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Graphical solution of a linear programming problem with constraints and optimal solution point.

Graph showing a system of linear inequalities with shaded feasible region and solution points.

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1. The system of inequalities is:
- y ≥ 0
- x ≥ 0
- x + y ≤ 6

2. The feasible region is the triangle bounded by the lines x=0, y=0, and x+y=6 in the first quadrant.

3. The vertices of the feasible region are (0,0), (6,0), and (0,6).

4. The objective function is P = 2x + 3y.

5. Evaluate P at each vertex:
- At (0,0): P = 2(0) + 3(0) = 0
- At (6,0): P = 2(6) + 3(0) = 12
- At (0,6): P = 2(0) + 3(6) = 18

6. The maximum value of P is 18 at the point (0,6).

7. The minimum value of P is 0 at the point (0,0).

Parent Tip: Review the logic above to help your child master the concept of linear programming algebra 1 worksheet.

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