To solve the given linear programming problem, follow these steps:

Problem:

Maximize $P = 3x + 3y$
Subject to:

1. $2x + y \leq 20$
2. $x + 2y \leq 16$
3. $x, y \geq 0$

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Step 1: Identify and Graph Constraints

The feasible region is defined by the constraints. We'll find the corner points where the constraints intersect.

Rearrange the inequalities into slope-intercept form:

1. $2x + y \leq 20$
   $y \leq -2x + 20$

2. $x + 2y \leq 16$
   $y \leq -\frac{1}{2}x + 8$

3. $x \geq 0, y \geq 0$ restricts the region to the first quadrant.

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Step 2: Find Intersection Points (Corner Points)

The corner points of the feasible region are found by solving the equations of the lines pairwise:

1. Intersection of $2x + y = 20$ and $x + 2y = 16$:
   Solve the system: $2x + y = 20$ $x + 2y = 16$ Multiply the second equation by 2: $2x + y = 20$ $2x + 4y = 32$ Subtract the first equation from the second: $3y = 12 \implies y = 4$ Substitute $y = 4$ into $2x + y = 20$: $2x + 4 = 20 \implies 2x = 16 \implies x = 8$ So, one corner point is $(8, 4)$.

2. Intersection of $2x + y = 20$ and $y = 0$:
   Substitute $y = 0$ into $2x + y = 20$: $2x + 0 = 20 \implies x = 10$ So, another corner point is $(10, 0)$.

3. Intersection of $x + 2y = 16$ and $x = 0$:
   Substitute $x = 0$ into $x + 2y = 16$: $0 + 2y = 16 \implies y = 8$ So, another corner point is $(0, 8)$.

4. Intersection of $x = 0$ and $y = 0$ is $(0, 0)$.

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Step 3: Evaluate $P = 3x + 3y$ at Each Corner Point

Evaluate $P$ at the four corner points:

1. At $(0, 0)$:
   $P = 3(0) + 3(0) = 0$
2. At $(10, 0)$:
   $P = 3(10) + 3(0) = 30$
3. At $(0, 8)$:
   $P = 3(0) + 3(8) = 24$
4. At $(8, 4)$:
   $P = 3(8) + 3(4) = 24 + 12 = 36$

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Step 4: Conclusion

The maximum value of $P$ is $\mathbf{36}$, which occurs at the point $\mathbf{(8, 4)}$.

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Answer to Part 1:

A. $P = 36$

Answer to Part 2:

A. The coordinates are $(8, 4)$.

Would you like to see a graphical representation of this solution or explore similar problems?

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Related Questions:

1. How is the feasible region identified geometrically?
2. Can a linear programming problem have no feasible solution?
3. How do changes in the coefficients of $P$ affect the optimal solution?
4. What if the inequalities included $\geq$ constraints—how would that change the approach?
5. How can simplex methods be applied to solve this problem?

Tip: Always check the feasibility of the intersection points by substituting them back into all constraints.