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What is the Feasible Region for Two Variables?
Grade Level:
Class 8
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Feasible Region for two variables is like a special 'allowed' area on a graph. It's the set of all possible points (x, y) that satisfy ALL the given conditions or inequalities at the same time. Think of it as the 'sweet spot' where all your rules are followed.
Simple Example
Quick Example
Imagine you want to buy pens and notebooks. A pen costs Rs 10 and a notebook costs Rs 20. You have only Rs 100. Also, you want to buy at least 2 pens and at least 1 notebook. The feasible region would show all the combinations of pens and notebooks you can buy that fit your budget and minimum quantity rules.
Worked Example
Step-by-Step
Let's find the feasible region for these conditions:
1) x >= 0
2) y >= 0
3) x + y <= 5
Step 1: Draw the x and y axes. Since x >= 0 and y >= 0, our region will be in the first quadrant (top-right part of the graph).
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Step 2: Consider the inequality x + y <= 5. First, treat it as an equation: x + y = 5. Find two points on this line. If x = 0, y = 5. So, (0, 5). If y = 0, x = 5. So, (5, 0).
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Step 3: Plot the points (0, 5) and (5, 0) on your graph and draw a straight line connecting them. This line represents x + y = 5.
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Step 4: Now, decide which side of the line x + y = 5 represents x + y <= 5. Pick a test point not on the line, for example, (0, 0). Substitute it into x + y <= 5: 0 + 0 <= 5, which is 0 <= 5. This is true.
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Step 5: Since (0, 0) satisfies the inequality, shade the region that includes (0, 0) and is below the line x + y = 5.
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Step 6: Combine this shaded region with the conditions x >= 0 and y >= 0. The area that is in the first quadrant AND below the line x + y = 5 is your feasible region.
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Answer: The feasible region is the triangle formed by the points (0,0), (5,0), and (0,5).
Why It Matters
Understanding feasible regions helps scientists and engineers find the best solutions when there are many limits. For example, in AI/ML, it helps train models efficiently. Economists use it to find the best production plans, and in data science, it helps optimize resource allocation for various tasks.
Common Mistakes
MISTAKE: Not shading the correct side of an inequality line. Students often pick the wrong side after plotting the line. | CORRECTION: Always use a test point (like (0,0) if it's not on the line) to check which side satisfies the inequality. If the test point makes the inequality true, shade that side; otherwise, shade the opposite side.
MISTAKE: Forgetting to consider all inequalities. Sometimes students only shade for one or two conditions. | CORRECTION: Make sure to identify the region that satisfies *every single* inequality given. It's the intersection of all individual shaded regions.
MISTAKE: Drawing solid lines for strict inequalities (like > or <) or dashed lines for non-strict inequalities (like >= or <=). | CORRECTION: Use a solid line for inequalities that include 'equal to' (>= or <=) to show points on the line are included. Use a dashed line for strict inequalities (> or <) to show points on the line are NOT included.
Practice Questions
Try It Yourself
QUESTION: Shade the feasible region for x >= 2 and y >= 1. | ANSWER: The region to the right of the vertical line x=2 and above the horizontal line y=1.
QUESTION: Find the feasible region for x + y <= 4, x >= 0, y >= 0. | ANSWER: The triangular region with vertices (0,0), (4,0), and (0,4).
QUESTION: Plot the feasible region for: x >= 0, y >= 0, x + y <= 6, and x <= 4. | ANSWER: A quadrilateral region with vertices (0,0), (4,0), (4,2), and (0,6).
MCQ
Quick Quiz
Which of these points lies in the feasible region defined by x >= 1 and y >= 2?
(0, 0)
(1, 1)
(2, 3)
(0, 2)
The Correct Answer Is:
C
For (2, 3), x=2 satisfies x >= 1, and y=3 satisfies y >= 2. So, (2, 3) is in the feasible region. Options A, B, and D fail at least one condition.
Real World Connection
In the Real World
Imagine a food delivery app like Swiggy or Zomato. When a delivery driver gets orders, the app uses feasible regions to find the best route. It considers factors like how far the restaurants are, how far the customers are, traffic, and the time limit for delivery. The feasible region helps identify all possible routes that meet these conditions, then finds the most efficient one.
Key Vocabulary
Key Terms
INEQUALITY: A mathematical statement comparing two expressions using symbols like <, >, <=, >=. | GRAPH: A diagram showing relationships between variables, usually on a coordinate plane. | COORDINATE PLANE: A 2D surface defined by an x-axis and a y-axis. | OPTIMIZATION: The process of finding the best possible solution under given conditions. | CONSTRAINT: A limit or restriction on variables in a problem.
What's Next
What to Learn Next
Once you understand feasible regions, you're ready to learn about 'Linear Programming'. This is where you use the feasible region to find the maximum or minimum value of an objective function, which is super useful for making the best decisions in real life!
