y being greater than or equal to zero in the first quadrant.

## Step 2: Identify the vertices of the feasible region

Now that we have graphed the inequalities and the first quadrant, the feasible region can be identified as the intersection points of the lines where all constraints are satisfied. The vertices of the feasible region are the following points: 1. (0, 0) 2. (0, 2) 3. (3, 2) 4. (4, 0)

## Step 3: Compute P for each vertex

Now we will find the value of the objective function P = 5x + y at each of the vertices of the feasible region: 1. P(0, 0) = 5(0) + (0) = 0 2. P(0, 2) = 5(0) + (2) = 2 3. P(3, 2) = 5(3) + (2) = 17 4. P(4, 0) = 5(4) + (0) = 20

## Step 4: Identify the maximum value of P

Now, it's time to compare the values of P at the vertices of the feasible region: 1. P(0, 0) = 0 2. P(0, 2) = 2 3. P(3, 2) = 17 4. P(4, 0) = 20 We can see that P(4, 0) has the highest value of 20 among all vertices. Therefore, the maximum value of P = 5x + y subject to the given constraints is 20, and it is achieved at the point (4, 0).

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