Question

The solution set for the system of linear inequations $x+y \ge 1 ; 7x+9y \le 63 ; y \le 5 ; x \le 6, x \ge 0$ and $y \ge 0$ is represented graphically in the figure. What is the correct option?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

To quickly find the correct region, use the test point $(0,0)$. For $x+y \ge 1$, $0 \ge 1$ is false, so shade away from origin. For $7x+9y \le 63$, $0 \le 63$ is true, so shade towards origin.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to identify the correct graphical representation of the feasible region defined by the given system of linear inequalities.

Step 2: Detailed Explanation:
Let's analyze each inequality to determine the bounded region:
1. $x + y \ge 1$: The line $x + y = 1$ passes through $(1, 0)$ and $(0, 1)$. The inequality represents the half-plane away from the origin.
2. $7x + 9y \le 63$: The line passes through $(9, 0)$ and $(0, 7)$. The inequality represents the half-plane containing the origin.
3. $y \le 5$: Represents the region below the horizontal line $y = 5$.
4. $x \le 6$: Represents the region to the left of the vertical line $x = 6$.
5. $x \ge 0, y \ge 0$: Restricts the feasible region entirely to the first quadrant.
The intersection of all these regions forms a closed polygon in the first quadrant, bounded by the axes, the two slanted lines, and the horizontal and vertical boundaries. Plotting these lines accurately will yield the required shaded region.

Step 3: Final Answer:
The correct graphical representation matches option (A).