Show A Separate Graph Of The Constraint Lines And The Soluti

Show a separate graph of the constraint lines and the solutions that satisfy each of the following constraints

Analyze and depict the constraint lines for the given inequalities:

  • 15X + 7Y ≥ 75
  • -5X + 15Y ≤ 150
  • 3Y ≥ 21
  • -4X ≥ 60
  • 5X + 3Y ≤ 0

Paper For Above instruction

Linear programming problems often involve multiple constraints that define feasible regions within a coordinate plane. Visualizing these constraints through their respective lines allows for better understanding of solution spaces and helps identify optimal solutions. This analysis requires plotting each constraint line individually and highlighting the areas that satisfy their respective inequalities. Below, we interpret each constraint line, convert inequalities into equalities for graphing, and determine the feasible region that satisfies the constraints.

1. 15X + 7Y ≥ 75:

To graph this constraint, first rewrite it as an equality: 15X + 7Y = 75. For plotting, identify two points:

- When X = 0 : 7Y = 75 → Y ≈ 10.71

- When Y = 0 : 15X = 75 → X = 5

The inequality 15X + 7Y ≥ 75 indicates the feasible region is on or above this line.

2. -5X + 15Y ≤ 150:

Rewrite as an equality: -5X + 15Y = 150. For plotting:

- When X = 0 : 15Y = 150 → Y = 10

- When Y = 0 : -5X = 150 → X = -30 (which may be outside the positive quadrant)

Since the inequality is ≤, the feasible region is on or below this line, considering the positive quadrant constraints that X, Y ≥ 0.

3. 3Y ≥ 21:

Rewrite as an equality: 3Y = 21 → Y = 7.

Since inequality is ≥, the feasible region is on or above the horizontal line Y = 7.

4. -4X ≥ 60:

Rewrite as: -4X = 60 → X = -15.

However, since the inequality is ≥, and X ≥ 0 in feasible regions (common in LP problems), the constraint may imply X ≥ -15, but since negative values are invalid in many LP contexts, it simplifies to X ≥ 0 with the region on or to the right of X = -15, which effectively is X ≥ 0.

5. 5X + 3Y ≤ 0:

In the positive orthant, the only solution satisfying this inequality is at or below the line 5X + 3Y = 0, which passes through the origin. Given X,Y ≥ 0, the only feasible point satisfying this inequality is at X=0, Y=0.

Plotting these lines and shading the regions consistent with each inequality produces an intersection area that represents all solutions satisfying every constraint simultaneously. The feasible region is determined by the overlapping regions, considering the inequalities' directions. Visualizing these constraints gives clarity about the potential optimal solutions when decision variables represent quantities in real-world problems, such as resource allocation or production planning.

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