Which one of the following set of constraints represents the shaded r…

The correct answer is Option (1) → $x + y ≤ 800, 2x + y ≤ 1000, x ≤ 400, y ≤ 700,x ≥ 0,y ≥ 0$

To find the system of inequalities representing the shaded region, observe the boundaries and intercepts of the lines:

1. The line passing through (800, 0) and (0, 800) has the equation:

$x + y = 800 \Rightarrow x + y \leq 800$ (since region is below this line)

2. The line passing through (500, 0) and (0, 1000) has the equation:

$2x + y = 1000 \Rightarrow 2x + y \leq 1000$ (since region is below this line)

3. The vertical boundary at $x = 400$ gives:

$x \leq 400$

4. The horizontal boundary at $y = 700$ gives:

$y \leq 700$

5. Since only the first quadrant is considered:

$x \geq 0$ and $y \geq 0$