Question

The constraints of a linear programming problem are x+2y≤10 and 6x+3y≤18. Which of the following points lie in the feasible region?

• A. (0,6)
• B. (4,3)
• C. (5,7)
• D. (1,7)
• E. (1,3)

Answer 1

Answer: (E)

The given constraints are:

1. \( x + 2y \leq 10 \) 

2. \( 6x + 3y \leq 18 \)

We need to check which of the points satisfy both constraints.

Checking for each point:

For \( (0, 6) \):

1. \( 0 + 2(6) = 12 \), which does not satisfy \( x + 2y \leq 10 \).

2. \( 6(0) + 3(6) = 18 \), which satisfies \( 6x + 3y \leq 18 \).

This point is not in the feasible region.

For \( (4, 3) \):

1. \( 4 + 2(3) = 10 \), which satisfies \( x + 2y \leq 10 \).

2. \( 6(4) + 3(3) = 24 + 9 = 33 \), which does not satisfy \( 6x + 3y \leq 18 \).

This point is not in the feasible region.

For \( (5, 7) \):

1. \( 5 + 2(7) = 5 + 14 = 19 \), which does not satisfy \( x + 2y \leq 10 \).

2. \( 6(5) + 3(7) = 30 + 21 = 51 \), which does not satisfy \( 6x + 3y \leq 18 \).

This point is not in the feasible region.

For \( (1, 7) \):

1. \( 1 + 2(7) = 1 + 14 = 15 \), which does not satisfy \( x + 2y \leq 10 \).

2. \( 6(1) + 3(7) = 6 + 21 = 27 \), which does not satisfy \( 6x + 3y \leq 18 \).

This point is not in the feasible region.

For \( (1, 3) \):

1. \( 1 + 2(3) = 1 + 6 = 7 \), which satisfies \( x + 2y \leq 10 \).

2. \( 6(1) + 3(3) = 6 + 9 = 15 \), which satisfies \( 6x + 3y \leq 18 \).

This point satisfies both constraints and is in the feasible region.

The correct answer is (1, 3).